Hamiltonian Chaos
Chaos and TimeSeries Analysis
10/24/00 Lecture #8 in Physics 505
This is probably the most technically difficult
lecture of the course.
Comments on Homework
#6 (Lyapunov Exponent)

Not everyone had a good graph of LE versus
C
for B = 0.3

Some had numerical troubles with unbounded orbits (C > 1.42)

BASIC code for doing part 3 has been put
on the WWW
Review (last
week)  Bifurcations

Bifurcation is a qualitative change in behavior at a critical
parameter value

Observation of a bifurcation verifies determinism

Flows are often analyzed using their maps (Poincaré
section)

Classifications:

Local  involves single equilibrium points

Global  equilibrium points appear or vanish

Continuous (subtle)  eigenvalues cross unit circle

Discontinuous (catastrophic)  eigenvalues appear or vanish

Explosive  like catastrophic but no hysteresis
(occur when attractor touches the basin boundary)

There are dozens of bifurcations, many not discovered

Terminology is not precise or universal (still evolving)

Transcritical Bifurcation

Pitchfork Bifurcation

Flip Bifurcation

Tangent (or SaddleNode or Blue Sky) Bifurcation

Catastrophe (1D example)

Hopf Bifurcation

Niemark (or Secondary Hopf) Bifurcation

A stable limit cycle becomes unstable and a 2torus is born

The Poincaré section exhibits a Hopf bifurcation

Main sequence (quasiperiodic route to chaos)

fixed point > limit cycle > 2torus > chaos

Ntorus with N > 2 not usually seen (Piexito's Theorem)
(3torus and higher are structurally unstable)

This contradicts the Landau theory of turbulence
(turbulence is a sum of very many periodic modes)

Also called the NewhouseRuelleTakens route

Probably the most common route to chaos at highD
Hamiltonian Systems 
Introduction and Motivation

These are systems that conserve mechanical energy

They have no dissipation (frictionless)

They are of historical interest and importance

Examples (all from physics):

Planetary motion (recall 3body problem)

Charged particles in magnetic fields

Incompressible fluid flows (liquids)

Trajectories of magnetic field lines

Quantum mechanics

Statistical mechanics
A Case Study  Mass on
a Spring (frictionless)

dx/dt = v

dv/dt = (k/m)x

This system has 1 spatial dimension (1 degree of freedom)

It has a 2D phase space however

Solution: kx^{2} + mv^{2} =
constant (conservation of energy)

Hamiltonian: H = kx^{2}/2 + mv^{2}/2
(total energy)

kx^{2}/2 is the potential energy (stored in spring)

mv^{2}/2 is the kinetic energy (energy of motion)

Let k = m = 1 for simplicity

Given the Hamiltonian, we can get the equations of motion:

dx/dt = dH/dv
= v

dv/dt = dH/dx
= x

where d is the partial derivative

The motion occurs along a 1D curve in 2D space

This curve is not a limit cycle (it is a center)

Such a system cannot exhibit chaos (even if driven)
Hamilton's Equations
(N Dimensions)

Generalize the above ideas to dimensions N > 1:

dq_{i}/dt = dH/dp_{i}
(q is a generalized coordinate)

dp_{i}/dt = dH/dq_{i}
(p is a generalized momentum = mv)

p and q constitute the phase space for
the dynamics

Ndimensional dynamics have a 2Ndimensional phase space

p_{i} and q_{i} (for i
= 1 to N) are the phase space variables

Note: dH/dt = dH/dp
dp/dt + dH/dq
dq/dt = 0

H is a constant of the motion (Hamiltonian)

There may be other constants (say a total of k)

The dynamics are constrained to a 2N  k dimensional
space

Hamiltonian's equations are just another dynamical ODE system:

dq/dt = f(p, q)

dp/dt = g(p, q)

...

Note: dV/dt / V = trace J = f_{q}
+ g_{p} + ...
= d/dq
[dH/dp]
+ d/dp
[dH/dq]
+ ... = 0

Phasespace volume is conserved
Properties of Hamiltonian
Systems

They have no dissipation (frictionless)

There are one or more conserved quantities (energy, ...)

They are described by a Hamiltonian function H

There are 2N dimensions for N degrees of freedom

Motion is on a 2N  k dimensional (hyper)surface

k + 1 Lyapunov exponents are equal to zero

There are no attractors (or attractor = basin)

Transients don't die away (no need to wait)

Equations are timereversible

Orbit returns arbitrarily close to the initial condition

Phasespace volume is conserved (Liouville's theorem)

The flow is incompressible (like water)

The Lyapunov exponents sum to zero

Chaos can occur only for N > 1 (at least 2 degrees
of freedom)

The dynamics occur in a space of integer dimension

This space may be a (fat)
fractal however (many holes)
2D Symplectic (AreaPreserving)
Maps

X_{n}_{+1} = f(X_{n}, Y_{n})

Y_{n}_{+1} = g(X_{n}, Y_{n})

A_{n}_{+1}/A_{n} = det J
= f_{x}g_{y}  f_{y}g_{x}
= 1

Example: Hénon map with B = 1

X_{n}_{+1} = 1  CX_{n}^{2}
+ Y_{n}

Y_{n}_{+1} = X_{n}

A_{n}_{+1}/A_{n} = 0  (1)(1) =
1

Computer demo (C = 0.3)

More general polynomial symplectic map:

X_{n}_{+1} = A + Y_{n} + F(X_{n})

Y_{n}_{+1} = B  X_{n}

One choice of F is C_{1} + C_{2}X
+ C_{3}X^{2} + ...

Verify that this has A_{n}_{+1}/A_{n}
= det J = 1

Slide show from Strange
Attractors book

Stochastic web maps:

These occur for charged particle in EM wave

X_{n}_{+1} = a_{1} + [X_{n}
+ a_{2}sin(a_{3}Y_{n} + a_{4})]cos
a
+ Y_{n}sin a

Y_{n}_{+1} = a_{5} + [X_{n}
+ a_{2}sin(a_{3}Y_{n} + a_{4})]sin
a
+ Y_{n}cos a

where a = 2p/N
(N is an integer)

Verify that this has A_{n}_{+1}/A_{n}
= det J = 1

l_{1} is positive but small

Exhibit minimal chaos or Arnol'd diffusion

Examples: case 1
(N = 9) / case 2 (N
= 5)
Simple Pendulum (2D
Conservative Flow)

dx/dt = v (v is really an angular
velocity)

dv/dt = sin x (x is really an
angle)

For x << 1, sin x > x and orbits
are circles around a center:

More generally equilibria are at v* = 0, x* = Np
(where N is an integer, N = 0, ±1, ±2,
±3, ...)

Phase space trajectories:

Opoints (centers) and Xpoints (saddle points)

Separatrix (homoclinic orbit) separates trapped (elliptic)
and passing (hyperbolic) orbits

Homoclinic orbits are sensitive to perturbations
Chirikov (Standard) Map

Start with the pendulum equations:

Solve by the leapfrog method:

v_{n}_{+1} = v_{n}  h_{1}sin
x_{n}

x_{n}_{+1} = x_{n} + h_{2}v_{n}_{+1}

Leap frog is symplectic if f_{x} = g_{v}
= 0

Let q = x/2p,
r = v/2p, h_{1}
= K, h_{2} = 1:

2pr_{n}_{+1} = 2pr_{n}
 K sin(2pq_{n})

2pq_{n+1} = 2pq_{n}
+ 2pr_{n}_{+1}

r_{n}_{+1} = [r_{n}  (K/2p)
sin(2pq_{n})] mod 1

q_{n+1} = [q_{n}
+ r_{n}_{+1}] mod 1

K is the nonlinearity parameter

This system also models ball bouncing on vibrating floor

Animation of Chirikov
map
J. C. Sprott  Physics 505
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