MEASUREMENT OF QUANTUM STATE: ADEQUATIO INTELLECTUS NOSTRI ... · MEASUREMENT OF QUANTUM STATE:...

International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 1

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MEASUREMENT OF QUANTUM STATE:

ADEQUATIO INTELLECTUS NOSTRI CUM

RE -A ZEITGEIST MODEL

Dr K N Prasanna Kumar, Prof B S Kiranagi, Prof C S Bagewadi

ABSTRACT: Serge Haroche and David Wineland were awarded Nobel Prize in

Physics for being able to resolve the conflict of Quantum measurement. We give a

concomitant model for the same. Study of concatenated equation is done in the next

paper.

INTRODUCTION:

Believe nothing on the faith of traditions,

even though they have been held in honor

for many generations and in diverse places.

Do not believe a thing because many people speak of it.

Do not believe on the faith of the sages of the past.

Do not believe what you yourself have imagined,

persuading yourself that a God inspires you.

Believe nothing on the sole authority of your masters and priests.

After examination, believe what you yourself have tested

and found to be reasonable, and conform your conduct thereto.

As the Buddha was dying,

Ananda asked

who would be their teacher after death.

He replied to his disciple -

"Be lamps unto yourselves.

Be refuges unto yourselves.

Take yourself no external refuge.

Hold fast to the truth as a lamp.

Hold fast to the truth as a refuge.

Look not for a refuge in anyone besides yourselves.

And those, Ananda, who either now or after I am dead,

Shall be a lamp unto themselves,

Shall betake themselves as no external refuge,

But holding fast to the truth as their lamp,

Holding fast to the truth as their refuge,

Shall not look for refuge to anyone else besides themselves,


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It is they who shall reach to the very topmost height;

But they must be anxious to learn."

Quoted in Joseph Goldstein, The Experience of Insight

Buddha

Unbounded by time and space.

If I look for you,

I will always find you.

But when I am not looking,

You may be somewhere else entirely.

For you are a pigment of my perception,

As I am of yours.

Together we form

The brilliant and harmonious

Rainbow of God's love.

-CONNEE

Acknowledgements: It is to be stated in unmistakable and unequivocal terms that the

literary expatiation, predicational integrity, Introductory remarks, character constitution,

ontological consonance, primordial exactitude, accolytish representation, atrophied

asseveration, anamensial alienisms, anchorite aperitif, Arcadian Atticism reflective in passages,

essential predications, primary focus on homologues receptiveness of the subject matter in

question, differentiated instrumental and dynamism of the projective development of the topics

consummate abstractions, rational representation, conferential extrinsicness, manifestation of

histories, standard remarks, professed developments(Google Search) interfacial interference

and syncopated justices, are taken from various sources Such as Wikipedia, author’s Home

Page, ask a Physicist Column, Abstracts of articles of various authors, papers of various

authors, Web Graphs, Google search photographs, Google search results and other sources

which included literally dialectic deliberation, polemical argumentation, conjugatory confatalia

with fellow Professors, and scholars of repute. I have to state that I have put all concerted

efforts , sustained struggles, and protracted endeavor to mention each and every source at the

cross reference or at the reference list at the end. In the eventuality of any act of omission or

commission it is to be stated that such an eventuality has occurred attributable to inadvertence

and in deliberation and I beg professedly and profusely and assiduously and avidly with all

fervor and can dour the persons concerned. I am not presenting any panacea for all the ills

despite the penance done for therefor, and it is attributable and ascribable to the fact that

many highly esteemed and eminent persons allowed me to piggy ride on their backs I have

been able to write summarily and expressly this paper. Explanation and deliberation of

concatenation equations are done in the next paper. Towards the end of consummation,

consolidations, concretization, reinforcement, revitalization, rejuvenation, resurrection, and

consubstantiation of this mammoth project, singlehandedly I have gone through millions of

pages and drafted and typed myself, and if by chance there are any repetition, I make a

sincere entreat, earnest beseech, and fervent appeal and obsequesial dedication and

consecration to kindly pardon me on that score


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(1) An ion trap is a combination of electric or magnetic fields that captures ions in a

region of a vacuum system or tube. Ion traps have a number of scientific uses

such as mass spectrometery and trapping ions while the ion's quantum state is

manipulated. The two most common types of ion traps are the Penning trap and

the Paul trap (quadrupole ion trap).

(2) When using ion traps for scientific studies of quantum state manipulation,

the Paul trap is most often used. This work may lead to a trapped ion quantum

computer and has already been used to create the world's most accurate atomic

clocks.

(3) An ion trap mass spectrometer may incorporate a Penning trap (Fourier

transform ion cyclotron resonance), Paul trap or the Kingdon trap. The Orbitrap,

introduced in 2005, is based on the Kingdon trap. Other types of mass

spectrometers may also use a linear quadrupole ion trap as a selective mass

filter. In an electron gun (a device emitting high-speed electrons, such as those

in CRTs), an ion trap may be implemented above the cathode (using an extra,

positively-charged electrode between the cathode and the extraction electrode)

to prevent its degradation by positive ions accelerated backward by the fields

intended to pull electrons away from the cathode. Penning traps are devices for

the storage of charged particles using a homogeneous static magnetic field and a

spatially inhomogeneous static electric field. This kind of trap is particularly

well suited to precision measurements of properties of ions and stable subatomic

particles which have a non-zero electric charge. Recently this trap has been used

in the physical realization of quantum computation and quantum information

processing as well. The Penning trap has also been used in the realization of

what is known as a geonium atom. Currently Penning traps are used in many

laboratories worldwide. For example, they are used at CERN to

store antiprotons.

(4) Penning traps use a strong homogeneous axial magnetic field to confine

particles radially and a quadrupole electric field to confine the particles axially.

The static electric potential can be generated using a set of three electrodes: a

ring and two end caps. In an ideal Penning trap the ring and end caps

are hyperboloids of revolution. For trapping of positive (negative) ions, the end

cap electrodes are kept at a positive (negative) potential relative to the ring. This

potential produces a saddle point in the centre of the trap, which traps ions along

http://en.wikipedia.org/w/index.php?title=File:Penning_Trap.svg&page=1


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the axial direction.

(5) The electric field causes ions to oscillate (harmonically in the case of an ideal

penning trap) along the trap axis. The magnetic field in combination with the

electric field causes charged particles to move in the radial plane with a motion

which traces out an epitrochoid. The orbital motion of ions in the radial plane is

composed of two modes at frequencies which are called the magnetron and

the modified cyclotron frequencies. These motions are similar to

the deferent and epicycle, respectively, of the Ptolemaic model of the solar

system.

A classical trajectory in the radial plane for (Courtesy: Google Search:

Wikipedia)

(6) The sum of these two frequencies is the cyclotron frequency, which depends

only on the ratio of electric charge to mass and on the strength of the magnetic

field. This frequency can be measured very accurately and can be used to

measure the masses of charged particles.

(7) Many of the highest-precision mass measurements (masses of the electron,

proton, 2H, 20Ne and 28Si) come from Penning traps. Buffer gas cooling,

resistive cooling and laser cooling are techniques to remove energy from ions in

a Penning trap. Buffer gas cooling relies on collisions between the ions and

neutral gas molecules that bring the ion energy closer the energy of the gas

molecules.

(8) In resistive cooling, moving image charges in the electrodes are made to do

work through an external resistor, effectively removing energy from the

ions. Laser cooling can be used to remove energy from some kinds of ions in

Penning traps. This technique requires ions with an appropriate electronic

structure. Radiative cooling is the process by which the ions lose energy by

creating electromagnetic waves by virtue of their acceleration in the magnetic

field. This process dominates the cooling of electrons in Penning traps, but is

very small and usually negligible for heavier particles.

(9) Using the Penning trap can have advantages over the radio frequency trap (Paul

trap). Firstly, in the Penning trap only static fields are applied and therefore


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there is no micro-motion and resultant heating of the ion due to the dynamic

fields. Also, the Penning trap can be made larger whilst maintaining strong

trapping. The trapped ion can then be held further away from the electrode

surfaces. Interaction with patch potentials on the electrode surfaces can be

responsible for heating and decoherence effects and these effects scale as a high

power of the inverse distance between the ion and the electrode.

(10) Fourier transform ion cyclotron resonance mass spectrometry (also

known as Fourier transform mass spectrometry), is a type of mass

spectrometry used for determining the mass-to-charge ratio (m/z) of ions based

on the cyclotron frequency of the ions in a fixed magnetic field. The ions are

trapped in a Penning trap where they are excited to a larger cyclotron radius by

an oscillating electric field perpendicular to the magnetic field. The excitation

also results in the ions moving in phase (in a packet). The signal is detected as

an image current on a pair of plates which the packet of ions passes close to as

they cyclotron. The resulting signal is called a free induction decay (fid),

transient or interferogram that consists of a superposition of sine waves. The

useful signal is extracted from this data by performing a Fourier transform to

give a mass spectrum. Single ions can be investigated in a Penning trap held at a

temperature of 4 K. For this the ring electrode is segmented and opposite

electrodes is connected to a superconducting coil and the source and the gate of

a field effect transistor. The coil and the parasitic capacitances of the circuit

form a LC circuit with a Q of about 50 000. The LC circuit is excited by an

external electric pulse. The segmented electrodes couple the motion of the single

electron to the LC circuit. Thus the energy in the LC circuit in resonance with

the ion slowly oscillates between the many electrons (10000) in the gate of the

field effect transistor and the single electron. This can be detected in the signal

at the drain of the field effect transistor.

Ions trapped by optical fields

Ion sees the light: charged particle is trapped by laser(Courtesy: Google Search :

Wikipedia)

(11) Physicists in Germany claimed to have trapped single ions using lasers

for the first time – an achievement that could open the door to advanced

simulations of quantum systems.


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(12) In the past, the trapping of atomic particles has followed a basic rule: use

radio-frequency (RF) electromagnetic fields for ions, and optical lasers for

neutral particles, such as atoms.

(13) This is because RF fields can only exert electric forces on charges; try to

use them on neutral particles and there's little effect.

(14) A laser, on the other hand, can draw the dipole moments of neutral

particles towards the centre of its beam. But the resultant optical trap is

relatively weak, and so ions – which are sensitive to stray electric fields – easily,

escape. Now, Tobias Schaetz and colleagues at the Max-Planck Institute for

Quantum Optics in Garching, Germany, claim to have got around this problem.

they describe an "experimental proof-of-principle" of how stronger, more

focused lasers can optically trap ions for over a millisecond.(Courtesy: Physics

World.com)

(15) In the experiment, Schaetz's group cooled a single 24Mg+ ion in a

standard RF trap before superimposing it with the field from a strong laser. The

physicists then gradually reduced the RF field to zero so the ion was contained

by the optical field alone. Finally, they observed the fluorescence light of the ion

through a CCD camera to check it had been trapped successfully. Over several

experiments, they calculated the trapping lifetime as about 1.8 ms. "The lifetime

of the ion in the optical dipole trap is limited by photon scattering and is thus

expected to be improvable by state of the art techniques," they note.

(16) Optical traps for ions could benefit simulations on quantum systems. If

optical traps are superimposed on RF traps, there comes the ability to simulate

in two or three dimensions, or collide particles in new ways.

(17) "The biggest advantage of being able to trap an ion optically, instead of

[using] an RF electrode, is that one can then study the collision between a

neutral atom and an ion at low energy," explains Yu-Ju Lin, a researcher at the

Joint Quantum Institute at the University of Maryland, US. "If an ion is still

trapped by an RF electrode, it always produces an RF-induced micro-motion –

i.e. kinetic energy – which limits how low the energy of the collisions [can go]."

(18) Andrew Steane, a quantum physicist at the University of Oxford, UK,

calls the research "excellent experimental physics", although he notes that it is

an extension of previous ideas and experiments. "Such [an optical trap] is of

course already widely used in experiments on neutral atoms," he adds.

"Applying that idea to ions extends the toolbox available to experiments in basic

atomic physics and quantum mechanics." (From the reports of Jon Cartwright is

a freelance journalist based in Bristol)

(19) A quadrupole ion trap or quadrupole ion storage trap (QUISTOR) exists

in both linear and 3D (Paul Trap, QIT) varieties and refers to an ion trap that

uses constant DC and radio frequency (RF) oscillating AC electric fields to trap

ions. It is commonly used as a component of a mass spectrometer. The invention

of the 3D quadrupole ion trap itself is attributed to Wolfgang Paul who shared

the Nobel Prize in Physics in 1989 for this work.


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Theory

The 3D trap itself generally consists of two hyperbolic metal electrodes with their foci

facing each other and a hyperbolic ring electrode halfway between the other two

electrodes. The ions are trapped in the space between these three electrodes by AC

(oscillating, non-static) and DC (non-oscillating, static) electric fields. The AC radio

frequency voltage oscillates between the two hyperbolic metal end cap electrodes if ion

excitation is desired; the driving AC voltage is applied to the ring electrode. The ions

are first pulled up and down axially while being pushed in radially. The ions are then

pulled out radially and pushed in axially (from the top and bottom). In this way the ions

move in a complex motion that generally involves the cloud of ions being long and

narrow and then short and wide, back and forth, oscillating between the two states.

Since the mid-1980s most 3D traps (Paul traps) have used ~1 mtorr of helium. The use

of damping gas and the mass-selective instability mode developed by Stafford et al. led

to the first commercial 3D ion traps.

Linear Ion Trap at the University of Calgary(Courtesy Google Search)

The quadrupole ion trap has two configurations: the three dimensional form described

above and the linear form made of 4 parallel electrodes. A

simplified rectilinear configuration has also been used. The advantage of the linear

design is in its simplicity, but this leaves a particular constraint on its modeling. To

understand how this originates, it is helpful to visualize the linear form. The Paul trap is

designed to create a saddle-shaped field to trap a charged ion, but with a quadrupole,

this saddle-shaped electric field cannot be rotated about an ion in the centre. It can only

'flap' the field up and down. For this reason, the motions of a single ion in the trap are

described by the Mathieu Equations. These equations can only be solved numerically or

equivalently by computer simulations.

The intuitive explanation and lowest order approximation is the same as strong

focusing in accelerator physics. Since the field affects the acceleration, the position lags

behind (to lowest order by half a period). So the particles are at defocused positions

when the field is focusing and vice versa. Being farther from center, they experience a

http://en.wikipedia.org/wiki/File:Lin_quad_thompson_ucalgary.JPG


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stronger field when the field is focusing than when it is defocusing.

Equations of motion

Further ,

And

The trapping of ions can be understood in terms of stability regions in and

space.

Linear ion trap

LTQ (Linear trap quadrupole)

The linear ion trap uses a set of quadrupole rods to confine ions radially and static

electrical potential on-end electrodes to confine the ions axially. The linear form of the

trap can be used as a selective mass filter, or as an actual trap by creating a potential

well for the ions along the axis of the electrodes Advantages of the linear trap design

are increased ion storage capacity, faster scan times, and simplicity of construction

(although quadrupole rod alignment is critical, adding a quality control constraint to

their production. This constraint is additionally present in the machining requirements

of the 3D trap).

Cylindrical ion trap

Cylindrical ion traps have a cylindrical rather than a hyperbolic ring electrode. This

configuration has been used in miniature arrays of traps.

The term Ion trapping is used to describe the build-up of a higher concentration of a

chemical across a cell membrane due to the pKa value of the chemical and difference

of pH across the cell membrane. Generally speaking, these results in basic chemicals

accumulate in acidic bodily fluids such as the cytosol, and acidic chemicals

accumulating in basic fluids such as mastitic milk.


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Many cells have other mechanisms to pump a molecule inside or outside the cell against

the concentration gradient, but these processes are active ones, meaning that they

require enzymes and consume cellular energy. In contrast, ion trapping does not require

any enzyme or energy. It is similar to osmosis in that they both involve

the semipermeable nature of the cell membrane.

Cells have a more acidic pH inside the cell than outside (gastric mucosal cells being an

exception). Therefore basic drugs (like bupivacaine, pyrimethamine) are more charged

inside the cell than outside. The cell membrane is permeable to non-ionized (fat-

soluble) molecules; ionized (water-soluble) molecules cannot cross it easily. Once a

non-charged molecule of a basic chemical crosses the cell membrane to enter the cell, it

becomes charged due to gaining a hydrogen ion because of the lower pH inside the cell,

and thus becomes unable to cross back. Because transmembrane equilibrium must be

maintained, another unionized molecule must diffuse into the cell to repeat the process.

Thus its concentration inside the cell increases many times that of the outside. The non-

charged molecules of the drug remain in equal concentration on either side of the cell

membrane.

The charge of a molecule depends upon the pH of its solution. In an acidic medium,

basic drugs are more charged and acidic drugs are less charged. The converse is true in

a basic medium. For example, Naproxen is a non-steroidal anti-inflammatory drug that

is a weak acid (its pKa value is 5.0). The gastric juice has a pH of 2.0. It is a three-fold

difference (due to log scale) between its pH and its pKa; therefore there is a 1000×

difference between the charged and uncharged concentrations. So, in this case, for every

one molecule of charged Naproxen, there are 1000 molecules of uncharged Naproxen at

a pH of 2. This is why weak acids are better absorbed from the stomach and weak bases

from intestine where the pH is alkaline. When pH of a solution is equal to pKa of

dissolved drug, then 50% of the drug is ionized, another 50% is unionized. Ion trapping

is the reason why basic (alkaline) drugs are secreted into the stomach (for

example morphine) where pH is acidic, and acidic drugs are excreted in urine when it is

alkaline. Similarly, ingesting sodium bicarbonate with amphetamine, a weak base,

causes better absorption of amphetamine (in stomach) and its lesser excretion (in urine),

thus prolonging its actions. Ion trapping can cause partial failure of certain anti-cancer

chemotherapies. Ion trapping is also important outside of pharmacology. For example it

causes weakly acidic hormones to accumulate in the cytosol of cells. This is important

in keeping the external concentration of the hormone low in the extracellular

environment where many hormones are sensed. Examples of plant hormones that are

subjected to ion trapping areabscisic acid, gibberellic acid and retinoic acid. Examples

of animal hormones subjected to ion trapping include Prostacyclin and Leukotrienes.

Ions are generally categorized into the following groups based on mobility

values and dimensions.


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(1) Free-floating electron

An free-floating electron exists by itself and weights only about 1/1800 of

the hydrogen atom. Its mobility values are as large as beta rays generated

by cathode rays or radiant substances. It is generally found at high altitudes

where the air is rarefied, or in highly purified nitrogen, helium and argon.

(2) Ionized atom

An atom, which has lost an electron, is a positively ionized atom. An electronically

neutral atom, which has obtained an electron, is a negatively ionized atom. Both

types of ions along with electrons exist only in the upper layers of the atmosphere.

(3) Small ion

Most ions found in the atmosphere belong to this group (also known as

Lightweight or Normal ion). As soon as an electron or ionized atom shows up in

the atmosphere, it attracts gaseous molecules and combines with them to form a

small ion molecule while positioning itself in the center. A small ion molecule

consists of 2 to 30 molecules. Generally, positive ions weigh more than negatively

charged ions, and mobility values are larger than 0.4-0.8 (cm2/Vs).

(4) Large ion

A large ion (also known as Heavy ion) is a negative or positive small ion

(molecule) absorbed by dust, mist or another tiny particle. While having the same

structure as small ions, it can weight 1,000 times more. Mobility values range from

0.0005 to 0.01 (cm2/Vs). Many exist in polluted air.

(5) Middle ion


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This group of ions was discovered by Pollock and exists only in low humidity

conditions, and does not exist near the earth's surface. Mobility values range from

0.01 to 0.1 (cm2/Vs).

Ion channels are pore-forming proteins that help establish and control

the voltage gradient across the plasma membrane of cells (see membrane potential) by

allowing the flow of ions down their electrochemical gradient. They are present in

the membranes that surround all biological cells. The study of ion channels involves

many scientific techniques such as voltage clamp electrophysiology (in particular patch

clamp), immunohistochemistry, and RT-PCR.

Basic features

Ion channels regulate the flow of ions across the membrane in all cells. Ion channels

are integral membrane proteins; or, more typically, an assembly of several proteins. They

are present on all membranes of cell (plasma membrane) and intracellular organelles

(nucleus, mitochondria, endoplasmic reticulum, golgi apparatus and so on). Such

"multi-subunit" assemblies usually involve a circular arrangement of identical

or homologous proteins closely packed around a water-filled pore through the plane of

the membrane or lipid bilayer. For most voltage-gated ion channels, the pore-forming

subunit(s) are called the α subunit, while the auxiliary subunits are denoted β, γ, and so

on. Some channels permit the passage of ions based solely on their charge of positive

(cation) or negative (anion). However, the archetypal channel pore is just one or two

atoms wide at its narrowest point and is selective for specific species of ion, such

as sodium or potassium. These ions move through the channel pore single file nearly as

quickly as the ions move through free fluid. In some ion channels, passage through the

pore is governed by a "gate," which may be opened or closed by chemical or electrical

signals, temperature, or mechanical force, depending on the variety of channel.

Biological role

Because channels underlie the nerve impulse and because "transmitter-activated"

channels mediate conduction across the synapses, channels are especially prominent

components of the nervous system. Indeed, most of the offensive and defensive toxins

that organisms have evolved for shutting down the nervous systems of predators and

prey (e.g., the venoms produced by spiders, scorpions, snakes, fish, bees, sea snails and

others) work by modulating ion channel conductance and/or kinetics. In addition, ion

channels are key components in a wide variety of biological processes that involve

rapid changes in cells, such as cardiac, skeletal, and smooth

muscle contraction, epithelial transport of nutrients and ions, T-cell activation

and pancreatic beta-cell insulin release. In the search for new drugs, ion channels are a

frequent target.


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Diversity

There are over 300 types of ion channels in a living cell. Ion channels may be classified

by the nature of their gating, the species of ions passing through those gates, the number

of gates (pores) and localization of proteins. Further heterogeneity of ion channels

arises when channels with different constitutive subunits give rise to a specific kind of

current. Absence or mutation of one or more of the contributing types of channel

subunits can result in loss of function and, potentially, underlie neurologic diseases.

Classification by gating

Ion channels may be classified by gating, i.e. what opens and closes the channels.

Voltage-gated ion channels open or close depending on the voltage gradient across the

plasma membrane, while ligand-gated ion channels open or close depending on binding

of ligands to the channel.

Voltage-gated ion channel

Voltage-gated ion channels open and close in response to membrane potential.

Voltage-gated sodium channels: This family contains at least 9 members and is largely

responsible for action potential creation and propagation. The pore-forming α subunits

are very large (up to 4,000 amino acids) and consist of four homologous repeat domains

(I-IV) each comprising six transmembrane segments (S1-S6) for a total of 24

transmembrane segments. The members of this family also coassemble with auxiliary β

subunits, each spanning the membrane once. Both α and β subunits are extensively

glycosylated.

Voltage-gated calcium channels: This family contains 10 members, though these members

are known to coassemble with α2δ, β, and γ subunits. These channels play an important

role in both linking muscle excitation with contraction as well as neuronal excitation

with transmitter release. The α subunits have an overall structural resemblance to those

of the sodium channels and are equally large.

Cation channels of sperm: This small family of channels, normally referred to as

Catsper channels, is related to the two-pore channels and distantly related to TRP

channels.

Voltage-gated potassium channels (KV): This family contains almost 40 members,

which are further divided into 12 subfamilies. These channels are known mainly for

their role in repolarizing the cell membrane following action potentials. The α subunits

have six transmembrane segments, homologous to a single domain of the sodium

channels. Correspondingly, they assemble as tetramers to produce a functioning

channel.

Some transient receptor potential channels: This group of channels, normally referred to

simply as TRP channels, is named after their role in Drosophila phototransduction. This


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family, containing at least 28 members, is incredibly diverse in its method of activation.

Some TRP channels seem to be constitutively open, while others are gated by voltage,

intracellular Ca2+, pH, redox state, osmolarity, and mechanical stretch. These channels

also vary according to the ion(s) they pass, some being selective for Ca2+ while others

are less selective, acting as cation channels. This family is subdivided into 6 subfamilies

based on homology: classical (TRPC), vanilloid receptors (TRPV), melastatin (TRPM),

polycystins (TRPP), mucolipins (TRPML), and ankyrin transmembrane protein 1

(TRPA).

Hyperpolarization-activated cyclic nucleotide-gated channels: The opening of these

channels is due to Hyperpolarization rather than the depolarization required for other

cyclic nucleotide-gated channels. These channels are also sensitive to the cyclic

nucleotides cAMP and cGMP, which alter the voltage sensitivity of the channel’s

opening. These channels are permeable to the monovalent cations K+ and Na+. There

are 4 members of this family, all of which form tetramers of six-transmembrane α

subunits. As these channels open under hyperpolarizing conditions, they function

as pacemaking channels in the heart, particularly the SA node.

Voltage-gated proton channels: Voltage-gated proton channels open with depolarization,

but in a strongly pH-sensitive manner. The result is that these channels open only when

the electrochemical gradient is outward, such that their opening will only allow protons

to leave cells. Their function thus appears to be acid extrusion from cells. Another

important function occurs in phagocytes (e.g. eosinophils, neutrophils, macrophages)

during the "respiratory burst." When bacteria or other microbes are engulfed by

phagocytes, the enzyme NADPH oxidase assembles in the membrane and begins to

produce reactive oxygen species (ROS) that help kill bacteria. NADPH oxidase is

electrogenic, moving electrons across the membrane, and proton channels open to allow

proton flux to balance the electron movement electrically.

Ligand-gated

Also known as ionotropic receptors, this group of channels open in response to specific

ligand molecules binding to the extracellular domain of the receptor protein. Ligand

binding causes a conformational change in the structure of the channel protein that

ultimately leads to the opening of the channel gate and subsequent ion flux across the

plasma membrane. Examples of such channels include the cation-permeable "nicotinic"

Acetylcholine receptor, ion tropic glutamate-gated receptors and ATP-gated P2X receptors,

and the anion-permeable γ-amino butyric acid-gated GABAA receptor.

Ion channels activated by second messengers may also be categorized in this group,

although ligands and second messengers are otherwise distinguished from each other.

Other gating

Other gating include activation/inactivation by e.g. second messengers from the inside of

the cell membrane, rather as from outside, as in the case for ligands. Ions may count to


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such second messengers, and then causes direct activation, rather than indirect, as in the

case were the electric potential of ions cause activation/inactivation of voltage-gated ion

channels.

Some potassium channels

Inward-rectifier potassium channels: These channels allow potassium to flow into the cell

in an inwardly rectifying manner, i.e., potassium flows effectively into, but not out of,

the cell. This family is composed of 15 official and 1 unofficial members and is further

subdivided into 7 subfamilies based on homology. These channels are affected by

intracellular ATP, PIP2, and G-protein βγ subunits. They are involved in important

physiological processes such as the pacemaker activity in the heart, insulin release, and

potassium uptake in glial cells. They contain only two transmembrane segments,

corresponding to the core pore-forming segments of the KV and KCa channels. Their α

subunits form tetramers.

Calcium-activated potassium channels: This family of channels is, for the most part,

activated by intracellular Ca2+ and contains 8 members.

Two-pore-domain potassium channels: This family of 15 members forms what is known

as leak channels, and they follow Goldman-Hodgkin-Katz (open) rectification.

Light-gated channels like channelrhodopsin are directly opened by the action of light.

Mechanosensitive ion channels are opening under the influence of stretch, pressure, shear,

displacement.

Cyclic nucleotide-gated channels: This superfamily of channels contains two families: the

cyclic nucleotide-gated (CNG) channels and the Hyperpolarization-activated, cyclic

nucleotide-gated (HCN) channels. It should be noted that this grouping is functional

rather than evolutionary.

Cyclic nucleotide-gated channels: This family of channels is characterized by activation

due to the binding of intracellular cAMP or cGMP, with specificity varying by member.

These channels are primarily permeable to monovalent cations such as K+ and Na+.

They are also permeable to Ca2+, though it acts to close them. There are 6 members of

this family, which is divided into 2 subfamilies.

Hyperpolarization-activated cyclic nucleotide-gated channels

Temperature Gated Channels: Members of the Transient Receptor Potential ion

channel superfamily, such as TRPV1 or TRPM8 are opened either by hot or cold

temperatures.

Classification by type of ions

Chloride channels: This superfamily of poorly-understood channels consists of

approximately 13 members. They include ClCs, CLICs, Bestrophins and CFTRs. These


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channels are non-selective for small anions; however chloride is the most abundant

anion, and hence they are known as chloride channels.

Potassium channels

Voltage-gated potassium channels e.g., Kvs, Kirs etc.

Calcium-activated potassium channels e.g., BKCa or MaxiK, SK, etc.

Inward-rectifier potassium channels

Two-pore-domain potassium channels: This family of 15 members forms what is

known as leak channels, and they follow Goldman-Hodgkin-Katz (open) rectification.

Sodium channels

Voltage-gated sodium channels NaVs

Epithelial sodium channels (ENaC)

Calcium channels CaVs

Proton channels

Voltage-gated proton channels

Non-selective cation channels: These let many types of cations, mainly Na+, K+ and

Ca2+ through the channel.

Most Transient receptor potential channels

Other classifications

There are other types of ion channel classifications that are based on less normal

characteristics, e.g. multiple pores and transient potentials.

Almost all ion channels have one single pore. However, there are also those with two:

Two-pore channels: This small family of 2 members putatively forms cation-selective

ion channels. They are predicted to contain two KV-style six-transmembrane domains,

suggesting they form a dimer in the membrane. These channels are related to catsper

channels channels and, more distantly, TRP channels.

There are channels that are classified by the duration of the response to stimuli:

Transient receptor potential channels: This group of channels, normally referred to

simply as TRP channels, is named after their role in Drosophila photo transduction.

This family, containing at least 28 members, is incredibly diverse in its method of

activation. Some TRP channels seem to be constitutively open, while others are gated

by voltage, intracellular Ca2+, pH, redox state, Cosmolarity, and mechanical stretch.

These channels also vary according to the ion(s) they pass, some being selective for


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Ca2+ while others are less selective, acting as cation channels. This family is

subdivided into 6 subfamilies based on homology: canonical (TRPC), vanilloid

receptors (TRPV), melastatin (TRPM), polycystins (TRPP), mucolipins (TRPML), and

ankyrin transmembrane protein 1 (TRPA).

Detailed structure

Channels differ with respect to the ion they let pass (for example, Na+, K+, Cl−), the

ways in which they may be regulated, the number of subunits of which they are

composed and other aspects of structure. A channel belonging to the largest class,

which includes the voltage-gated channels that underlie the nerve impulse, consists of

four subunits with six transmembrane helices each. On activation, these helices move

about and open the pore. Two of these six helices are separated by a loop that lines the

pore and is the primary determinant of ion selectivity and conductance in this channel

class and some others. The existence and mechanism for ion selectivity was first

postulated in the 1960s by Clay Armstrong. He suggested that the pore lining could

efficiently replace the water molecules that normally shield potassium ions, but that

sodium ions were too small to allow such shielding, and therefore could not pass

through. This mechanism was finally confirmed when the structure of the channel was

elucidated. The channel subunits of one such other class, for example, consist of just

this "P" loop and two transmembrane helices. The determination of their molecular

structure by Roderick MacKinnon using X-ray crystallography won a share of the

2003 Nobel Prize in Chemistry.

Because of their small size and the difficulty of crystallizing integral membrane

proteins for X-ray analysis, it is only very recently that scientists have been able to

directly examine what channels "look like." Particularly in cases where the

crystallography required removing channels from their membranes with detergent,

many researchers regard images that have been obtained as tentative. An example is the

long-awaited crystal structure of a voltage-gated potassium channel, which was

reported in May 2003. One inevitable ambiguity about these structures relates to the

strong evidence that channels change conformation as they operate (they open and

close, for example), such that the structure in the crystal could represent any one of

these operational states. Most of what researchers have deduced about channel

operation so far they have established through

electrophysiology, biochemistry, gene sequence comparison and mutagenesis.

Channels can have single (CLICs) to multiple transmembrane (K channels, P2X

receptors, Na channels) domains which span plasma membrane to form pores. Pore can

determine the selectivity of the channel. Gate can be formed either inside or outside the

pore region.

Diseases of ion channels

There are a number of chemicals and genetic disorders which disrupt normal

functioning of ion channels and have disastrous consequences for the organism. Genetic


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disorders of ion channels and their modifiers are known as Channelopathies..

Chemicals

Tetrodotoxin (TTX), used by puffer fish and some types of newts for defense. It blocks

sodium channels.

Saxitoxin is produced by a dinoflagellate also known as "red tide". It blocks voltage

dependent sodium channels.

Conotoxin, is used by cone snails to hunt prey.

Lidocaine and Novocaine belong to a class of local anesthetics which block sodium ion

channels.

Dendrotoxin is produced by mamba snakes, and blocks potassium channels.

Iberiotoxin is produced by the Buthus tamulus (Eastern Indian scorpion) and blocks

potassium channels.

Heteropodatoxin is produced by Heteropoda venatoria (brown huntsman spider or laya)

and blocks potassium channels.

Genetic

Shaker gene mutations cause a defect in the voltage gated ion channels, slowing down

the repolarization of the cell.

Equine hyperkalaemic periodic paralysis as well as Human hyperkalaemic periodic

paralysis (HyperPP) is caused by a defect in voltage dependent sodium channels.

Paramyotonia congenita (PC) and potassium aggravated myotonias (PAM)

Generalized epilepsy with febrile seizures plus (GEFS+)

Episodic Ataxia (EA), characterized by sporadic bouts of severe dis coordination with

or without myokymia, and can be provoked by stress, startle, or heavy exertion such as

exercise.

Familial hemiplegic migraine (FHM)

Spinocerebellar ataxia type 13

Long QT syndrome is a ventricular arrhythmia syndrome caused by mutations in one or

more of presently ten different genes, most of which are potassium channels and all of

which affect cardiac repolarization.

Brugada syndrome is another ventricular arrhythmia caused by voltage-gated sodium

channel gene mutations.

Cystic fibrosis is caused by mutations in the CFTR gene, which is a chloride channel.


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Mucolipidosis type IV is caused by mutations in the gene encoding

the TRPML1 channel

References

Neural networks were trained using whole ion mobility spectra from a standardized

database of 3137 spectra for 204 chemicals at various concentrations. Performance of

the network was measured by the success of classification into ten chemical classes.

Eleven stages for evaluation of spectra and of spectral pre-processing were employed

and minimums established for response thresholds and spectral purity. After

optimization of the database, network, and pre-processing routines, the fraction of

successful classifications by functional group was 0.91 throughout a range of

concentrations. Network classification relied on a combination of features, including

drift times, number of peaks, relative intensities, and other factors apparently including

peak shape. The network was opportunistic, exploiting different features within

different chemical classes. Application of neural networks in a two-tier design where

chemicals were first identified by class and then individually eliminated all but one

false positive out of 161 test spectra. These findings establish that ion mobility spectra,

even with low resolution instrumentation, contain sufficient detail to permit the

development of automated identification systems.

(Adapted from lectures by Dr. Richard F.W. Bader

Professor of Chemistry / McMaster University / Hamilton, Ontario)

Classification of Chemical Bonds

To make a quantitative assessment of the type of binding present in a particular

molecule it is necessary to have a measure of the extent of charge transfer present in the

molecule relative to the charge distributions of the separated atoms. This information is

contained in the density difference or bond density distribution, the distribution

obtained by subtracting the atomic densities from the molecular charge distribution.

Such a distribution provides a detailed measure of the net reorganization of the charge

densities of the separated atoms accompanying the formation of the molecule.

The density distribution resulting from the overlap of the undistorted atomic densities

(the distribution which is subtracted from the molecular distribution) does not place

sufficient charge density in the binding region to balance the nuclear forces

of repulsion. The regions of charge increase in a bond density map are, therefore, the

regions to which charge is transferred relative to the separated atoms to obtain a state of

electrostatic equilibrium and hence a chemical bond. Thus we may use the location of

this charge increase relative to the positions of the nuclei to characterize the bond and to

obtain an explanation for its electrostatic stability.


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In covalent binding we shall find that the forces binding the nuclei are exerted by an

increase in the charge density which is shared mutually between them. In ionic binding

both nuclei are bound by a charge increase which is localized in the region of a single

nucleus.

Covalent Binding

The bond density map of the nitrogen molecule (Fig. 7-2) is illustrative of the

characteristics of covalent binding.

Fig. 7-2. Bond density (or density difference) maps and their profiles along the

internuclear axis for N2 and LiF. The solid and dashed lines represent an increase and a

decrease respectively in the molecular charge density relative to the overlapped atomic

distributions. These maps contrast the two possible extremes of the manner in which the

original atomic charge densities may be redistributed to obtain a chemical bond. Click

here for contour values.

The principal feature of this map is a large accumulation of charge density in the

binding region, corresponding in this case to a total increase of one quarter of an

electronic charge. As noted in the study of the total charge distribution, charge density

is also

transferred to the antibinding regions of the nuclei but the amount transferred to either

region, 0.13 e-, is less than is accumulated in the binding region. The charge density of

the original atoms is decreased in regions perpendicular to the bond at the positions of

the nuclei. In three dimensions, the regions of charge deficit correspond to two

continuous rings or roughly doughnut-shaped regions encircling the bond axis.

The increase in charge density in the antibinding regions and the removal of charge

density from the immediate regions of the nuclei result in an increase in the forces of

repulsion exerted on the nuclei, forces resulting from the close approach of the two

atoms and from the partial overlap of their density distributions. The repulsive forces


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are obviously balanced by the forces exerted on the nuclei by the shared increase in

charge density located in the binding region. A bond is classified as covalent when the

bond density distribution indicates that the charge increase responsible for the binding

of the nuclei is shared by both nuclei. It is not necessary for covalent binding that the

density increase in the binding region be shared equally as in the completely

symmetrical case of N2. We shall encounter heteronuclear molecules (molecules with

different nuclei) in which the net force binding the nuclei is exerted by a density

increase which, while shared, is not shared equally between the two nuclei. The

pattern of charge rearrangement in the bond density map for N2 is, aside from the

accumulation of charge density in the binding region, quite distinct from that found for

H2 , another but simpler example of covalent binding. The pattern observed for

nitrogen, a charge increase concentrated along the bond axis in both the binding and

antibinding regions and a removal of charge density from a region perpendicular to the

axis, is characteristic of atoms which in the orbital model of bonding employ p atomic

orbitals in forming the bond. Since a p orbital concentrates charge density on opposite

sides of a nucleus, the large buildup of charge density in the antibinding regions is to be

expected. In the orbital theory of the hydrogen molecule, the bond is the result of the

overlap of s orbitals. The bond density map in this case is characterized by a simple

transfer of charge from the antibinding to the binding region since s orbitals do not

possess the strong directional or nodal properties of p orbitals. Further examples of both

types of charge rearrangements or polarizations will be illustrated below.

Ionic Binding

We shall preface our discussion of the bond density map for ionic binding with a

calculation of the change in energy associated with the formation of the bond in LiF.

While the calculation will be relatively crude and based on a very simple model, it will

illustrate that the complete transfer of valence charge density from one atom to another

in forming a molecule is in certain cases energetically possible.

Lithium possesses the electronic configuration 1s22s1 and is from group IA of the

periodic table. It possesses a very low ionization potential and an electron affinity

which is zero for all practical purposes. Fluorine is from group VIIA and has a

configuration 1s22s22p5. It possesses a high ionization potential and a high electron

affinity. The following calculation will illustrate that the 2s electron of Li could

conceivably be transferred completely to the 2p shell of orbitals on F in which there is a

single vacancy. This would result in the formation of a molecule best described as

Li+F-, and in the electron configurations 1s2 for Li+ and 1s22s22p6 for F-.

(2)

The two ions are oppositely charged and will attract one another. The energy released

when the two ions approach one another from infinity to form the LiF molecule is

easily estimated. To a first approximation it is simply -e2/R where R is the final

equilibrium distance between the two ions in the molecule:


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The transfer of charge density from lithium to fluorine is very evident in the bond

density map for LiF (Fig. 7-2). The charge density of the 2s electron on the lithium atom

is a very diffuse distribution and consequently the negative contours in the bond density

map denoting its removal are of large spatial extent but small in magnitude. The

principal charge increase is nearly symmetrically arranged about the fluorine nucleus

and is completely encompassed by a single nodal surface. The total charge increase on

fluorine amounts to approximately one electronic charge. The charge increase in the

antibinding region of the lithium nucleus corresponds to only 0.01 electronic charges.

(The great disparity in the magnitudes of the charge increases on lithium and fluorine

are most strikingly portrayed in the profile of the bond density map, also shown in Fig.

7-2) It is equally important to realize that the charge increase on lithium occurs within

the region of the 1s inner shell or core density and not in the region of the valence

density. Thus the slight charge increase on lithium is primarily a result of a polarization

of its core density and not of an accumulation of valence density.

The pattern of charge increase and charge removal in the region of the fluorine, while

similar to that for a nitrogen nucleus in N2, is much more symmetrical, and the charge

density corresponds very closely to the distribution obtained from a single

2p electron. Thus the simple orbital model of the bond in LiF which describes the

bond as a transfer of the 2s electron on lithium to the single 2p vacancy on fluorine is

a remarkably good one.

While the bond density map for LiF substantiates the concept of charge transfer and

the formation of Li+ and F- ions it also indicates that the charge distributions of both

ions are polarized. The charge increase in the binding region of fluorine exceeds

slightly that in its antibinding region (the F- ion is polarized towards the Li+ ion) and

the charge distribution of the Li+ ion is polarized away from the fluorine. A

consideration of the forces exerted on the nuclei in this case will demonstrate that these

polarizations are a necessary requirement for the attainment of electrostatic equilibrium

in the face of a complete charge transfer from lithium to fluorine.

Consider first the forces acting on the nuclei in the simple model of the ionic bond,

the model which ignores the polarizations of the ions and pictures the molecule as two

closed-shell spherical ions in mutual contact. If the charge density of the Li+ ion is

spherical it will exert no net force on the lithium nucleus. The F- ion possesses ten

electrons and, since the charge density on the F- ion is also considered to be spherical,

the attractive force this density exerts on the Li nucleus is the same as that obtained for

all ten electrons concentrated at the fluorine nucleus. Nine of these electrons will screen

the nine positive nuclear charges on fluorine from the lithium nucleus. The net force on

the lithium nucleus is, therefore, one of attraction because of the one excess negative

charge on F.

For the molecule to be stable, the final force on the lithium nucleus must be zero. This

can be achieved by a distortion of the spherical charge distribution of the Li+ ion. If a

small amount of the 1s charge density on lithium is removed from the region adjacent to


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fluorine and placed on the side of the lithium nucleus away from the fluorine, i.e., the

charge distribution is polarized away from the fluorine, it will exert a force on the

lithium nucleus in a direction away from the fluorine. Thus the force on the lithium

nucleus in an ionic bond can be zero only if the charge density of the Li+ ion is

polarized away from the negative end of the molecule.

A similar consideration of the forces exerted on the fluorine nucleus demonstrates that

the F- ion density must also be polarized. The fluorine nucleus experiences a net force

of repulsion because of the presence of the lithium ion. The two negative charges

centred on lithium screen only two of its three nuclear charges. Therefore, the charge

density of the F- ion must be polarized towards the lithium in order to exert an attractive

force on the fluorine nucleus which will balance the repulsive force arising from the

presence of the Li+ ion. Thus both nuclei in the LiF molecule are bound by the increase

in charge density localized in the region of the fluorine.

The charge distribution of a molecule with an ionic bond will necessarily be

characterized not only by the transfer of electronic charge from one atom to another, but

also by a polarization of each of the resulting ions in a direction counter to the transfer

of charge, as indicated in the bond density map for LiF.

The bond density maps for N2 and LiF are shown side by side to provide a contrast of

the changes in the atomic charge densities responsible for the two extremes of chemical

binding. In a covalent bond the increase in charge density which binds both nuclei is

shared between them. In an ionic bond both nuclei are bound by the forces exerted by

the charge density localized on a single nucleus. It must be stressed that there is no

fundamental difference between the forces responsible for a covalent or an ionic bond.

They are electrostatic in each case.

FORMULATION OF THE PROBLEM:

NOTATION :

𝐺13 : Category one of Field (We are classifying based on the characteristics of the systems to which the

ion separation is executed)

𝐺14 : Category Two of Field

𝐺15 : Category Three of Field

𝑇13 :Category one of Ions (Again classification is based on systems under investigation)

𝑇14 : Category Two of Ions


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𝑇15 : Category Three of Ions

𝐺16 : Category one of Laser (We are again classifying based on the different laser types used in the

investigatory process)

𝐺17 : Category Two of Laser

𝐺18 : Category Three of Laser

𝑇16 :Category one: Suppression of thermal conduction of ions and hence decrease in temperature

(cooling) and study of properties of ions WITH photons

𝑇17 : Category Two: Suppression of thermal conduction of ions and hence decrease in temperature


𝑇18 : Category Three: Suppression of thermal conduction of ions and hence decrease in temperature


𝑎13 1 , 𝑎14

1 , 𝑎15 1 , 𝑏13

1 , 𝑏14 1 , 𝑏15

1 𝑎16 2 , 𝑎17

2 , 𝑎18 2

𝑏16 2 , 𝑏17

2 , 𝑏18 2 : are Accentuation coefficients

𝑎13′ 1 , 𝑎14

′ 1 , 𝑎15′ 1 , 𝑏13

′ 1 , 𝑏14′ 1 , 𝑏15

′ 1 , 𝑎16′ 2 , 𝑎17

′ 2 , 𝑎18′ 2 ,

𝑏16′ 2 , 𝑏17

′ 2 , 𝑏18′ 2 are Dissipation coefficients

FIELD-ION SYSTEM:

GOVERNING EQUATIONS:

The differential system of this model is now

𝑑𝐺13

𝑑𝑡= 𝑎13

1 𝐺14 − 𝑎13′ 1 + 𝑎13

′′ 1 𝑇14 , 𝑡 𝐺13 1

𝑑𝐺14

𝑑𝑡= 𝑎14

1 𝐺13 − 𝑎14′ 1 + 𝑎14

′′ 1 𝑇14 , 𝑡 𝐺14 2

𝑑𝐺15

𝑑𝑡= 𝑎15

1 𝐺14 − 𝑎15′ 1 + 𝑎15

′′ 1 𝑇14 , 𝑡 𝐺15 3

𝑑𝑇13

𝑑𝑡= 𝑏13

1 𝑇14 − 𝑏13′ 1 − 𝑏13

′′ 1 𝐺, 𝑡 𝑇13 4

𝑑𝑇14

𝑑𝑡= 𝑏14

1 𝑇13 − 𝑏14′ 1 − 𝑏14

′′ 1 𝐺, 𝑡 𝑇14 5

𝑑𝑇15

𝑑𝑡= 𝑏15

1 𝑇14 − 𝑏15′ 1 − 𝑏15

′′ 1 𝐺, 𝑡 𝑇15 6

+ 𝑎13′′ 1 𝑇14 , 𝑡 = First augmentation factor 7

− 𝑏13′′ 1 𝐺, 𝑡 = First detrition factor 8

LASER AND ION COOLOING SYSTEM AND STUDY OF CONCOMITANT PROPERTIES:

GOVERNING EQUATIONS:

The differential system of this model is now

9

𝑑𝐺16

𝑑𝑡= 𝑎16

2 𝐺17 − 𝑎16′ 2 + 𝑎16

′′ 2 𝑇17 , 𝑡 𝐺16 10

𝑑𝐺17

𝑑𝑡= 𝑎17

2 𝐺16 − 𝑎17′ 2 + 𝑎17

′′ 2 𝑇17 , 𝑡 𝐺17 11


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𝑑𝐺18

𝑑𝑡= 𝑎18

2 𝐺17 − 𝑎18′ 2 + 𝑎18

′′ 2 𝑇17 , 𝑡 𝐺18 12

𝑑𝑇16

𝑑𝑡= 𝑏16

2 𝑇17 − 𝑏16′ 2 − 𝑏16

′′ 2 𝐺19 , 𝑡 𝑇16 13

𝑑𝑇17

𝑑𝑡= 𝑏17

2 𝑇16 − 𝑏17′ 2 − 𝑏17

′′ 2 𝐺19 , 𝑡 𝑇17 14

𝑑𝑇18

𝑑𝑡= 𝑏18

2 𝑇17 − 𝑏18′ 2 − 𝑏18

′′ 2 𝐺19 , 𝑡 𝑇18 15

+ 𝑎16′′ 2 𝑇17 , 𝑡 = First augmentation factor 16

− 𝑏16′′ 2 𝐺19 , 𝑡 = First detrition factor 17

CONCATENATED EQUATIONS: 18

𝑑𝐺13

𝑑𝑡= 𝑎13

1 𝐺14 − 𝑎13′ 1 + 𝑎13

′′ 1 𝑇14 , 𝑡 + 𝑎16′′ 2,2 𝑇17 , 𝑡 𝐺13 19

𝑑𝐺14

𝑑𝑡= 𝑎14

1 𝐺13 − 𝑎14′ 1 + 𝑎14

′′ 1 𝑇14 , 𝑡 + 𝑎17′′ 2,2 𝑇17 , 𝑡 𝐺14 20

𝑑𝐺15

𝑑𝑡= 𝑎15

1 𝐺14 − 𝑎15′ 1 + 𝑎15

′′ 1 𝑇14 , 𝑡 + 𝑎18′′ 2,2 𝑇17 , 𝑡 𝐺15 21

Where 𝑎13′′ 1 𝑇14 , 𝑡 , 𝑎14

′′ 1 𝑇14 , 𝑡 , 𝑎15′′ 1 𝑇14 , 𝑡 are first augmentation coefficients for

category 1, 2 and 3

+ 𝑎16′′ 2,2 𝑇17 , 𝑡 , + 𝑎17

′′ 2,2 𝑇17 , 𝑡 , + 𝑎18′′ 2,2 𝑇17 , 𝑡 are second augmentation coefficients

for category 1, 2 and 3

22

23

24

𝑑𝑇13

𝑑𝑡= 𝑏13

1 𝑇14 − 𝑏13′ 1 − 𝑏13

′′ 1 𝐺, 𝑡 + 𝑏16′′ 2,2 𝐺19 , 𝑡 𝑇13 25

𝑑𝑇14

𝑑𝑡= 𝑏14

1 𝑇13 − 𝑏14′ 1 − 𝑏14

′′ 1 𝐺, 𝑡 + 𝑏17′′ 2,2 𝐺19, 𝑡 𝑇14 26

𝑑𝑇15

𝑑𝑡= 𝑏15

1 𝑇14 − 𝑏15′ 1 − 𝑏15

′′ 1 𝐺, 𝑡 + 𝑏18′′ 2,2 𝐺19 , 𝑡 𝑇15 27

Where − 𝑏13′′ 1 𝐺, 𝑡 , − 𝑏14

′′ 1 𝐺, 𝑡 , − 𝑏15′′ 1 𝐺, 𝑡 are first detrition coefficients for

category 1, 2 and 3

+ 𝑏16′′ 2,2 𝐺19, 𝑡 , + 𝑏17

′′ 2,2 𝐺19 , 𝑡 , + 𝑏18′′ 2,2 𝐺19 , 𝑡 are second augmentation coefficients


28

29

𝑑𝐺16

𝑑𝑡= 𝑎16

2 𝐺17 − 𝑎16′ 2 + 𝑎16

′′ 2 𝑇17 , 𝑡 + 𝑎13′′ 1,1 𝑇14 , 𝑡 𝐺16 30

𝑑𝐺17

𝑑𝑡= 𝑎17

2 𝐺16 − 𝑎17′ 2 + 𝑎17

′′ 2 𝑇17 , 𝑡 + 𝑎14′′ 1,1 𝑇14 , 𝑡 𝐺17 31

𝑑𝐺18

𝑑𝑡= 𝑎18

2 𝐺17 − 𝑎18′ 2 + 𝑎18

′′ 2 𝑇17 , 𝑡 + 𝑎15′′ 1,1 𝑇14 , 𝑡 𝐺18 32

Where + 𝑎16′′ 2 𝑇17 , 𝑡 , + 𝑎17

′′ 2 𝑇17 , 𝑡 , + 𝑎18′′ 2 𝑇17 , 𝑡 are first augmentation coefficients


+ 𝑎13′′ 1,1 𝑇14 , 𝑡 , + 𝑎14

′′ 1,1 𝑇14 , 𝑡 , + 𝑎15′′ 1,1 𝑇14 , 𝑡 are second detrition coefficients for

33


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category 1, 2 and 3

34

𝑑𝑇16

𝑑𝑡= 𝑏16

2 𝑇17 − 𝑏16′ 2 − 𝑏16

′′ 2 𝐺19, 𝑡 − 𝑏13′′ 1,1 𝐺, 𝑡 𝑇16 35

𝑑𝑇17

𝑑𝑡= 𝑏17

2 𝑇16 − 𝑏17′ 2 − 𝑏17

′′ 2 𝐺19, 𝑡 − 𝑏14′′ 1,1 𝐺, 𝑡 𝑇17 36

𝑑𝑇18

𝑑𝑡= 𝑏18

2 𝑇17 − 𝑏18′ 2 − 𝑏18

′′ 2 𝐺19, 𝑡 − 𝑏15′′ 1,1 𝐺, 𝑡 𝑇18 37

Where − 𝑏16′′ 2 𝐺19 , 𝑡 , − 𝑏17

′′ 2 𝐺19 , 𝑡 , − 𝑏18′′ 2 𝐺19 , 𝑡 are first detrition coefficients for

category 1, 2 and 3

− 𝑏13′′ 1,1 𝐺, 𝑡 , − 𝑏14

′′ 1,1 𝐺, 𝑡 , − 𝑏15′′ 1,1 𝐺, 𝑡 are second detrition coefficients for

category 1, 2 and 3

38

39

40

Where we suppose

(A) 𝑎𝑖 1 , 𝑎𝑖

′ 1 , 𝑎𝑖′′ 1 , 𝑏𝑖

1 , 𝑏𝑖′ 1 , 𝑏𝑖

′′ 1 > 0,

𝑖, 𝑗 = 13,14,15

(B) The functions 𝑎𝑖′′ 1 , 𝑏𝑖

′′ 1 are positive continuous increasing and bounded.

Definition of (𝑝𝑖) 1 , (𝑟𝑖)

1 :

𝑎𝑖′′ 1 (𝑇14 , 𝑡) ≤ (𝑝𝑖)

1 ≤ ( 𝐴 13 )(1)

𝑏𝑖′′ 1 (𝐺, 𝑡) ≤ (𝑟𝑖)

1 ≤ (𝑏𝑖′) 1 ≤ ( 𝐵 13 )(1)

41

42

(C) 𝑙𝑖𝑚𝑇2→∞ 𝑎𝑖′′ 1 𝑇14 , 𝑡 = (𝑝𝑖)

1

limG→∞ 𝑏𝑖′′ 1 𝐺, 𝑡 = (𝑟𝑖)

1

Definition of ( 𝐴 13 )(1), ( 𝐵 13 )(1) :

Where ( 𝐴 13 )(1), ( 𝐵 13 )(1), (𝑝𝑖) 1 , (𝑟𝑖)

1 are positive constants

and 𝑖 = 13,14,15

43

They satisfy Lipschitz condition:

|(𝑎𝑖′′ ) 1 𝑇14

′ , 𝑡 − (𝑎𝑖′′ ) 1 𝑇14 , 𝑡 | ≤ ( 𝑘 13 )(1)|𝑇14 − 𝑇14

′ |𝑒−( 𝑀 13 )(1)𝑡

|(𝑏𝑖′′ ) 1 𝐺 ′ , 𝑡 − (𝑏𝑖

′′ ) 1 𝐺, 𝑡 | < ( 𝑘 13 )(1)||𝐺 − 𝐺 ′ ||𝑒−( 𝑀 13 )(1)𝑡

44

45

46

With the Lipschitz condition, we place a restriction on the behavior of functions

(𝑎𝑖′′ ) 1 𝑇14

′ , 𝑡 and(𝑎𝑖′′ ) 1 𝑇14 , 𝑡 . 𝑇14

′ , 𝑡 and 𝑇14 , 𝑡 are points belonging to the interval

( 𝑘 13 )(1), ( 𝑀 13 )(1) . It is to be noted that (𝑎𝑖′′ ) 1 𝑇14 , 𝑡 is uniformly continuous. In the eventuality of

the fact, that if ( 𝑀 13 )(1) = 1 then the function (𝑎𝑖′′ ) 1 𝑇14 , 𝑡 , the first augmentation coefficient

attributable to terrestrial organisms, would be absolutely continuous.

47

Definition of ( 𝑀 13 )(1), ( 𝑘 13 )(1) :

(D) ( 𝑀 13 )(1), ( 𝑘 13 )(1), are positive constants

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(𝑎𝑖) 1

( 𝑀 13 )(1) ,(𝑏𝑖) 1

( 𝑀 13 )(1) < 1

Definition of ( 𝑃 13 )(1), ( 𝑄 13 )(1) :

(E) There exists two constants ( 𝑃 13 )(1) and ( 𝑄 13 )(1) which together with

( 𝑀 13 )(1), ( 𝑘 13 )(1), (𝐴 13)(1)𝑎𝑛𝑑 ( 𝐵 13 )(1) and the constants

(𝑎𝑖) 1 , (𝑎𝑖

′) 1 , (𝑏𝑖) 1 , (𝑏𝑖

′) 1 , (𝑝𝑖) 1 , (𝑟𝑖)

1 , 𝑖 = 13,14,15,

satisfy the inequalities

1

( 𝑀 13 )(1) [ (𝑎𝑖) 1 + (𝑎𝑖

′) 1 + ( 𝐴 13 )(1) + ( 𝑃 13 )(1) ( 𝑘 13 )(1)] < 1

1

( 𝑀 13 )(1) [ (𝑏𝑖) 1 + (𝑏𝑖

′) 1 + ( 𝐵 13 )(1) + ( 𝑄 13 )(1) ( 𝑘 13 )(1)] < 1

49

50

51

52

53

Where we suppose 54

(F) 𝑎𝑖 2 , 𝑎𝑖

′ 2 , 𝑎𝑖′′ 2 , 𝑏𝑖

2 , 𝑏𝑖′ 2 , 𝑏𝑖

′′ 2 > 0, 𝑖, 𝑗 = 16,17,18 55

(G) The functions 𝑎𝑖′′ 2 , 𝑏𝑖

′′ 2 are positive continuous increasing and bounded. 56

Definition of (pi) 2 , (ri)

2 : 57

𝑎𝑖′′ 2 𝑇17 , 𝑡 ≤ (𝑝𝑖)

2 ≤ 𝐴 16 2

58

𝑏𝑖′′ 2 (𝐺, 𝑡) ≤ (𝑟𝑖)

2 ≤ (𝑏𝑖′) 2 ≤ ( 𝐵 16 )(2) 59

(H) lim𝑇2→∞ 𝑎𝑖′′ 2 𝑇17 , 𝑡 = (𝑝𝑖)

2 60

lim𝐺→∞ 𝑏𝑖′′ 2 𝐺19 , 𝑡 = (𝑟𝑖)

2 61

Definition of ( 𝐴 16 )(2), ( 𝐵 16 )(2) :

Where ( 𝐴 16 )(2), ( 𝐵 16 )(2), (𝑝𝑖) 2 , (𝑟𝑖)

2 are positive constants and 𝑖 = 16,17,18

62

They satisfy Lipschitz condition: 63

|(𝑎𝑖′′ ) 2 𝑇17

′ , 𝑡 − (𝑎𝑖′′ ) 2 𝑇17 , 𝑡 | ≤ ( 𝑘 16 )(2)|𝑇17 − 𝑇17

′ |𝑒−( 𝑀 16 )(2)𝑡 64

|(𝑏𝑖′′ ) 2 𝐺19

′ , 𝑡 − (𝑏𝑖′′ ) 2 𝐺19 , 𝑡 | < ( 𝑘 16 )(2)|| 𝐺19 − 𝐺19

′ ||𝑒−( 𝑀 16 )(2)𝑡 65

With the Lipschitz condition, we place a restriction on the behavior of functions (𝑎𝑖′′ ) 2 𝑇17

′ , 𝑡

and(𝑎𝑖′′ ) 2 𝑇17 , 𝑡 . 𝑇17

′ , 𝑡 and 𝑇17 , 𝑡 are points belonging to the interval ( 𝑘 16 )(2), ( 𝑀 16 )(2) . It is

to be noted that (𝑎𝑖′′ ) 2 𝑇17 , 𝑡 is uniformly continuous. In the eventuality of the fact, that if ( 𝑀 16 )(2) =

1 then the function (𝑎𝑖′′ ) 2 𝑇17 , 𝑡 , the first augmentation coefficient attributable to terrestrial

organisms, would be absolutely continuous.

66

Definition of ( 𝑀 16 )(2), ( 𝑘 16 )(2) : 67

(I) ( 𝑀 16 )(2), ( 𝑘 16 )(2), are positive constants

(𝑎𝑖) 2

( 𝑀 16 )(2) ,(𝑏𝑖)

2

( 𝑀 16 )(2) < 1

68

Definition of ( 𝑃 13 )(2), ( 𝑄 13 )(2) :

There exists two constants ( 𝑃 16 )(2) and ( 𝑄 16 )(2) which together

with ( 𝑀 16 )(2), ( 𝑘 16 )(2), (𝐴 16)(2)𝑎𝑛𝑑 ( 𝐵 16 )(2) and the constants

69


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(𝑎𝑖) 2 , (𝑎𝑖

′) 2 , (𝑏𝑖) 2 , (𝑏𝑖

′) 2 , (𝑝𝑖) 2 , (𝑟𝑖)

2 , 𝑖 = 16,17,18,

satisfy the inequalities

1

( M 16 )(2) [ (ai) 2 + (ai

′) 2 + ( A 16 )(2) + ( P 16 )(2) ( k 16 )(2)] < 1 70

1

( 𝑀 16 )(2) [ (𝑏𝑖) 2 + (𝑏𝑖

′) 2 + ( 𝐵 16 )(2) + ( 𝑄 16 )(2) ( 𝑘 16 )(2)] < 1 71

72

Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions

Definition of 𝐺𝑖 0 , 𝑇𝑖 0 :

𝐺𝑖 𝑡 ≤ 𝑃 13 1

𝑒 𝑀 13 1 𝑡 , 𝐺𝑖 0 = 𝐺𝑖0 > 0

𝑇𝑖(𝑡) ≤ ( 𝑄 13 )(1)𝑒( 𝑀 13 )(1)𝑡 , 𝑇𝑖 0 = 𝑇𝑖0 > 0

73

74

Theorem 1: if the conditions (A)-(E) above are fulfilled, there exists a solution satisfying the conditions

Definition of 𝐺𝑖 0 , 𝑇𝑖 0

𝐺𝑖 𝑡 ≤ ( 𝑃 16 )(2)𝑒( 𝑀 16 )(2)𝑡 , 𝐺𝑖 0 = 𝐺𝑖0 > 0

𝑇𝑖(𝑡) ≤ ( 𝑄 16 )(2)𝑒( 𝑀 16 )(2)𝑡 , 𝑇𝑖 0 = 𝑇𝑖0 > 0

75

76

PROOF:

Consider operator 𝒜(1) defined on the space of sextuples of continuous functions 𝐺𝑖 , 𝑇𝑖 : ℝ+ → ℝ+

which satisfy

77

𝐺𝑖 0 = 𝐺𝑖0 , 𝑇𝑖 0 = 𝑇𝑖

0 , 𝐺𝑖0 ≤ ( 𝑃 13 )(1) , 𝑇𝑖

0 ≤ ( 𝑄 13 )(1), 78

0 ≤ 𝐺𝑖 𝑡 − 𝐺𝑖0 ≤ ( 𝑃 13 )(1)𝑒( 𝑀 13 )(1)𝑡 79

0 ≤ 𝑇𝑖 𝑡 − 𝑇𝑖0 ≤ ( 𝑄 13 )(1)𝑒( 𝑀 13 )(1)𝑡 80

By

𝐺 13 𝑡 = 𝐺130 + (𝑎13) 1 𝐺14 𝑠 13 − (𝑎13

′ ) 1 + 𝑎13′′ ) 1 𝑇14 𝑠 13 , 𝑠 13 𝐺13 𝑠 13 𝑑𝑠 13

𝑡

0

81

𝐺 14 𝑡 = 𝐺140 + (𝑎14 ) 1 𝐺13 𝑠 13 − (𝑎14

′ ) 1 + (𝑎14′′ ) 1 𝑇14 𝑠 13 , 𝑠 13 𝐺14 𝑠 13 𝑑𝑠 13

𝑡

0 82

𝐺 15 𝑡 = 𝐺150 + (𝑎15) 1 𝐺14 𝑠 13 − (𝑎15

′ ) 1 + (𝑎15′′ ) 1 𝑇14 𝑠 13 , 𝑠 13 𝐺15 𝑠 13 𝑑𝑠 13

𝑡

0 83

𝑇 13 𝑡 = 𝑇130 + (𝑏13 ) 1 𝑇14 𝑠 13 − (𝑏13

′ ) 1 − (𝑏13′′ ) 1 𝐺 𝑠 13 , 𝑠 13 𝑇13 𝑠 13 𝑑𝑠 13

𝑡

0 84

𝑇 14 𝑡 = 𝑇140 + (𝑏14 ) 1 𝑇13 𝑠 13 − (𝑏14

′ ) 1 − (𝑏14′′ ) 1 𝐺 𝑠 13 , 𝑠 13 𝑇14 𝑠 13 𝑑𝑠 13

𝑡

0 85

T 15 t = T150 + (𝑏15) 1 𝑇14 𝑠 13 − (𝑏15

′ ) 1 − (𝑏15′′ ) 1 𝐺 𝑠 13 , 𝑠 13 𝑇15 𝑠 13 𝑑𝑠 13

𝑡

0

Where 𝑠 13 is the integrand that is integrated over an interval 0, 𝑡

86


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87

PROOF:

Consider operator 𝒜(2) defined on the space of sextuples of continuous functions 𝐺𝑖 , 𝑇𝑖 : ℝ+ → ℝ+

which satisfy

88

𝐺𝑖 0 = 𝐺𝑖0 , 𝑇𝑖 0 = 𝑇𝑖

0 , 𝐺𝑖0 ≤ ( 𝑃 16 )(2) , 𝑇𝑖

0 ≤ ( 𝑄 16 )(2), 89

0 ≤ 𝐺𝑖 𝑡 − 𝐺𝑖0 ≤ ( 𝑃 16 )(2)𝑒( 𝑀 16 )(2)𝑡 90

0 ≤ 𝑇𝑖 𝑡 − 𝑇𝑖0 ≤ ( 𝑄 16 )(2)𝑒( 𝑀 16 )(2)𝑡 91

By

𝐺 16 𝑡 = 𝐺160 + (𝑎16) 2 𝐺17 𝑠 16 − (𝑎16

′ ) 2 + 𝑎16′′ ) 2 𝑇17 𝑠 16 , 𝑠 16 𝐺16 𝑠 16 𝑑𝑠 16

𝑡

0

92

𝐺 17 𝑡 = 𝐺170 + (𝑎17 ) 2 𝐺16 𝑠 16 − (𝑎17

′ ) 2 + (𝑎17′′ ) 2 𝑇17 𝑠 16 , 𝑠 17 𝐺17 𝑠 16 𝑑𝑠 16

𝑡

0 93

𝐺 18 𝑡 = 𝐺180 + (𝑎18 ) 2 𝐺17 𝑠 16 − (𝑎18

′ ) 2 + (𝑎18′′ ) 2 𝑇17 𝑠 16 , 𝑠 16 𝐺18 𝑠 16 𝑑𝑠 16

𝑡

0 94

𝑇 16 𝑡 = 𝑇160 + (𝑏16 ) 2 𝑇17 𝑠 16 − (𝑏16

′ ) 2 − (𝑏16′′ ) 2 𝐺 𝑠 16 , 𝑠 16 𝑇16 𝑠 16 𝑑𝑠 16

𝑡

0 95

𝑇 17 𝑡 = 𝑇170 + (𝑏17 ) 2 𝑇16 𝑠 16 − (𝑏17

′ ) 2 − (𝑏17′′ ) 2 𝐺 𝑠 16 , 𝑠 16 𝑇17 𝑠 16 𝑑𝑠 16

𝑡

0 96

𝑇 18 𝑡 = 𝑇180 + (𝑏18 ) 2 𝑇17 𝑠 16 − (𝑏18

′ ) 2 − (𝑏18′′ ) 2 𝐺 𝑠 16 , 𝑠 16 𝑇18 𝑠 16 𝑑𝑠 16

𝑡

0

Where 𝑠 16 is the integrand that is integrated over an interval 0, 𝑡

97

(a) The operator 𝒜(1) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious

that

𝐺13 𝑡 ≤ 𝐺130 + (𝑎13) 1 𝐺14

0 +( 𝑃 13 )(1)𝑒( 𝑀 13 )(1)𝑠 13 𝑡

0𝑑𝑠 13 =

1 + (𝑎13 ) 1 𝑡 𝐺140 +

(𝑎13 ) 1 ( 𝑃 13 )(1)

( 𝑀 13 )(1) 𝑒( 𝑀 13 )(1)𝑡 − 1

98

From which it follows that

𝐺13 𝑡 − 𝐺130 𝑒−( 𝑀 13 )(1)𝑡 ≤

(𝑎13 ) 1

( 𝑀 13 )(1) ( 𝑃 13 )(1) + 𝐺140 𝑒

− ( 𝑃 13 )(1)+𝐺14

0

𝐺140

+ ( 𝑃 13 )(1)

𝐺𝑖0 is as defined in the statement of theorem 1

99

Analogous inequalities hold also for 𝐺14 , 𝐺15 , 𝑇13 , 𝑇14 , 𝑇15 100

(b) The operator 𝒜(2) maps the space of functions satisfying 34,35,36 into itself .Indeed it is obvious

that

101

𝐺16 𝑡 ≤ 𝐺160 + (𝑎16) 2 𝐺17

0 +( 𝑃 16 )(6)𝑒( 𝑀 16 )(2)𝑠 16 𝑡

0𝑑𝑠 16 =

1 + (𝑎16 ) 2 𝑡 𝐺170 +

(𝑎16 ) 2 ( 𝑃 16 )(2)

( 𝑀 16 )(2) 𝑒( 𝑀 16 )(2)𝑡 − 1

102

From which it follows that 103


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𝐺16 𝑡 − 𝐺160 𝑒−( 𝑀 16 )(2)𝑡 ≤

(𝑎16 ) 2

( 𝑀 16 )(2) ( 𝑃 16 )(2) + 𝐺170 𝑒

− ( 𝑃 16 )(2)+𝐺17

0

𝐺170

+ ( 𝑃 16 )(2)

Analogous inequalities hold also for 𝐺17 , 𝐺18 , 𝑇16 , 𝑇17 , 𝑇18 104

It is now sufficient to take (𝑎𝑖)

1

( 𝑀 13 )(1) ,(𝑏𝑖)

1

( 𝑀 13 )(1) < 1 and to choose

( P 13 )(1) and ( Q 13 )(1) large to have

105

(𝑎𝑖) 1

(𝑀 13 ) 1 ( 𝑃 13) 1 + ( 𝑃 13 )(1) + 𝐺𝑗0 𝑒

− ( 𝑃 13 )(1)+𝐺𝑗

0

𝐺𝑗0

≤ ( 𝑃 13 )(1)

106

(𝑏𝑖) 1

(𝑀 13 ) 1 ( 𝑄 13 )(1) + 𝑇𝑗0 𝑒

− ( 𝑄 13 )(1)+𝑇𝑗

0

𝑇𝑗0

+ ( 𝑄 13 )(1) ≤ ( 𝑄 13 )(1)

107

In order that the operator 𝒜(1) transforms the space of sextuples of functions 𝐺𝑖 , 𝑇𝑖 satisfying 34,35,36

into itself

108

The operator 𝒜(1) is a contraction with respect to the metric

𝑑 𝐺 1 , 𝑇 1 , 𝐺 2 , 𝑇 2 =

𝑠𝑢𝑝𝑖

{𝑚𝑎𝑥𝑡∈ℝ+

𝐺𝑖 1 𝑡 − 𝐺𝑖

2 𝑡 𝑒−(𝑀 13 ) 1 𝑡 ,𝑚𝑎𝑥𝑡∈ℝ+

𝑇𝑖 1 𝑡 − 𝑇𝑖

2 𝑡 𝑒−(𝑀 13 ) 1 𝑡}

109

110

Indeed if we denote

Definition of 𝐺 , 𝑇 : 𝐺 , 𝑇 = 𝒜(1)(𝐺, 𝑇)

It results

𝐺 13 1

− 𝐺 𝑖 2

≤ (𝑎13) 1 𝑡

0 𝐺14

1 − 𝐺14

2 𝑒−( 𝑀 13 ) 1 𝑠 13 𝑒( 𝑀 13 ) 1 𝑠 13 𝑑𝑠 13 +

{(𝑎13′ ) 1 𝐺13

1 − 𝐺13

2 𝑒−( 𝑀 13 ) 1 𝑠 13 𝑒−( 𝑀 13 ) 1 𝑠 13

𝑡

0+

(𝑎13′′ ) 1 𝑇14

1 , 𝑠 13 𝐺13

1 − 𝐺13

2 𝑒−( 𝑀 13 ) 1 𝑠 13 𝑒( 𝑀 13 ) 1 𝑠 13 +

𝐺13 2

|(𝑎13′′ ) 1 𝑇14

1 , 𝑠 13 − (𝑎13

′′ ) 1 𝑇14 2

, 𝑠 13 | 𝑒−( 𝑀 13) 1 𝑠 13 𝑒( 𝑀 13 ) 1 𝑠 13 }𝑑𝑠 13

Where 𝑠 13 represents integrand that is integrated over the interval 0, t

From the hypotheses on 25,26,27,28 and 29 it follows

111

𝐺 1 − 𝐺 2 𝑒−( 𝑀 13) 1 𝑡 ≤1

( 𝑀 13 ) 1 (𝑎13) 1 + (𝑎13′ ) 1 + ( 𝐴 13) 1 + ( 𝑃 13) 1 ( 𝑘 13) 1 𝑑 𝐺 1 , 𝑇 1 ; 𝐺 2 , 𝑇 2

And analogous inequalities for 𝐺𝑖 𝑎𝑛𝑑 𝑇𝑖 . Taking into account the hypothesis (34,35,36) the result

follows

112

Remark 1: The fact that we supposed (𝑎13′′ ) 1 and (𝑏13

′′ ) 1 depending also on t can be considered as not 113


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conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition

necessary to prove the uniqueness of the solution bounded by ( 𝑃 13) 1 𝑒( 𝑀 13 ) 1 𝑡 𝑎𝑛𝑑 ( 𝑄 13) 1 𝑒( 𝑀 13 ) 1 𝑡

respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it

suffices to consider that (𝑎𝑖′′ ) 1 and (𝑏𝑖

′′ ) 1 , 𝑖 = 13,14,15 depend only on T14 and respectively on

𝐺(𝑎𝑛𝑑 𝑛𝑜𝑡 𝑜𝑛 𝑡) and hypothesis can replaced by a usual Lipschitz condition.

Remark 2: There does not exist any 𝑡 where 𝐺𝑖 𝑡 = 0 𝑎𝑛𝑑 𝑇𝑖 𝑡 = 0

From 19 to 24 it results

𝐺𝑖 𝑡 ≥ 𝐺𝑖0𝑒 − (𝑎𝑖

′ ) 1 −(𝑎𝑖′′ ) 1 𝑇14 𝑠 13 ,𝑠 13 𝑑𝑠 13

𝑡0 ≥ 0

𝑇𝑖 𝑡 ≥ 𝑇𝑖0𝑒 −(𝑏𝑖

′ ) 1 𝑡 > 0 for t > 0

114

Definition of ( 𝑀 13) 1 1

, ( 𝑀 13) 1 2

𝑎𝑛𝑑 ( 𝑀 13) 1 3 :

Remark 3: if 𝐺13 is bounded, the same property have also 𝐺14 𝑎𝑛𝑑 𝐺15 . indeed if

𝐺13 < ( 𝑀 13) 1 it follows 𝑑𝐺14

𝑑𝑡≤ ( 𝑀 13) 1

1− (𝑎14

′ ) 1 𝐺14 and by integrating

𝐺14 ≤ ( 𝑀 13) 1 2

= 𝐺140 + 2(𝑎14 ) 1 ( 𝑀 13) 1

1/(𝑎14

′ ) 1

In the same way , one can obtain

𝐺15 ≤ ( 𝑀 13) 1 3

= 𝐺150 + 2(𝑎15 ) 1 ( 𝑀 13) 1

2/(𝑎15

′ ) 1

If 𝐺14 𝑜𝑟 𝐺15 is bounded, the same property follows for 𝐺13 , 𝐺15 and 𝐺13 , 𝐺14 respectively.

115

116

Remark 4: If 𝐺13 𝑖𝑠 bounded, from below, the same property holds for 𝐺14 𝑎𝑛𝑑 𝐺15 . The proof is

analogous with the preceding one. An analogous property is true if 𝐺14 is bounded from below.

117

Remark 5: If T13 is bounded from below and lim𝑡→∞((𝑏𝑖′′ ) 1 (𝐺 𝑡 , 𝑡)) = (𝑏14

′ ) 1 then 𝑇14 → ∞.

Definition of 𝑚 1 and 𝜀1 :

Indeed let 𝑡1 be so that for 𝑡 > 𝑡1

(𝑏14) 1 − (𝑏𝑖′′ ) 1 (𝐺 𝑡 , 𝑡) < 𝜀1, 𝑇13 (𝑡) > 𝑚 1

118

Then 𝑑𝑇14

𝑑𝑡≥ (𝑎14) 1 𝑚 1 − 𝜀1𝑇14 which leads to

𝑇14 ≥ (𝑎14 ) 1 𝑚 1

𝜀1 1 − 𝑒−𝜀1𝑡 + 𝑇14

0 𝑒−𝜀1𝑡 If we take t such that 𝑒−𝜀1𝑡 = 1

2 it results

𝑇14 ≥ (𝑎14 ) 1 𝑚 1

2 , 𝑡 = 𝑙𝑜𝑔

2

𝜀1 By taking now 𝜀1 sufficiently small one sees that T14 is unbounded.

The same property holds for 𝑇15 if lim𝑡→∞(𝑏15′′ ) 1 𝐺 𝑡 , 𝑡 = (𝑏15

′ ) 1

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to

42

119

120

It is now sufficient to take (𝑎𝑖)

2

( 𝑀 16 )(2) ,(𝑏𝑖)

2

( 𝑀 16 )(2) < 1 and to choose

( 𝑃 16 )(2) 𝑎𝑛𝑑 ( 𝑄 16 )(2) large to have

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(𝑎𝑖) 2

(𝑀 16 ) 2 ( 𝑃 16) 2 + ( 𝑃 16 )(2) + 𝐺𝑗0 𝑒

− ( 𝑃 16 )(2)+𝐺𝑗

0

𝐺𝑗0

≤ ( 𝑃 16 )(2)

122

(𝑏𝑖) 2

(𝑀 16 ) 2 ( 𝑄 16 )(2) + 𝑇𝑗0 𝑒

− ( 𝑄 16 )(2)+𝑇𝑗

0

𝑇𝑗0

+ ( 𝑄 16 )(2) ≤ ( 𝑄 16 )(2)

123

In order that the operator 𝒜(2) transforms the space of sextuples of functions 𝐺𝑖 , 𝑇𝑖 satisfying 34,35,36

into itself

124

The operator 𝒜(2) is a contraction with respect to the metric

𝑑 𝐺19 1 , 𝑇19

1 , 𝐺19 2 , 𝑇19

2 =

𝑠𝑢𝑝𝑖

{𝑚𝑎𝑥𝑡∈ℝ+

𝐺𝑖 1 𝑡 − 𝐺𝑖

2 𝑡 𝑒−(𝑀 16 ) 2 𝑡 ,𝑚𝑎𝑥𝑡∈ℝ+

𝑇𝑖 1 𝑡 − 𝑇𝑖

2 𝑡 𝑒−(𝑀 16 ) 2 𝑡}

125

Indeed if we denote

Definition of 𝐺19 , 𝑇19

: 𝐺19 , 𝑇19

= 𝒜(2)(𝐺19 , 𝑇19)

126

It results

𝐺 16 1

− 𝐺 𝑖 2

≤ (𝑎16) 2 𝑡

0 𝐺17

1 − 𝐺17

2 𝑒−( 𝑀 16 ) 2 𝑠 16 𝑒( 𝑀 16 ) 2 𝑠 16 𝑑𝑠 16 +

{(𝑎16′ ) 2 𝐺16

1 − 𝐺16

2 𝑒−( 𝑀 16 ) 2 𝑠 16 𝑒−( 𝑀 16 ) 2 𝑠 16

𝑡

0+

(𝑎16′′ ) 2 𝑇17

1 , 𝑠 16 𝐺16

1 − 𝐺16

2 𝑒−( 𝑀 16 ) 2 𝑠 16 𝑒( 𝑀 16 ) 2 𝑠 16 +

𝐺16 2

|(𝑎16′′ ) 2 𝑇17

1 , 𝑠 16 − (𝑎16

′′ ) 2 𝑇17 2

, 𝑠 16 | 𝑒−( 𝑀 16) 2 𝑠 16 𝑒( 𝑀 16 ) 2 𝑠 16 }𝑑𝑠 16

127

128

Where 𝑠 16 represents integrand that is integrated over the interval 0, 𝑡

From the hypotheses on 25,26,27,28 and 29 it follows

129

𝐺19 1 − 𝐺19

2 e−( M 16 ) 2 t ≤1

( M 16 ) 2 (𝑎16 ) 2 + (𝑎16′ ) 2 + ( A 16) 2 + ( P 16) 2 ( 𝑘 16) 2 d 𝐺19

1 , 𝑇19 1 ; 𝐺19

2 , 𝑇19 2

130

And analogous inequalities for G𝑖 and T𝑖 . Taking into account the hypothesis (34,35,36) the result follows 131

Remark 1: The fact that we supposed (𝑎16′′ ) 2 and (𝑏16

′′ ) 2 depending also on t can be considered as not

conformal with the reality, however we have put this hypothesis ,in order that we can postulate condition

necessary to prove the uniqueness of the solution bounded by ( P 16) 2 e( M 16 ) 2 t and ( Q 16) 2 e( M 16 ) 2 t

respectively of ℝ+.

If instead of proving the existence of the solution on ℝ+, we have to prove it only on a compact then it

suffices to consider that (𝑎𝑖′′ ) 2 and (𝑏𝑖

′′ ) 2 , 𝑖 = 16,17,18 depend only on T17 and respectively on

𝐺19 (and not on t) and hypothesis can replaced by a usual Lipschitz condition.

132

Remark 2: There does not exist any t where G𝑖 t = 0 and T𝑖 t = 0

From 19 to 24 it results

G𝑖 t ≥ G𝑖0e − (𝑎𝑖

′ ) 2 −(𝑎𝑖′′ ) 2 T17 𝑠 16 ,𝑠 16 d𝑠 16

t0 ≥ 0

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T𝑖 t ≥ T𝑖0e −(𝑏𝑖

′ ) 2 t > 0 for t > 0

Definition of ( M 16) 2 1

, ( M 16) 2 2

and ( M 16) 2 3 :

Remark 3: if G16 is bounded, the same property have also G17 and G18 . indeed if

G16 < ( M 16) 2 it follows dG17

dt≤ ( M 16) 2

1− (𝑎17

′ ) 2 G17 and by integrating

G17 ≤ ( M 16) 2 2

= G170 + 2(𝑎17) 2 ( M 16) 2

1/(𝑎17

′ ) 2

In the same way , one can obtain

G18 ≤ ( M 16) 2 3

= G180 + 2(𝑎18) 2 ( M 16) 2

2/(𝑎18

′ ) 2

If G17 or G18 is bounded, the same property follows for G16 , G18 and G16 , G17 respectively.

134

Remark 4: If G16 is bounded, from below, the same property holds for G17 and G18 . The proof is

analogous with the preceding one. An analogous property is true if G17 is bounded from below.

135

Remark 5: If T16 is bounded from below and limt→∞((𝑏𝑖′′ ) 2 ( 𝐺19 t , t)) = (𝑏17

′ ) 2 then T17 → ∞.

Definition of 𝑚 2 and ε2 :

Indeed let t2 be so that for t > t2

(𝑏17) 2 − (𝑏𝑖′′ ) 2 ( 𝐺19 t , t) < ε2, T16 (t) > 𝑚 2

136

137

Then dT17

dt≥ (𝑎17 ) 2 𝑚 2 − ε2T17 which leads to

T17 ≥ (𝑎17 ) 2 𝑚 2

ε2 1 − e−ε2t + T17

0 e−ε2t If we take t such that e−ε2t = 1

2 it results

138

T17 ≥ (𝑎17 ) 2 𝑚 2

2 , 𝑡 = log

2

ε2 By taking now ε2 sufficiently small one sees that T17 is unbounded.

The same property holds for T18 if lim𝑡→∞(𝑏18′′ ) 2 𝐺19 t , t = (𝑏18

′ ) 2

We now state a more precise theorem about the behaviors at infinity of the solutions of equations 37 to

42

139

140

Behavior of the solutions of equation 37 to 42

Theorem 2: If we denote and define

Definition of (𝜎1) 1 , (𝜎2) 1 , (𝜏1) 1 , (𝜏2) 1 :

(a) 𝜎1) 1 , (𝜎2) 1 , (𝜏1) 1 , (𝜏2) 1 four constants satisfying

−(𝜎2) 1 ≤ −(𝑎13′ ) 1 + (𝑎14

′ ) 1 − (𝑎13′′ ) 1 𝑇14 , 𝑡 + (𝑎14

′′ ) 1 𝑇14 , 𝑡 ≤ −(𝜎1) 1

−(𝜏2) 1 ≤ −(𝑏13′ ) 1 + (𝑏14

′ ) 1 − (𝑏13′′ ) 1 𝐺, 𝑡 − (𝑏14

′′ ) 1 𝐺, 𝑡 ≤ −(𝜏1) 1

141

Definition of (𝜈1) 1 , (𝜈2) 1 , (𝑢1) 1 , (𝑢2) 1 , 𝜈 1 , 𝑢 1 :

(b) By (𝜈1) 1 > 0 , (𝜈2) 1 < 0 and respectively (𝑢1) 1 > 0 , (𝑢2) 1 < 0 the roots of the equations

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(𝑎14) 1 𝜈 1 2

+ (𝜎1) 1 𝜈 1 − (𝑎13 ) 1 = 0 and (𝑏14) 1 𝑢 1 2

+ (𝜏1) 1 𝑢 1 − (𝑏13 ) 1 = 0

Definition of (𝜈 1) 1 , , (𝜈 2) 1 , (𝑢 1) 1 , (𝑢 2) 1 :

By (𝜈 1) 1 > 0 , (𝜈 2) 1 < 0 and respectively (𝑢 1) 1 > 0 , (𝑢 2) 1 < 0 the roots of the equations

(𝑎14) 1 𝜈 1 2

+ (𝜎2) 1 𝜈 1 − (𝑎13) 1 = 0 and (𝑏14) 1 𝑢 1 2

+ (𝜏2) 1 𝑢 1 − (𝑏13) 1 = 0

143

Definition of (𝑚1) 1 , (𝑚2) 1 , (𝜇1) 1 , (𝜇2) 1 , (𝜈0) 1 :-

(c) If we define (𝑚1) 1 , (𝑚2) 1 , (𝜇1) 1 , (𝜇2) 1 by

(𝑚2) 1 = (𝜈0) 1 , (𝑚1) 1 = (𝜈1) 1 , 𝑖𝑓 (𝜈0) 1 < (𝜈1) 1

(𝑚2) 1 = (𝜈1) 1 , (𝑚1) 1 = (𝜈 1) 1 , 𝑖𝑓 (𝜈1) 1 < (𝜈0) 1 < (𝜈 1) 1 ,

and (𝜈0) 1 =𝐺13

0

𝐺140

( 𝑚2) 1 = (𝜈1) 1 , (𝑚1) 1 = (𝜈0) 1 , 𝑖𝑓 (𝜈 1) 1 < (𝜈0) 1

144

and analogously

(𝜇2) 1 = (𝑢0) 1 , (𝜇1) 1 = (𝑢1) 1 , 𝑖𝑓 (𝑢0) 1 < (𝑢1) 1

(𝜇2) 1 = (𝑢1) 1 , (𝜇1) 1 = (𝑢 1) 1 , 𝑖𝑓 (𝑢1) 1 < (𝑢0) 1 < (𝑢 1) 1 ,

and (𝑢0) 1 =𝑇13

0

𝑇140

( 𝜇2) 1 = (𝑢1) 1 , (𝜇1) 1 = (𝑢0) 1 , 𝑖𝑓 (𝑢 1) 1 < (𝑢0) 1 where (𝑢1) 1 , (𝑢 1) 1

are defined by 59 and 61 respectively

145

146

Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities

𝐺130 𝑒 (𝑆1) 1 −(𝑝13 ) 1 𝑡 ≤ 𝐺13(𝑡) ≤ 𝐺13

0 𝑒(𝑆1) 1 𝑡

where (𝑝𝑖) 1 is defined by equation 25

1

(𝑚1) 1 𝐺130 𝑒 (𝑆1) 1 −(𝑝13 ) 1 𝑡 ≤ 𝐺14(𝑡) ≤

1

(𝑚2) 1 𝐺130 𝑒(𝑆1) 1 𝑡

147

( (𝑎15 ) 1 𝐺13

0

(𝑚1) 1 (𝑆1) 1 −(𝑝13 ) 1 −(𝑆2) 1 𝑒 (𝑆1) 1 −(𝑝13 ) 1 𝑡 − 𝑒−(𝑆2) 1 𝑡 + 𝐺15

0 𝑒−(𝑆2) 1 𝑡 ≤ 𝐺15 (𝑡) ≤

(𝑎15 ) 1 𝐺130

(𝑚2) 1 (𝑆1) 1 −(𝑎15′ ) 1

[𝑒(𝑆1) 1 𝑡 − 𝑒−(𝑎15′ ) 1 𝑡] + 𝐺15

0 𝑒−(𝑎15′ ) 1 𝑡)

148

𝑇130 𝑒(𝑅1) 1 𝑡 ≤ 𝑇13 (𝑡) ≤ 𝑇13

0 𝑒 (𝑅1) 1 +(𝑟13 ) 1 𝑡 149

1

(𝜇1) 1 𝑇130 𝑒(𝑅1) 1 𝑡 ≤ 𝑇13 (𝑡) ≤

1

(𝜇2) 1 𝑇130 𝑒 (𝑅1) 1 +(𝑟13 ) 1 𝑡 150

(𝑏15 ) 1 𝑇130

(𝜇1) 1 (𝑅1) 1 −(𝑏15′ ) 1

𝑒(𝑅1) 1 𝑡 − 𝑒−(𝑏15′ ) 1 𝑡 + 𝑇15

0 𝑒−(𝑏15′ ) 1 𝑡 ≤ 𝑇15 (𝑡) ≤

(𝑎15 ) 1 𝑇130

(𝜇2) 1 (𝑅1) 1 +(𝑟13 ) 1 +(𝑅2) 1 𝑒 (𝑅1) 1 +(𝑟13 ) 1 𝑡 − 𝑒−(𝑅2) 1 𝑡 + 𝑇15

0 𝑒−(𝑅2) 1 𝑡

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Definition of (𝑆1) 1 , (𝑆2) 1 , (𝑅1) 1 , (𝑅2) 1 :-

Where (𝑆1) 1 = (𝑎13) 1 (𝑚2) 1 − (𝑎13′ ) 1

(𝑆2) 1 = (𝑎15) 1 − (𝑝15 ) 1

(𝑅1) 1 = (𝑏13) 1 (𝜇2) 1 − (𝑏13′ ) 1

(𝑅2) 1 = (𝑏15′ ) 1 − (𝑟15 ) 1

152

Behavior of the solutions of equation 37 to 42

Theorem 2: If we denote and define

153

Definition of (σ1) 2 , (σ2) 2 , (τ1) 2 , (τ2) 2 :

(d) σ1) 2 , (σ2) 2 , (τ1) 2 , (τ2) 2 four constants satisfying

154

−(σ2) 2 ≤ −(𝑎16′ ) 2 + (𝑎17

′ ) 2 − (𝑎16′′ ) 2 T17 , 𝑡 + (𝑎17

′′ ) 2 T17 , 𝑡 ≤ −(σ1) 2 155

−(τ2) 2 ≤ −(𝑏16′ ) 2 + (𝑏17

′ ) 2 − (𝑏16′′ ) 2 𝐺19 , 𝑡 − (𝑏17

′′ ) 2 𝐺19 , 𝑡 ≤ −(τ1) 2 156

Definition of (𝜈1) 2 , (ν2) 2 , (𝑢1) 2 , (𝑢2) 2 : 157

By (𝜈1) 2 > 0 , (ν2) 2 < 0 and respectively (𝑢1) 2 > 0 , (𝑢2) 2 < 0 the roots 158

(e) of the equations (𝑎17 ) 2 𝜈 2 2

+ (σ1) 2 𝜈 2 − (𝑎16 ) 2 = 0 159

and (𝑏14) 2 𝑢 2 2

+ (τ1) 2 𝑢 2 − (𝑏16) 2 = 0 and 160

Definition of (𝜈 1) 2 , , (𝜈 2) 2 , (𝑢 1) 2 , (𝑢 2) 2 : 161

By (𝜈 1) 2 > 0 , (ν 2) 2 < 0 and respectively (𝑢 1) 2 > 0 , (𝑢 2) 2 < 0 the 162

roots of the equations (𝑎17 ) 2 𝜈 2 2

+ (σ2) 2 𝜈 2 − (𝑎16 ) 2 = 0 163

and (𝑏17) 2 𝑢 2 2

+ (τ2) 2 𝑢 2 − (𝑏16 ) 2 = 0 164

Definition of (𝑚1) 2 , (𝑚2) 2 , (𝜇1) 2 , (𝜇2) 2 :- 165

(f) If we define (𝑚1) 2 , (𝑚2) 2 , (𝜇1) 2 , (𝜇2) 2 by 166

(𝑚2) 2 = (𝜈0) 2 , (𝑚1) 2 = (𝜈1) 2 , 𝑖𝑓 (𝜈0) 2 < (𝜈1) 2 167

(𝑚2) 2 = (𝜈1) 2 , (𝑚1) 2 = (𝜈 1) 2 , 𝑖𝑓 (𝜈1) 2 < (𝜈0) 2 < (𝜈 1) 2 ,

and (𝜈0) 2 =G16

0

G170

168

( 𝑚2) 2 = (𝜈1) 2 , (𝑚1) 2 = (𝜈0) 2 , 𝑖𝑓 (𝜈 1) 2 < (𝜈0) 2 169

and analogously

(𝜇2) 2 = (𝑢0) 2 , (𝜇1) 2 = (𝑢1) 2 , 𝑖𝑓 (𝑢0) 2 < (𝑢1) 2

(𝜇2) 2 = (𝑢1) 2 , (𝜇1) 2 = (𝑢 1) 2 , 𝑖𝑓 (𝑢1) 2 < (𝑢0) 2 < (𝑢 1) 2 ,

and (𝑢0) 2 =T16

0

T170

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( 𝜇2) 2 = (𝑢1) 2 , (𝜇1) 2 = (𝑢0) 2 , 𝑖𝑓 (𝑢 1) 2 < (𝑢0) 2 171

Then the solution of 19,20,21,22,23 and 24 satisfies the inequalities

G160 e (S1) 2 −(𝑝16 ) 2 t ≤ 𝐺16 𝑡 ≤ G16

0 e(S1) 2 t

172

(𝑝𝑖) 2 is defined by equation 25 173

1

(𝑚1) 2 G160 e (S1) 2 −(𝑝16 ) 2 t ≤ 𝐺17(𝑡) ≤

1

(𝑚2) 2 G160 e(S1) 2 t 174

( (𝑎18 ) 2 G16

0

(𝑚1) 2 (S1) 2 −(𝑝16 ) 2 −(S2) 2 e (S1) 2 −(𝑝16 ) 2 t − e−(S2) 2 t + G18

0 e−(S2) 2 t ≤ G18(𝑡) ≤

(𝑎18 ) 2 G160

(𝑚2) 2 (S1) 2 −(𝑎18′ ) 2

[e(S1) 2 t − e−(𝑎18′ ) 2 t] + G18

0 e−(𝑎18′ ) 2 t)

175

T160 e(R1) 2 𝑡 ≤ 𝑇16(𝑡) ≤ T16

0 e (R1) 2 +(𝑟16 ) 2 𝑡 176

1

(𝜇1) 2 T160 e(R1) 2 𝑡 ≤ 𝑇16 (𝑡) ≤

1

(𝜇2) 2 T160 e (R1) 2 +(𝑟16 ) 2 𝑡 177

(𝑏18 ) 2 T160

(𝜇1) 2 (R1) 2 −(𝑏18′ ) 2

e(R1) 2 𝑡 − e−(𝑏18′ ) 2 𝑡 + T18

0 e−(𝑏18′ ) 2 𝑡 ≤ 𝑇18 (𝑡) ≤

(𝑎18 ) 2 T160

(𝜇2) 2 (R1) 2 +(𝑟16 ) 2 +(R2) 2 e (R1) 2 +(𝑟16 ) 2 𝑡 − e−(R2) 2 𝑡 + T18

0 e−(R2) 2 𝑡

178

Definition of (S1) 2 , (S2) 2 , (R1) 2 , (R2) 2 :- 179

Where (S1) 2 = (𝑎16) 2 (𝑚2) 2 − (𝑎16′ ) 2

(S2) 2 = (𝑎18) 2 − (𝑝18 ) 2

180

(𝑅1) 2 = (𝑏16) 2 (𝜇2) 1 − (𝑏16′ ) 2

(R2) 2 = (𝑏18′ ) 2 − (𝑟18 ) 2

181

182

PROOF : From 19,20,21,22,23,24 we obtain

𝑑𝜈 1

𝑑𝑡= (𝑎13 ) 1 − (𝑎13

′ ) 1 − (𝑎14′ ) 1 + (𝑎13

′′ ) 1 𝑇14 , 𝑡 − (𝑎14′′ ) 1 𝑇14 , 𝑡 𝜈 1 − (𝑎14 ) 1 𝜈 1

Definition of 𝜈 1 :- 𝜈 1 =𝐺13

𝐺14

It follows

− (𝑎14) 1 𝜈 1 2

+ (𝜎2) 1 𝜈 1 − (𝑎13) 1 ≤𝑑𝜈 1

𝑑𝑡≤ − (𝑎14 ) 1 𝜈 1

2+ (𝜎1) 1 𝜈 1 − (𝑎13) 1

From which one obtains

Definition of (𝜈 1) 1 , (𝜈0) 1 :-

(a) For 0 < (𝜈0) 1 =𝐺13

0

𝐺140 < (𝜈1) 1 < (𝜈 1) 1

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𝜈 1 (𝑡) ≥(𝜈1) 1 +(𝐶) 1 (𝜈2) 1 𝑒

− 𝑎14 1 (𝜈1) 1 −(𝜈0) 1 𝑡

1+(𝐶) 1 𝑒 − 𝑎14 1 (𝜈1) 1 −(𝜈0) 1 𝑡

, (𝐶) 1 =(𝜈1) 1 −(𝜈0) 1

(𝜈0) 1 −(𝜈2) 1

it follows (𝜈0) 1 ≤ 𝜈 1 (𝑡) ≤ (𝜈1) 1

In the same manner , we get

𝜈 1 (𝑡) ≤(𝜈 1) 1 +(𝐶 ) 1 (𝜈 2) 1 𝑒

− 𝑎14 1 (𝜈 1) 1 −(𝜈 2) 1 𝑡

1+(𝐶 ) 1 𝑒 − 𝑎14 1 (𝜈 1) 1 −(𝜈 2) 1 𝑡

, (𝐶 ) 1 =(𝜈 1) 1 −(𝜈0) 1

(𝜈0) 1 −(𝜈 2) 1

From which we deduce (𝜈0) 1 ≤ 𝜈 1 (𝑡) ≤ (𝜈 1) 1

184

(b) If 0 < (𝜈1) 1 < (𝜈0) 1 =𝐺13

0

𝐺140 < (𝜈 1) 1 we find like in the previous case,

(𝜈1) 1 ≤(𝜈1) 1 + 𝐶 1 (𝜈2) 1 𝑒

− 𝑎14 1 (𝜈1) 1 −(𝜈2) 1 𝑡

1+ 𝐶 1 𝑒 − 𝑎14 1 (𝜈1) 1 −(𝜈2) 1 𝑡

≤ 𝜈 1 𝑡 ≤

(𝜈 1) 1 + 𝐶 1 (𝜈 2) 1 𝑒

− 𝑎14 1 (𝜈 1) 1 −(𝜈 2) 1 𝑡

1+ 𝐶 1 𝑒 − 𝑎14 1 (𝜈 1) 1 −(𝜈 2) 1 𝑡

≤ (𝜈 1) 1

185

(c) If 0 < (𝜈1) 1 ≤ (𝜈 1) 1 ≤ (𝜈0) 1 =𝐺13

0

𝐺140 , we obtain

(𝜈1) 1 ≤ 𝜈 1 𝑡 ≤(𝜈 1) 1 + 𝐶 1 (𝜈 2) 1 𝑒

− 𝑎14 1 (𝜈 1) 1 −(𝜈 2) 1 𝑡

1+ 𝐶 1 𝑒 − 𝑎14 1 (𝜈 1) 1 −(𝜈 2) 1 𝑡

≤ (𝜈0) 1

And so with the notation of the first part of condition (c) , we have

Definition of 𝜈 1 𝑡 :-

(𝑚2) 1 ≤ 𝜈 1 𝑡 ≤ (𝑚1) 1 , 𝜈 1 𝑡 =𝐺13 𝑡

𝐺14 𝑡

In a completely analogous way, we obtain

Definition of 𝑢 1 𝑡 :-

(𝜇2) 1 ≤ 𝑢 1 𝑡 ≤ (𝜇1) 1 , 𝑢 1 𝑡 =𝑇13 𝑡

𝑇14 𝑡

Now, using this result and replacing it in 19, 20,21,22,23, and 24 we get easily the result stated in the

theorem.

Particular case :

If (𝑎13′′ ) 1 = (𝑎14

′′ ) 1 , 𝑡𝑕𝑒𝑛 (𝜎1) 1 = (𝜎2) 1 and in this case (𝜈1) 1 = (𝜈 1) 1 if in addition (𝜈0) 1 =

(𝜈1) 1 then 𝜈 1 𝑡 = (𝜈0) 1 and as a consequence 𝐺13(𝑡) = (𝜈0) 1 𝐺14(𝑡) this also defines (𝜈0) 1 for

the special case

Analogously if (𝑏13′′ ) 1 = (𝑏14

′′ ) 1 , 𝑡𝑕𝑒𝑛 (𝜏1) 1 = (𝜏2) 1 and then

(𝑢1) 1 = (𝑢 1) 1 if in addition (𝑢0) 1 = (𝑢1) 1 then 𝑇13(𝑡) = (𝑢0) 1 𝑇14 (𝑡) This is an important

consequence of the relation between (𝜈1) 1 and (𝜈 1) 1 , and definition of (𝑢0) 1 .

186

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PROOF : From 19,20,21,22,23,24 we obtain

d𝜈 2

dt= (𝑎16) 2 − (𝑎16

′ ) 2 − (𝑎17′ ) 2 + (𝑎16

′′ ) 2 T17 , t − (𝑎17′′ ) 2 T17 , t 𝜈 2 − (𝑎17) 2 𝜈 2

188

Definition of 𝜈 2 :- 𝜈 2 =G16

G17

189

It follows

− (𝑎17 ) 2 𝜈 2 2

+ (σ2) 2 𝜈 2 − (𝑎16 ) 2 ≤d𝜈 2

dt≤ − (𝑎17 ) 2 𝜈 2

2+ (σ1) 2 𝜈 2 − (𝑎16 ) 2

190

From which one obtains

Definition of (𝜈 1) 2 , (𝜈0) 2 :-

(d) For 0 < (𝜈0) 2 =G16

0

G170 < (𝜈1) 2 < (𝜈 1) 2

𝜈 2 (𝑡) ≥(𝜈1) 2 +(C) 2 (𝜈2) 2 𝑒

− 𝑎17 2 (𝜈1) 2 −(𝜈0) 2 𝑡

1+(C) 2 𝑒 − 𝑎17 2 (𝜈1) 2 −(𝜈0) 2 𝑡

, (C) 2 =(𝜈1) 2 −(𝜈0) 2

(𝜈0) 2 −(𝜈2) 2

it follows (𝜈0) 2 ≤ 𝜈 2 (𝑡) ≤ (𝜈1) 2

191

In the same manner , we get

𝜈 2 (𝑡) ≤(𝜈 1) 2 +(C ) 2 (𝜈 2) 2 𝑒

− 𝑎17 2 (𝜈 1) 2 −(𝜈 2) 2 𝑡

1+(C ) 2 𝑒 − 𝑎17 2 (𝜈 1) 2 −(𝜈 2) 2 𝑡

, (C ) 2 =(𝜈 1) 2 −(𝜈0) 2

(𝜈0) 2 −(𝜈 2) 2

192

From which we deduce (𝜈0) 2 ≤ 𝜈 2 (𝑡) ≤ (𝜈 1) 2 193

(e) If 0 < (𝜈1) 2 < (𝜈0) 2 =G16

0

G170 < (𝜈 1) 2 we find like in the previous case,

(𝜈1) 2 ≤(𝜈1) 2 + C 2 (𝜈2) 2 𝑒

− 𝑎17 2 (𝜈1) 2 −(𝜈2) 2 𝑡

1+ C 2 𝑒 − 𝑎17 2 (𝜈1) 2 −(𝜈2) 2 𝑡

≤ 𝜈 2 𝑡 ≤

(𝜈 1) 2 + C 2 (𝜈 2) 2 𝑒

− 𝑎17 2 (𝜈 1) 2 −(𝜈 2) 2 𝑡

1+ C 2 𝑒 − 𝑎17 2 (𝜈 1) 2 −(𝜈 2) 2 𝑡

≤ (𝜈 1) 2

194

(f) If 0 < (𝜈1) 2 ≤ (𝜈 1) 2 ≤ (𝜈0) 2 =G16

0

G170 , we obtain

(𝜈1) 2 ≤ 𝜈 2 𝑡 ≤(𝜈 1) 2 + C 2 (𝜈 2) 2 𝑒

− 𝑎17 2 (𝜈 1) 2 −(𝜈 2) 2 𝑡

1+ C 2 𝑒 − 𝑎17 2 (𝜈 1) 2 −(𝜈 2) 2 𝑡

≤ (𝜈0) 2

And so with the notation of the first part of condition (c) , we have

195

Definition of 𝜈 2 𝑡 :-

(𝑚2) 2 ≤ 𝜈 2 𝑡 ≤ (𝑚1) 2 , 𝜈 2 𝑡 =𝐺16 𝑡

𝐺17 𝑡

196

In a completely analogous way, we obtain

Definition of 𝑢 2 𝑡 :-

(𝜇2) 2 ≤ 𝑢 2 𝑡 ≤ (𝜇1) 2 , 𝑢 2 𝑡 =𝑇16 𝑡

𝑇17 𝑡

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Now, using this result and replacing it in 19, 20,21,22,23, and 24 we get easily the result stated in the

theorem.

198

Particular case :

If (𝑎16′′ ) 2 = (𝑎17

′′ ) 2 , 𝑡𝑕𝑒𝑛 (σ1) 2 = (σ2) 2 and in this case (𝜈1) 2 = (𝜈 1) 2 if in addition (𝜈0) 2 =

(𝜈1) 2 then 𝜈 2 𝑡 = (𝜈0) 2 and as a consequence 𝐺16(𝑡) = (𝜈0) 2 𝐺17(𝑡)

Analogously if (𝑏16′′ ) 2 = (𝑏17

′′ ) 2 , 𝑡𝑕𝑒𝑛 (τ1) 2 = (τ2) 2 and then

(𝑢1) 2 = (𝑢 1) 2 if in addition (𝑢0) 2 = (𝑢1) 2 then 𝑇16(𝑡) = (𝑢0) 2 𝑇17 (𝑡) This is an important

consequence of the relation between (𝜈1) 2 and (𝜈 1) 2

199

We can prove the following

Theorem 3: If (𝑎𝑖′′ ) 1 𝑎𝑛𝑑 (𝑏𝑖

′′ ) 1 are independent on 𝑡 , and the conditions (with the notations

25,26,27,28)

(𝑎13′ ) 1 (𝑎14

′ ) 1 − 𝑎13 1 𝑎14

1 < 0

(𝑎13′ ) 1 (𝑎14

′ ) 1 − 𝑎13 1 𝑎14

1 + 𝑎13 1 𝑝13

1 + (𝑎14′ ) 1 𝑝14

1 + 𝑝13 1 𝑝14

1 > 0

(𝑏13′ ) 1 (𝑏14

′ ) 1 − 𝑏13 1 𝑏14

1 > 0 ,

(𝑏13′ ) 1 (𝑏14

′ ) 1 − 𝑏13 1 𝑏14

1 − (𝑏13′ ) 1 𝑟14

1 − (𝑏14′ ) 1 𝑟14

1 + 𝑟13 1 𝑟14

1 < 0

𝑤𝑖𝑡𝑕 𝑝13 1 , 𝑟14

1 as defined by equation 25 are satisfied , then the system

200

Theorem 3: If (𝑎𝑖′′ ) 2 𝑎𝑛𝑑 (𝑏𝑖

′′ ) 2 are independent on t , and the conditions (with the notations

25,26,27,28)

201

202

(𝑎16′ ) 2 (𝑎17

′ ) 2 − 𝑎16 2 𝑎17

2 < 0 203

(𝑎16′ ) 2 (𝑎17

′ ) 2 − 𝑎16 2 𝑎17

2 + 𝑎16 2 𝑝16

2 + (𝑎17′ ) 2 𝑝17

2 + 𝑝16 2 𝑝17

2 > 0 204

(𝑏16′ ) 2 (𝑏17

′ ) 2 − 𝑏16 2 𝑏17

2 > 0 , 205

(𝑏16′ ) 2 (𝑏17

′ ) 2 − 𝑏16 2 𝑏17

2 − (𝑏16′ ) 2 𝑟17

2 − (𝑏17′ ) 2 𝑟17

2 + 𝑟16 2 𝑟17

2 < 0

𝑤𝑖𝑡𝑕 𝑝16 2 , 𝑟17

2 as defined by equation 25 are satisfied , then the system

206

207

𝑎13 1 𝐺14 − (𝑎13

′ ) 1 + (𝑎13′′ ) 1 𝑇14 𝐺13 = 0 208

𝑎14 1 𝐺13 − (𝑎14

′ ) 1 + (𝑎14′′ ) 1 𝑇14 𝐺14 = 0 209

𝑎15 1 𝐺14 − (𝑎15

′ ) 1 + (𝑎15′′ ) 1 𝑇14 𝐺15 = 0 210

𝑏13 1 𝑇14 − [(𝑏13

′ ) 1 − (𝑏13′′ ) 1 𝐺 ]𝑇13 = 0 211

𝑏14 1 𝑇13 − [(𝑏14

′ ) 1 − (𝑏14′′ ) 1 𝐺 ]𝑇14 = 0 212

𝑏15 1 𝑇14 − [(𝑏15

′ ) 1 − (𝑏15′′ ) 1 𝐺 ]𝑇15 = 0 213

has a unique positive solution , which is an equilibrium solution for the system (19 to 24) 214

𝑎16 2 𝐺17 − (𝑎16

′ ) 2 + (𝑎16′′ ) 2 𝑇17 𝐺16 = 0 215


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𝑎17 2 𝐺16 − (𝑎17

′ ) 2 + (𝑎17′′ ) 2 𝑇17 𝐺17 = 0 216

𝑎18 2 𝐺17 − (𝑎18

′ ) 2 + (𝑎18′′ ) 2 𝑇17 𝐺18 = 0 217

𝑏16 2 𝑇17 − [(𝑏16

′ ) 2 − (𝑏16′′ ) 2 𝐺19 ]𝑇16 = 0 218

𝑏17 2 𝑇16 − [(𝑏17

′ ) 2 − (𝑏17′′ ) 2 𝐺19 ]𝑇17 = 0 219

𝑏18 2 𝑇17 − [(𝑏18

′ ) 2 − (𝑏18′′ ) 2 𝐺19 ]𝑇18 = 0 220

has a unique positive solution , which is an equilibrium solution for (19 to 24) 221

222

Proof:

(a) Indeed the first two equations have a nontrivial solution 𝐺13 , 𝐺14 if

𝐹 𝑇 = (𝑎13′ ) 1 (𝑎14

′ ) 1 − 𝑎13 1 𝑎14

1 + (𝑎13′ ) 1 (𝑎14

′ ′ ) 1 𝑇14 + (𝑎14′ ) 1 (𝑎13

′′ ) 1 𝑇14 +

(𝑎13′′ ) 1 𝑇14 (𝑎14

′′ ) 1 𝑇14 = 0

223

Proof:

(a) Indeed the first two equations have a nontrivial solution 𝐺16 , 𝐺17 if

F 𝑇19 = (𝑎16′ ) 2 (𝑎17

′ ) 2 − 𝑎16 2 𝑎17

2 + (𝑎16′ ) 2 (𝑎17

′′ ) 2 𝑇17 + (𝑎17′ ) 2 (𝑎16

′′ ) 2 𝑇17 +

(𝑎16′′ ) 2 𝑇17 (𝑎17

′′ ) 2 𝑇17 = 0

224

Definition and uniqueness of T14∗ :-

After hypothesis 𝑓 0 < 0, 𝑓 ∞ > 0 and the functions (𝑎𝑖′′ ) 1 𝑇14 being increasing, it follows that

there exists a unique 𝑇14∗ for which 𝑓 𝑇14

∗ = 0. With this value , we obtain from the three first

equations

𝐺13 = 𝑎13 1 𝐺14

(𝑎13′ ) 1 +(𝑎13

′′ ) 1 𝑇14∗

, 𝐺15 = 𝑎15 1 𝐺14

(𝑎15′ ) 1 +(𝑎15

′′ ) 1 𝑇14∗

225

226

Definition and uniqueness of T17∗ :-

After hypothesis 𝑓 0 < 0, 𝑓 ∞ > 0 and the functions (𝑎𝑖′′ ) 2 𝑇17 being increasing, it follows that

there exists a unique T17∗ for which 𝑓 T17

∗ = 0. With this value , we obtain from the three first

equations

227

𝐺16 = 𝑎16 2 G17

(𝑎16′ ) 2 +(𝑎16

′′ ) 2 T17∗

, 𝐺18 = 𝑎18 2 G17

(𝑎18′ ) 2 +(𝑎18

′′ ) 2 T17∗

228

229

(c) By the same argument, the equations 92,93 admit solutions 𝐺13 , 𝐺14 if

𝜑 𝐺 = (𝑏13′ ) 1 (𝑏14

′ ) 1 − 𝑏13 1 𝑏14

1 −

(𝑏13′ ) 1 (𝑏14

′′ ) 1 𝐺 + (𝑏14′ ) 1 (𝑏13

′′ ) 1 𝐺 +(𝑏13′′ ) 1 𝐺 (𝑏14

′′ ) 1 𝐺 = 0

Where in 𝐺 𝐺13 , 𝐺14 , 𝐺15 , 𝐺13 , 𝐺15 must be replaced by their values from 96. It is easy to see that φ is a

decreasing function in 𝐺14 taking into account the hypothesis 𝜑 0 > 0 , 𝜑 ∞ < 0 it follows that there

exists a unique 𝐺14∗ such that 𝜑 𝐺∗ = 0

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(d) By the same argument, the equations 92,93 admit solutions 𝐺16 , 𝐺17 if

φ 𝐺19 = (𝑏16′ ) 2 (𝑏17

′ ) 2 − 𝑏16 2 𝑏17

2 −

(𝑏16′ ) 2 (𝑏17

′′ ) 2 𝐺19 + (𝑏17′ ) 2 (𝑏16

′′ ) 2 𝐺19 +(𝑏16′′ ) 2 𝐺19 (𝑏17

′′ ) 2 𝐺19 = 0

231

Where in 𝐺19 𝐺16 , 𝐺17 , 𝐺18 , 𝐺16 , 𝐺18 must be replaced by their values from 96. It is easy to see that φ

is a decreasing function in 𝐺17 taking into account the hypothesis φ 0 > 0 , 𝜑 ∞ < 0 it follows that

there exists a unique G14∗ such that φ 𝐺19

∗ = 0

232

233

Finally we obtain the unique solution of 89 to 94

𝐺14∗ given by 𝜑 𝐺∗ = 0 , 𝑇14

∗ given by 𝑓 𝑇14∗ = 0 and

𝐺13∗ =

𝑎13 1 𝐺14∗

(𝑎13′ ) 1 +(𝑎13

′′ ) 1 𝑇14∗

, 𝐺15∗ =

𝑎15 1 𝐺14∗

(𝑎15′ ) 1 +(𝑎15

′′ ) 1 𝑇14∗

𝑇13∗ =

𝑏13 1 𝑇14∗

(𝑏13′ ) 1 −(𝑏13

′′ ) 1 𝐺∗ , 𝑇15

∗ = 𝑏15 1 𝑇14

∗

(𝑏15′ ) 1 −(𝑏15

′′ ) 1 𝐺∗

Obviously, these values represent an equilibrium solution of 19,20,21,22,23,24

234

Finally we obtain the unique solution of 89 to 94 235

G17∗ given by φ 𝐺19

∗ = 0 , T17∗ given by 𝑓 T17

∗ = 0 and 236

G16∗ =

a16 2 G17∗

(a16′ ) 2 +(a16

′′ ) 2 T17∗

, G18∗ =

a18 2 G17∗

(a18′ ) 2 +(a18

′′ ) 2 T17∗

237

T16∗ =

b16 2 T17∗

(b16′ ) 2 −(b16

′′ ) 2 𝐺19 ∗ , T18

∗ = b18 2 T17

∗

(b18′ ) 2 −(b18

′′ ) 2 𝐺19 ∗

238

Obviously, these values represent an equilibrium solution of 19,20,21,22,23,24 239

240

ASYMPTOTIC STABILITY ANALYSIS

Theorem 4: If the conditions of the previous theorem are satisfied and if the functions

(𝑎𝑖′′ ) 1 𝑎𝑛𝑑 (𝑏𝑖

′′ ) 1 Belong to 𝐶 1 ( ℝ+) then the above equilibrium point is asymptotically stable.

Proof: Denote

Definition of 𝔾𝑖 , 𝕋𝑖 :-

𝐺𝑖 = 𝐺𝑖∗ + 𝔾𝑖 , 𝑇𝑖 = 𝑇𝑖

∗ + 𝕋𝑖

𝜕(𝑎14

′′ ) 1

𝜕𝑇14 𝑇14

∗ = 𝑞14 1 ,

𝜕(𝑏𝑖′′ ) 1

𝜕𝐺𝑗 𝐺∗ = 𝑠𝑖𝑗

241

Then taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24 242

𝑑𝔾13

𝑑𝑡= − (𝑎13

′ ) 1 + 𝑝13 1 𝔾13 + 𝑎13

1 𝔾14 − 𝑞13 1 𝐺13

∗ 𝕋14 243

𝑑𝔾14

𝑑𝑡= − (𝑎14

′ ) 1 + 𝑝14 1 𝔾14 + 𝑎14

1 𝔾13 − 𝑞14 1 𝐺14

∗ 𝕋14 244


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𝑑𝔾15

𝑑𝑡= − (𝑎15

′ ) 1 + 𝑝15 1 𝔾15 + 𝑎15

1 𝔾14 − 𝑞15 1 𝐺15

∗ 𝕋14 245

𝑑𝕋13

𝑑𝑡= − (𝑏13

′ ) 1 − 𝑟13 1 𝕋13 + 𝑏13

1 𝕋14 + 𝑠 13 𝑗 𝑇13∗ 𝔾𝑗

15𝑗=13 246

𝑑𝕋14

𝑑𝑡= − (𝑏14

′ ) 1 − 𝑟14 1 𝕋14 + 𝑏14

1 𝕋13 + 𝑠 14 (𝑗 )𝑇14∗ 𝔾𝑗

15𝑗=13 247

𝑑𝕋15

𝑑𝑡= − (𝑏15

′ ) 1 − 𝑟15 1 𝕋15 + 𝑏15

1 𝕋14 + 𝑠 15 (𝑗 )𝑇15∗ 𝔾𝑗

15𝑗=13 248

249

ASYMPTOTIC STABILITY ANALYSIS

Theorem 4: If the conditions of the previous theorem are satisfied and if the functions

(a𝑖′′ ) 2 and (b𝑖

′′ ) 2 Belong to C 2 ( ℝ+) then the above equilibrium point is asymptotically stable

250

Proof: Denote

Definition of 𝔾𝑖 , 𝕋𝑖 :-

251

G𝑖 = G𝑖∗ + 𝔾𝑖 , T𝑖 = T𝑖

∗ + 𝕋𝑖 252

∂(𝑎17′′ ) 2

∂T17 T17

∗ = 𝑞17 2 ,

∂(𝑏𝑖′′ ) 2

∂G𝑗 𝐺19

∗ = 𝑠𝑖𝑗 253

taking into account equations 89 to 94 and neglecting the terms of power 2, we obtain from 19 to 24 254

d𝔾16

dt= − (𝑎16

′ ) 2 + 𝑝16 2 𝔾16 + 𝑎16

2 𝔾17 − 𝑞16 2 G16

∗ 𝕋17 255

d𝔾17

dt= − (𝑎17

′ ) 2 + 𝑝17 2 𝔾17 + 𝑎17

2 𝔾16 − 𝑞17 2 G17

∗ 𝕋17 256

d𝔾18

dt= − (𝑎18

′ ) 2 + 𝑝18 2 𝔾18 + 𝑎18

2 𝔾17 − 𝑞18 2 G18

∗ 𝕋17 257

d𝕋16

dt= − (𝑏16

′ ) 2 − 𝑟16 2 𝕋16 + 𝑏16

2 𝕋17 + 𝑠 16 𝑗 T16∗ 𝔾𝑗

18𝑗 =16 258

d𝕋17

dt= − (𝑏17

′ ) 2 − 𝑟17 2 𝕋17 + 𝑏17

2 𝕋16 + 𝑠 17 (𝑗 )T17∗ 𝔾𝑗

18𝑗 =16 259

d𝕋18

dt= − (𝑏18

′ ) 2 − 𝑟18 2 𝕋18 + 𝑏18

2 𝕋17 + 𝑠 18 (𝑗 )T18∗ 𝔾𝑗

18𝑗 =16 260

261

The characteristic equation of this system is

𝜆 1 + (𝑏15′ ) 1 − 𝑟15

1 { 𝜆 1 + (𝑎15′ ) 1 + 𝑝15

1

𝜆 1 + (𝑎13′ ) 1 + 𝑝13

1 𝑞14 1 𝐺14

∗ + 𝑎14 1 𝑞13

1 𝐺13∗

𝜆 1 + (𝑏13′ ) 1 − 𝑟13

1 𝑠 14 , 14 𝑇14∗ + 𝑏14

1 𝑠 13 , 14 𝑇14∗

+ 𝜆 1 + (𝑎14′ ) 1 + 𝑝14

1 𝑞13 1 𝐺13

∗ + 𝑎13 1 𝑞14

1 𝐺14∗

𝜆 1 + (𝑏13′ ) 1 − 𝑟13

1 𝑠 14 , 13 𝑇14∗ + 𝑏14

1 𝑠 13 , 13 𝑇13∗

262


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𝜆 1 2

+ (𝑎13′ ) 1 + (𝑎14

′ ) 1 + 𝑝13 1 + 𝑝14

1 𝜆 1

𝜆 1 2

+ (𝑏13′ ) 1 + (𝑏14

′ ) 1 − 𝑟13 1 + 𝑟14

1 𝜆 1

+ 𝜆 1 2

+ (𝑎13′ ) 1 + (𝑎14

′ ) 1 + 𝑝13 1 + 𝑝14

1 𝜆 1 𝑞15 1 𝐺15

+ 𝜆 1 + (𝑎13′ ) 1 + 𝑝13

1 𝑎15 1 𝑞14

1 𝐺14∗ + 𝑎14

1 𝑎15 1 𝑞13

1 𝐺13∗

𝜆 1 + (𝑏13′ ) 1 − 𝑟13

1 𝑠 14 , 15 𝑇14∗ + 𝑏14

1 𝑠 13 , 15 𝑇13∗ } = 0

+

𝜆 2 + (𝑏18′ ) 2 − 𝑟18

2 { 𝜆 2 + (𝑎18′ ) 2 + 𝑝18

2

𝜆 2 + (𝑎16′ ) 2 + 𝑝16

2 𝑞17 2 G17

∗ + 𝑎17 2 𝑞16

2 G16∗

𝜆 2 + (𝑏16′ ) 2 − 𝑟16

2 𝑠 17 , 17 T17∗ + 𝑏17

2 𝑠 16 , 17 T17∗

+ 𝜆 2 + (𝑎17′ ) 2 + 𝑝17

2 𝑞16 2 G16

∗ + 𝑎16 2 𝑞17

2 G17∗

𝜆 2 + (𝑏16′ ) 2 − 𝑟16

2 𝑠 17 , 16 T17∗ + 𝑏17

2 𝑠 16 , 16 T16∗

𝜆 2 2

+ (𝑎16′ ) 2 + (𝑎17

′ ) 2 + 𝑝16 2 + 𝑝17

2 𝜆 2

𝜆 2 2

+ (𝑏16′ ) 2 + (𝑏17

′ ) 2 − 𝑟16 2 + 𝑟17

2 𝜆 2

+ 𝜆 2 2

+ (𝑎16′ ) 2 + (𝑎17

′ ) 2 + 𝑝16 2 + 𝑝17

2 𝜆 2 𝑞18 2 G18

+ 𝜆 2 + (𝑎16′ ) 2 + 𝑝16

2 𝑎18 2 𝑞17

2 G17∗ + 𝑎17

2 𝑎18 2 𝑞16

2 G16∗

𝜆 2 + (𝑏16′ ) 2 − 𝑟16

2 𝑠 17 , 18 T17∗ + 𝑏17

2 𝑠 16 , 18 T16∗ } = 0

And as one sees, all the coefficients are positive. It follows that all the roots have negative real part, and

this proves the theorem.

Acknowledgments:

The introduction is a collection of information from various articles, Books, News Paper reports, Home

Pages Of authors, Journal Reviews, the internet including Wikipedia. We acknowledge all authors who

have contributed to the same. In the eventuality of the fact that there has been any act of omission on the

part of the authors, We regret with great deal of compunction, contrition, and remorse. As Newton said, it

is only because erudite and eminent people allowed one to piggy ride on their backs; probably an attempt

has been made to look slightly further. Once again, it is stated that the references are only illustrative and

not comprehensive

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