Chaos

Chaos in Non-Linear Dynamics-Shweta Tripathi-Rajan SinghOutline Introduction to chaosDefinition of chaosProperties of ChaosStudy of Dynamic System

Introduction to ChaosNormal everyday meaning of the word is "a condition or place of great disorder or confusion", which sounds similar to the meaning of randomness: "having no specific pattern". However, when we use the word chaos in a mathematical or scientific sense, it means something very different.a chaotic system is not a random system. A chaotic system sometimes SEEMS random if you do not recognize that it is chaotic.Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions.

Definition of Chaos Irregular motion of a dynamical system that is deterministic, sensitive to initial conditions, and impossible to predict in the long term with anything less than an innite and perfect representation of analog values. G.W. FlakeChaos is sustained and disorderly-looking long-term evolution that satises certain special mathematical criteria and that occur deterministic non-linear system. G.P. Williams Chaos is sustained and disorderly-looking long-term evolution that satises certain special mathematical criteria and that occur deterministic non-linear system. G.P. WilliamsProperties of Chaotic SystemsChaotic systems are not random. They may appear to be. They have some simple defining features:1. Chaotic systems are deterministic. This means they have some determining equation ruling their behaviour.2. Chaotic systems are very sensitive to the initial conditions. A very slight change in the starting point can lead to enormously different outcomes. This makes the system fairly unpredictable.3. Chaotic systems appear to be disorderly, even random. But they are not.Beneath the seemingly random behaviour is a sense of order and pattern. Truly random systems are not chaotic.

Study of Dynamic SystemAny system that moves or changes in time.Examples: Motion of Planets Whether Stock market Chemical ReactionsCan one predict what will happen? Some of these dynamical systems are clearly predictable, like Motion of planets and Chemical reactions.While some of them are totally unpredictable, like Whether and Stock market.Even a simplest system which is dependent on only one variable can be very much unpredictable.Mathematical Dynamic SystemsSimplest of mathematical dynamical system is iterated functions. Example: Start with xCompute x Compute xCompute x...Can we predict what will happen?Applying x iteratively after sometimes starts giving 1 continuously, whatever is the input number given initially.In similar fashion, if we apply x iteratively, we get after sometime whatever be the input initially.Similarly sin(x) leads to 0 after some iterations.cos(x) leads to a strange quantity 0.73908 in the same way.

Now how about iterating a simple quadratic expression like 4x*(1-x)? Example: If we start with x=0.4we get 0.96, 0.154, 0.521, 0.998, 0.008, 0.032, 0.123, 0.431, 0.980, 0.078, 0.288, 0.823As the results dont follow any sort of pattern, or in other words, as these are totally unpredictable, we can say that this function is chaotic.

Orbit of X X -> f(X) -> f(f(X)) -> f(f(f(X))).Can one predict the fate of orbits?For root(x):

Sin(x) Fixed point Iteration:

Cos(x) Fixed point Iteration:

Geometric representation

xf(x)0110.5Evolution of a map: 1) Choose initial conditions2) Proceed vertically until you hit f(x) 3) Proceed horizontally until you hit y=x4) Repeat 2)5) Repeat 3) . :

Evolution of the logistic map

fixed point ?131) In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation WHY??!!!Phenomenology of the logistic map

y=xf(x)0110.5

y=xf(x)0110.5

010.51

010.51

fixed pointfixed point2-cycle?chaos?a)b)c)d)Whats going on? Analyze first a) b) b) c) , 14In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation WHY??!!! Recall the DEFINITION OF A FIXED POINT, from Pag.7 of the notes (special point where the velocity field is zero. The system is then in mechanical equilibrium)Geometrical representation

xf(x)0110.5

xf(x)0110.5

fixed pointEvolution of the logistic mapHow do we analyze the existence/stability of a fixed point?15In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation WHY??!!! Recall the DEFINITION OF A FIXED POINT, from Pag.7 of the notes (special point where the velocity field is zero. The system is then in mechanical equilibrium)Fixed points

- Condition for existence:

- Logistic map:

- Notice: since the second fixed point exists only for

Stability

- Define the distance of from the fixed point

- Consider a neighborhood of

- The requirement implies

Logistic map?Taylor expansion161) In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation WHY??!!!Stability and the Logistic Map

- Stability condition:- First fixed point: stable (attractor) for

- Second fixed point: stable (attractor) for

xf(x)0110.5

xf(x)0110.5 No coexistence of 2 stable fixed points for these parameters (transcritical biforcation)What about ?

17

Period doubling

xf(x)0110.5

Evolution of the logistic map

010.51) The map oscillatesbetween two values of x2) Period doubling:Observations:

What is it happening?18Check WHAT IS THE EXACT DEFINITION OF PITCHFORK BIFURCATION??!!! FROM egwald-online. MAKE A LIST OF POSSIBLE BIFURCATIONS?! Add that since x_{n+2}=x_n it is natural to regards these fixed points as stable fixed points (attractors) for f(f(x)).Period doubling

010.5and thus:- At the fixed point becomes unstable, since

Observation: an attracting 2-cycle starts (flip)-bifurcation The points are found solving the equations

These points form a 2-cycle forHowever, the relation suggeststhey are fixed points for the iterated map

Stability analysis for :

and thus: For , loss of stability and bifurcation to a 4-cycleNow, graphically..

>Why do these points appear? 19 Rifare the stability analysis as in pag.22 of Rasband EXPLAIN SOMEWHERE HOW TO GET THE FEILGELBAUM NUMBER, \mu_{\infty}, AND SHOW THE TABLE WITH THE BIFURCATION NUMBERS AS IN Rasband PAG.23

Bifurcation diagram

Plot of fixed points vs

20Check WHAT IS THE EXACT DEFINITION OF PITCHFORK BIFURCATION??!!! FROM egwald-online. MAKE A LIST OF POSSIBLE BIFURCATIONS?! _DEFINE_ PITCHFORK BIFURCATIONS!! AS IN PAG.31 OF Rasband 3) SHOW THAT THE LYAPUNOV EXPONENT SHOWS CHAOS..Bifurcation diagram

Plot of fixed points vs

Observations:Infinite series of period doublings at pitchfork-like (flip) bifurcationsAfter a point

chaos seems to appear3) Regions where stable periodic cycles exist occur for

What is general?21Check WHAT IS THE EXACT DEFINITION OF PITCHFORK BIFURCATION??!!! FROM egwald-online. MAKE A LIST OF POSSIBLE BIFURCATIONS?! _DEFINE_ PITCHFORK BIFURCATIONS!! AS IN PAG.31 OF Rasband 3) SHOW THAT THE LYAPUNOV EXPONENT SHOWS CHAOS..Fractals and ChaosWe have just looked at some simple function iteration. However, no matter how simple the function, iteration can produce complicated and unexpected results. All the functions we explored used real numbers. Results of iteration are even more unusual and exciting when complex numbers are used instead. Beautiful and intricate graphic images are produced when iterating a function with a complex number domain and range. This leads to a new field calledFRACTALSA few of the thousands of such fractals are shown below.

Thank You..