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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