Route to chaos

Awadhesh Prasad

University of Delhi, Delhi, India

Systems

Linear Nonlinear

-- Models -- Functions

-- Objects

Length of lightening?

Length of a tree?

Length of coastline

Area of surface?

Examples

Linear: Point, Line, Plane, Cube

Nonlinear: Sphere, length of a tree

Linear vs Nonlinear

inputoutput dinput)(output

Linear

Nonlinear

Differential equation: ),( tXFdt

dx

xdt

dx

xdt

dxt

dt

xd

2

2

22

2

nxdt

dxt

dt

xd n

xdt

dxx

dt

xd

2

2

Linear Nonlinear2x

dt

dx

Differential equation: ),( tXFdt

dx

xdt

dx

xdt

dxt

dt

xd

2

2

22

2

nxdt

dxt

dt

xd n

xdt

dxx

dt

xd

2

2

Linear Nonlinear2x

dt

dx

Pendulum

Individual/single

)sin(..

l

g

Pendulum: Linear vs Nonlinear

Linear

)sin(

l

g

..

)cos(0 tl

g ]});([{sin2 01 ktt

l

gksn

)2/sin( 0k

Nonlinear

sn Jacobian Elliptic function

)sin(..

l

g

Pendulum: Linear vs Nonlinear

Linear

)sin(..

l

g

)sin(

l

g

..

Nonlinear

Linear Pendulum

Undamped Damped

Center Stable

l

g

.....

l

g

Nonlinear pendulum

Undapmed

Center/saddle

)sin(..

l

g

Damped

Stable & Unstable

...

)sin( l

g

Solutions

Dissipation Nonlinearity+ = ?

1D Systems

0.

xFixed/stationary points00 x

0

|)(

xdx

xdf

xxfx )(.

)exp()0()( txtx

stable0

unstable0

1D Systems

0.

xFixed/stationary points 0x

02|)(

0x

dx

xdfx

2.

)( xxfx

U UUS S S

SU

2D Systems

.

.

)......3)(2)(1()( rrrrrfr

>2D Systems

xyzz

zyxy

xyx

3/8

28

)(

.

.

.

Solutions

Systems

Conservative Dissipative

Invariants Attractors

Solutions

Systems

Linear Nonlinear

-- Fixed point-- Periodic-- Quasiperiodic-- Chaotic

Solutions

-- Fixed point

-- Periodic

-- Chaotic

Lin

ea r

Non

line

ar

-- Quasiperiodic

Fixed point Solutions

Periodic Solutions

Unstable fixed points

Periodic solutions

Quasiperiodic

Quasiperiodic

Stable/unstable periodic orbits

21 nn xx

121 nn xx

)1(41 nnn xxx

Chaos!!

P3

P1

P2

P0

Chaotic Solutions

Chaotic Solutions

Properties of chaos

-- geometrically strange -- temporally irregular-- sensitive to initial conditions Due to UPO?

Unstable Periodic Orbits (UPOs)

Bifurcation

Sand piling

Heartbeat

)(ZFz

)(1 nn xfx

Qualitative change in dynamics as parameter varies

Chaos to Periodic: Heart Attack

Christini D J et al. PNAS 98, 5827(2001)

Montage

Onset of a temporal lobe epileptic seizure

Ref. L. Iasemidis

EEG: Epileptic Patient (temporal lobe epilepsy)temporal lobe epilepsy)

Preictal

minutes

ictal postictal

Aim

Periodic Chaotic

Routes?

Chaos Chaotic

)1(1 nnn xxx

Period doubling: Laser

End