Chaotic dynamics Perturbation theoryQuasi integrable Hamiltonians Small divisor problemKAM theorem, Melnikov’s integralNumerical measurements of Lyapunov exponentsNekhoroshev TheormChirikov criterion for onset of chaos (resonance overlap) Control of chaos

Perturbation theory

• Can try to find new variables so that is in action angle variables to first then n-th order in ε

• Generating function

Unperturbed frequencies

Unperturbed Hamiltonian is in action angle variables

Perturbation theory

• As long as we choose our coordinates so that the second two terms cancel we can action angle variables to first order in ε

• So we chose f such that• Not too hard to do if we work with polynomial

expansions

Birkhoff-Gustavson normal form

• Consider polynomial expansion of Hamiltonian in p,q and powers

• Successively apply canonical transformations• Hamiltonian is in a Birkhoff normal form if it can

be written

• where H(p,q) and I=(1/2)(p2 + q2), Hk(I) is a polynomial of order k in p,q and Rk is a polynomial of order > k in p,q

The uselessness of the Birhkoff normal form

• As long as no resonances it is possible to put it in a Birkhoff normal form

• It may not be possible to find an infinite series that converges

• There may be small resonant free neighborhoods where the Birkhoff normal form is useful

Expansion in Fourier series

• Likewise we can expand the function we use to do our canonical transformation

• Our condition to put Hamiltonian in action angle variables to order εbecomes

• Solving • We cannot find coefficient dk if there is an integer vector k such that• small divisors problem

Restrictions

• The small divisor problem implies that it is not possible to find an expansion everywhere at the same time

• Neighborhoods exist where there are no resonances (Diophantine condition)

• In small regions then Birkhoff normal forms can be found• Going beyond first order: Can keep repeating procedure

until you run into a resonance condition. Might be a maximum order of expansion that is best representative of Hamiltonian. May never converge

Generating higher order harmonics

• consider perturbation with 4 resonances

• Attempt to find action angle coordinates away from resonances will introduce terms like cos(q1-2q2) or cos(2q1) that are second order in ε

Secular normal forms

• Consider perturbations from planets but away from resonance

• Expand in Fourier coefficients• Search for new coordinates that put the Hamiltonian in

action angle variables to first order• As long as not near any resonances new coordinates

exhibit weak perturbations about old ones• This procedure sometimes called “averaging over fast

angles”• Recent 100,000 term expansions! (Bretagnon, Laskar)

Away from resonance• Hamiltonian with perturbation

• Generating function• Coordinates

• changing coordinates

• Slightly oscillating coordinates to achieve an integrable Hamiltonian

• If perturbation also depends on I, then similar procedure works to first order in ε. Procedure can also be taken to second order

KAM theorem

• Locally non resonant (satisfies diophantine condition) in neighborhood p such that

• For some parameters γ,τ • non degenerate locally• Then there is some ε below which the Hamiltonian in this

neighborhood is integrable(There is a canonical transformation such that the Hamiltonian can be put in action angle variables)

Resonance Normal form

• Small angle

• Chose to expand about exact resonance• After expansion system will have a second order term

dependent on momenta and a cosine term depend on Φ1.

• Always looks like a pendulum

Action angle variables for pendulum

Pendulum stable and unstable pointsto first order near unstable hyperbolic fixed point

Stable and Unstable manifolds

Adding a perturbation• The separatrix which has trajectories

that take an infinite length of time still contains stable and unstable manifolds and they must intersect otherwise the map is not area preserving

• Difference in H can be computed with an integral of the perturbation along the separatrix known as the Poincare-Melnikov integral

• Expect amount of “chaos” related to size of this integral which tells you the size of separatrix splitting

Homoclinic means a trajectory joining a saddle point to itself (at positive and negative time). The above phenomena is sometimes called the homoclinic tangle.

Numerical measurements of “chaos”

Properties of chaotic orbits• Topology of trajectories: Area filling trajectories

in 2D (or plot surfaces of section for 3D systems)• Exponential divergence of nearby orbits

(measure maximum Lyapunov exponent)• Non discrete frequency spectrum (tori should

have a finite number of fixed frequencies)– do a frequency analysis

Maximum Lyapunov exponent

• Difficulties is that you can’t let trajectories get too far apart and you need to integrate for a while as the exponent is defined only after a long time

• Bennetin’s method: Compute ratio, s, after time T.

• Renormalize starting condition by this ratio. Take the limit

Mean Exponential Growth factor

• Define a new quantity, the mean exponential growth factor of nearby orbits – MEGNO

• On average Y(t) oscillates about Lt. If non chaotic then Y(t) oscillates around 2.

If orbits exponentially diverge then

If orbits separate linearly then

Take the time average of Y(t)Apparently is faster to compute than directly the Lyapunov exponent because weights later times more strongly Proposed by Cincotta & Simo (2000)

Frequency Analysis

• Look for frequency of peak of this function

• If there is a single dominant peak frequency then we expect not chaotic

• Laskar looked primarily at secular evolution however later work considered averaging over short period phenomena

Resonance OverlapChirikov Criterion for onset of global chaos

• Because H only depends on p2, dq2/dt =2π is constant

• First resonance at p1=0, centered at q1=0 has width

• Second resonance at q1=2πt at p1=2π, same width

Surface of section created by plotting a point every time q2=0

Chirikov criterion and diffusion• Chirikov found ε=2.47 insured

global chaos in overlap region. This is higher than a computed “golden” KAM ratio which gave ε~1.

• Heterolinic tangle. Heteroclinic orbit joins to fixed points in positive and negative time but not necessarily the same fixed points

• Criteria for global chaos seems not be as well defined mathematically, nevertheless resonance overlap is widely applied to predict location of chaos.

Chirikov diffusion, using resonant perturbations and fact that they are overlapped to predict rate of diffusion in action space

Nekhoroshev Theorem• How big a change over what period of time do perturbations

cause? We don’t really care if a system if chaotic if the perturbations are very small

• Points in G, nearby neighborhood Δ• Matrix C that does not project any vectors to zero (convexivity

hypothesis?)• There exists constants, α,β,a,b,ε* such that

||p(t)-p(0)||<αεa for all ε<ε* and all |t|<β(ε*/ε)1/2 exp[(ε*/ε)b]

• Divergence occurs in a timescale depending on 1/ε

Nekhoroshev Theorem• There exists constants, α,β,a,b,ε* such that

||p(t)-p(0)||<αεa for all ε<ε* and all |t|<β(ε*/ε)1/2 exp[(ε*/ε)b]

• ε sets width of resonances likely to reside in domain G• Higher order resonances arise if Hamiltonian expanded but their width

also depends on ε• Diffusion is faster if resonances overlap• Fraction of phase space covered by overlapping resonances also set by

ε• See discussion in Chap 6 of Morbidelli’s book• I would like to understand how estimate these constants and powers for

specific examples ….• -

Controlling chaosGoal is to keep a particle near an unstable fixed point• Consider a pendulum• We can look for fixed points finding them at p=0, Φ=0,π• The fixed point at 0 is unstable• Linearize around unstable fixed point

• Equations of motion

• Eigenvalues and eigenvectors stable

stable unstable

unstable

Modify the System

• Unstable fixed point now at p=λ• If we can vary λ we can vary the position of the fixed

point w.r.t. to a body moving in the system• Varying λ moves the fixed point up and down• OGY control: (Ott, Grebogi & Yorke)• Adjust λ so that point is nearer stable trajectories and

moving toward the fixed point rather than diverging away

stable

stable unstable

unstable

OGY Control

• x

s

u

Want to move trajectory in direction perpendicular to unstable eigenvector (fu)Want to move it a distance that is set by how far away from fixed point orbit is. Original BGY was linear and with an adjustment once per cycle

Control

• In the absence of noise the system can be brought very close to the fixed point where infinitely small perturbations will keep it there

• Alternative approaches to control theory and many examples of successful control

Chaos in the solar system

• Two approaches:– Secular drift (e.g., Laskar)– Resonance overlap useful model for predicting

Lyapunov timescales --- including role of 3 body resonances in outer solar system (Holman & Murray)

• Individual resonances can be strongly influenced by secular resonances (e.g., 3:1 and ν6)

Reading

• Morbidelli’s book Chap 2-6• Chap 24 on Controlling Chaos from Michael

Cross’s 2000 class http://crossgroup.caltech.edu/Chaos_Course/Outline.html

• Lecar, M. et al., 2001, ARA&A, 39, 581 Chaos in the Solar System

Problems:• Using the OGY method of control as a guideline, consider

adding a resonant perturbation to an existing chaotic system, for example an overlapped forced pendulum with global chaos in the overlap region– How big a resonance term is required for a stable resonant island to

appear in the overlap region?