Hamiltonian Chaos and the standard map
What happens for small perturbation? Questions
of long time stability?
Outline:
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 tori
Homoclinic tangle or why chaos begins at hyperbolic fixed points.
Rational approximants and the KAM theorem.
Poincare section and twist maps
Area preserving maps
describe Hamiltonian system
intersecting with Poincare section
Area preserving property of twist maps
Poincare-Cartan invarient
Can be derived most elegantly from
notions of sympleptic geometry.
Conserving system
Poincare section
This can be generalised to the case of several variables and shows that
the Poincare sections generate a “sympleptic mapping”.
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 action
variables are constants of
motion; all tori are unperturbed.
e = 0.00
Phase portraits of the standard map
No (apparent) chaotic behaviour
but appearance of elliptic and
hyperbolic fixed points.
e = 0.05
Hyperbolic fixed point
Elliptic fixed point.
Phase portraits of the standard map
Many fixed points of the mapping
appear in elliptic/hyperbolic pairs.
e = 0.47
Phase portraits of the standard map
At a critical value of the non-linearity
parameter (or “kicking parameter”),
orbits can traverse in all directions;
this is the point at which the last
tori is destroyed.
Chaotic layer clearly seen at seperatrix.
e = 0.97
Phase portraits of the standard map
Chaotic “sea” now dominates
phase space; the system is
globally chaotic. Momentum is
unbounded by any tori.
e = 2.00
Analysis of fixed point types for
two dimensional maps
Tangent map at
fixed point; can prove
there is an invertable
diffeomorphism
connecting tangent map
with local phase space.
Elliptic fixed point
Hyperbolic fixed point
Parabolic fixed point
T
Analysis of the elliptic fixed point
Since these vectors are
linearly independent they
completely describe the motion
near the fixed point.
Analysis of the hyperbolic fixed point
Hyperbolic
Hyperbolic with
reflection
Fixed points of the standard map
Hyperbolic fixed point
Elliptic fixed point
Eigenvectors of hyperbolic point real; elliptic point complex.
Fixed points of the standard map
Note the change in the orientation
of the “elliptic island” as the perturbation
increases.
Poincare-Birkhoff theorem
Standard map
Perturbed map
Poincare-Birkhoff theorem
Standard map
Perturbed map
Elliptic
Hyperbolic
Rational tori in the standard map
Destruction of rational tori can clearly be seen in the standard map. Note that as
the theorem predicts hyperbolic and elliptic fixed points come in pairs.
2
3
4
5
1
Motion near a hyperbolic fixed point: The homoclinic tangle
Unstable manifold
Intuition suggests this is not the generic case!
– a simple “homoclinic connection” between
the stable and unstable branches of the
hyperbolic fixed point.
May also have
“heteroclinic connection”
Stable manifold
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?
Motion near a hyperbolic fixed point: The homoclinic tangle
Unstable manifold
Intuition suggests this is not the generic case!
– a simple “homoclinic connection” between
the stable and unstable branches of the
hyperbolic fixed point.
Stable manifold
Only possibility is a loop!
Motion near a hyperbolic fixed point: The homoclinic tangle
Unstable manifold
Intuition suggests this is not the generic case!
– a simple “homoclinic connection” between
the stable and unstable branches of the
hyperbolic fixed point.
Stable manifold
Which must then be repeated infinitely
many times!.
Motion near a hyperbolic fixed point: The homoclinic tangle
Area preservation implies
that oscillations increase in
amplitude near fixed point.
Branches from different hyperbolic fixed points
can also cross – “heteroclinic tangle”
Chaotic behaviour has its origin near the hyperbolic
fixed points; as seen in the standard map.
Motion near a hyperbolic fixed point: The homoclinic tangle
Iterate a line section; used 40,000 points online indicated by red line in graphs
for hyperbolic fixed point 23 iterations, for elliptic fixed point 811 and 2000 iterations
Hyperbolic fixed point
Initial line section just visible
here
Elliptic fixed point
Motion near a hyperbolic fixed point: The homoclinic tangle
Tori generated by action with simple rational
frequency are destoryed first by perturbation.
However in general this phase space structure
will 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 gaps
in the distribution of matter.
KAM tori: or why some tori last longer than others
Cantorous
3/2
5/8
Tori which survives the onset of chaos
in phase space the longest has action
given by the “golden mean”.
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 last
by chaos.
The shortest KAM theorem slide ever
Sufficiently irrational tori (i.e., winding number is irrational) are perserved
for small enough perturbations.
Preserved tori sattisfy:
Summary of classical chaos
Topologically transitive (mixing)
Initial trajectories diverge rapidily, but the attractor of motion is fractal and has
dimension greater than that of non-chaotic attractor
Dynamics 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?