Hamiltonian Chaos and the standard map

Poincare section and twist maps.Area preserving mappings.Standard map as time sections of kicked oscillator (link with quantum chaos).Phase portraits of the standard map. Fixed points in two dimensions. Fixed points of standard map.Poincare-Birkhoff theorem and destruction of rational toriHomoclinic tangle or why chaos begins at hyperbolic fixed points.Rational approximants and the KAM theorem.

Outline:

What happens for small perturbation? Questionsof long time stability?

Poincare section and twist maps

Area preserving mapsdescribe Hamiltonian systemintersecting with Poincare section

Area preserving property of twist maps

Poincare-Cartan invarient

This can be generalised to the case of several variables and shows thatthe Poincare sections generate a “sympleptic mapping”.

Conserving system

Poincare section

Can be derived most elegantly fromnotions of sympleptic geometry.

Discrete Hamiltonian approach to twist maps

Map is area perserving

The standard map

“Kicked oscillator”

Phase portraits of the standard map

With no “kicking” the actionvariables are constants ofmotion; all tori are unperturbed.

e = 0.00

Phase portraits of the standard map

No (apparent) chaotic behaviourbut appearance of elliptic and hyperbolic fixed points.

e = 0.05

Elliptic fixed point.

Hyperbolic fixed point

Phase portraits of the standard map

Many fixed points of the mappingappear in elliptic/hyperbolic pairs.

e = 0.47

Phase portraits of the standard map

At a critical value of the non-linearityparameter (or “kicking parameter”), orbits can traverse in all directions;this is the point at which the lasttori is destroyed.

Chaotic layer clearly seen at seperatrix.

e = 0.97

Phase portraits of the standard map

Chaotic “sea” now dominatesphase space; the system isglobally chaotic. Momentum isunbounded by any tori.

e = 2.00

Analysis of fixed point types for two dimensional maps

Tangent map atfixed point; can provethere is an invertablediffeomorphismconnecting tangent mapwith local phase space.

T

Elliptic fixed point

Hyperbolic fixed point

Parabolic fixed point

Analysis of the elliptic fixed point

Since these vectors arelinearly independent theycompletely describe the motionnear the fixed point.

Analysis of the hyperbolic fixed point

Hyperbolic

Hyperbolic withreflection

Hyperbolic fixed point

Elliptic fixed point

Fixed points of the standard map

Eigenvectors of hyperbolic point real; elliptic point complex.

Note the change in the orientationof the “elliptic island” as the perturbationincreases.

Fixed points of the standard map

Poincare-Birkhoff theorem

Standard map

Perturbed map

Poincare-Birkhoff theorem

Perturbed map

Elliptic

Hyperbolic

Standard map

Rational tori in the standard map

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Destruction of rational tori can clearly be seen in the standard map. Note that asthe theorem predicts hyperbolic and elliptic fixed points come in pairs.

Motion near a hyperbolic fixed point: The homoclinic tangle

Intuition suggests this is not the generic case! – a simple “homoclinic connection” between the stable and unstable branches of the hyperbolic fixed point.

Since the map has inversion stable and unstable branches cannot cross: could construct a many-one map if this was the case.

BUT: which point does X map to?

Unstable manifold

Stable manifold

May also have“heteroclinic connection”

Motion near a hyperbolic fixed point: The homoclinic tangle

Intuition suggests this is not the generic case! – a simple “homoclinic connection” between the stable and unstable branches of the hyperbolic fixed point.

Only possibility is a loop!

Unstable manifold

Stable manifold

Motion near a hyperbolic fixed point: The homoclinic tangle

Intuition suggests this is not the generic case! – a simple “homoclinic connection” between the stable and unstable branches of the hyperbolic fixed point.

Which must then be repeated infinitelymany times!.

Unstable manifold

Stable manifold

Branches from different hyperbolic fixed pointscan also cross – “heteroclinic tangle”

Area preservation impliesthat oscillations increase inamplitude near fixed point.

Chaotic behaviour has its origin near the hyperbolicfixed points; as seen in the standard map.

Motion near a hyperbolic fixed point: The homoclinic tangle

Motion near a hyperbolic fixed point: The homoclinic tangle

Iterate a line section; used 40,000 points online indicated by red line in graphsfor hyperbolic fixed point 23 iterations, for elliptic fixed point 811 and 2000 iterations

Hyperbolic fixed point Elliptic fixed point

Initial line section just visible here

Tori generated by action with simple rationalfrequency are destoryed first by perturbation.

However in general this phase space structurewill be self-similar!

Motion near a hyperbolic fixed point: The homoclinic tangle

Chaos in rational tori can be seen in the solar system in asteroid belt, and also rings of Saturn: leads to gapsin the distribution of matter.

Motion near a hyperbolic fixed point: The homoclinic tangle

KAM tori: or why some tori last longer than others

5/8

3/2

Tori which survives the onset of chaosin phase space the longest has actiongiven by the “golden mean”.

Cantorous

Continued fractions and rational approximants

(Known to Zu Congzhi in 5th centuary China)

Continued fractions and rational approximants

Golden mean is the “most irrational number”; torus destroyed lastby chaos.

The shortest KAM theorem slide ever

Preserved tori sattisfy:

Sufficiently irrational tori (i.e., winding number is irrational) are perservedfor small enough perturbations.

Summary of classical chaos

Topologically transitive (mixing)Initial trajectories diverge rapidily, but the attractor of motion is fractal and hasdimension greater than that of non-chaotic attractorDynamics are unpredicatble.

Universality in the standard map and different routes to chaos: period doubling route to chaos, intermittency, crisis.

Fine structure of phase space near hyperbolic fixed points plays a crucial role in chaos.

So what happens in quantum systems?