Chaos in Hamiltonian systems

Teemu Laakso

April 26, 2013

Course material:Chapter 7 from Ott 1993/2002, Chaos in Dynamical Systems, Cambridgehttp://matriisi.ee.tut.fi/courses/MAT-35006

Useful reading:Goldstein 2002, Classical Mechanics, Addison WesleyHand & Finch 1998, Analytical Mechanics, Cambridge

Advanced:Lichtenberg & Lieberman 1992, Regular and Chaotic Dynamics, SpringerArnold 1989, Mathematical Methods of Classical Mechanics, Springer

Contents

1 Introduction: classical mechanics 2

2 Symplectic structure 4

3 Canonical changes of variables 6

4 Hamiltonian maps 7

5 Integrable systems 10

6 Perturbations and the KAM theorem 13

7 The fate of resonant tori 15

8 Transition to global chaos 17

1

1 Introduction: classical mechanics

Newtonian mechanics

Consider a particle of mass m, subject to a force field F , in a d-dimensionalEuclidean space (one-body system). Newton’s second law; mr = F =⇒ asystem of 2d first order ODEs for position r ∈ Rd and velocity r ∈ Rd. Theseare equations of motion in phase space (the space Rd × Rd where (r, r) arecoordinates). The number of degrees of freedom of the mechanical system isN = d.

Example (harmonic oscillator). Set d = 1, F = −kr =⇒ r = −kr/m[draw a picture].

dr

dt= r

dr

dt= − k

mr.

The solution for initial values r(0) = r0, r(0) = 0 is r(t) = r0 cos(t√k/m

).

[Characteristic equation z2 + k/m = 0 =⇒ z1,2 = ±i√k/m = ±iωh =⇒

r(t) = a cosωht+ b sinωht.]

Lagrangian mechanics

For an unconstrained system of n bodies: N = dn (2N ODEs). Under holo-nomic constraints fj(r1, r2, . . . , rn, t) = 0, j = 1, . . . , k we can define gener-alized coordinates qi, i = 1, . . . , N , where N = dn− k, using transformationequations

r1 = r1(q1, q2, . . . , qN , t)

... (1)

rn = rn(q1, q2, . . . , qN , t).

Definition. The Lagrangian (function) is L = T−V , where T =∑n

i=1mi(ri·ri)/2 is the kinetic energy, and V = V (r1, r2, . . . , rn, t) is the potential energy.

2

Through the transformation equations (1), we have L = L(q, q, t). TheHamilton’s principle

δI = 0, I =

∫ t2

t1

L(q, q, t)dt

is a variational equation for finding a path q(t) from t1 to t2 for which the lineintegral I (action) is stationary [draw a picture]. Solution [e.g., Goldstein]yields the (Euler-)Lagrange equations of motion:

d

dt

∂L

∂qi− ∂L

∂qi= 0, i = 1, . . . , N.

[For all possible paths, the system takes the one requiring the least action.]

Example (harmonic oscillator). L = T − V = mr2/2− kr2/2,

d

dt

∂L

∂r= mr,

∂L

∂r= −kr =⇒ r = − k

mr.

Hamiltonian mechanics

Definition. The conjugate momenta pi = ∂L/∂qi and Hamiltonian H =qipi − L (Einstein summation convention).

We do a Legendre transformation;

dL =∂L

∂qidqi +

∂L

∂qidqi +

∂L

∂tdt = pidqi + pidqi +

∂L

∂tdt

dH = qidpi − pidqi −∂L

∂tdt

Since we wish H = H(q, p, t),

dH =∂H

∂qidqi +

∂H

∂pidpi +

∂H

∂tdt.

Equating terms, we have the Hamiltons equations of motion:

pi = −∂H∂qi

qi =∂H

∂pi.

(2)

The (q, p) ∈ RN × RN are canonical phase-space variables.

3

Example (harmonic oscillator). p = ∂L/∂r = mr, H = mr2−L = mr2/2+kr2/2 = T + V = E.

In general, if the transformation equations (1) do not depend on t explic-itly, and the forces are conservative (of the form F = −∇Φ), Hamiltonian isthe total energy.

For a time-independent Hamiltonian H(p, q) the total energy is conserved;

dH

dt=∂H

∂qq +

∂H

∂pp =

∂H

∂q

∂H

∂p− ∂H

∂p

∂H

∂q= 0.

=⇒ H(p, q) = E is a constant of motion.

2 Symplectic structure

The class of Hamiltonian systems is very special. Let x = (p, q)T and

f(x, t) = Ω

(∂H

∂x

)T, where Ω =

[0 −II 0

].

Now, Hamilton’s equations are x = f(x, t). Note that f(x, t) ∈ R2N (theHamiltonian vector field) is fully determined by H(x, t) ∈ R.

Theorem (Liouville’s theorem). Hamilton’s equations preserve 2N-dimensionalvolumes.

Proof.∂

∂x· f =

∂

∂p·(−∂H∂q

)+

∂

∂q·(∂H

∂p

)= 0,

d

dt

∫V

d2Nx =

∮S

dx

dt· dS =

∮S

f · dS =

∫V

(∂

∂x· f)

d2Nx = 0.

[differentiation under integral sign, Hamilton’s equations, divergence (Gauss’)theorem].

Corollary. There are no attractors in Hamiltonian systems.

4

Hamilton’s equations are symplectic. Consider three orbits (p, q), (p +δp, q+ δq), and (p+ δp′, q+ δq′). The differential symplectic area is the sumof parallelogram areas: [Ott, fig. 7.1]

N∑i=1

∣∣∣∣δpi δqiδp′i δq′i

∣∣∣∣ =N∑i=1

(δpiδq′i − δqiδp′i) = δp · δq′ − δq · δp′ = δxTΩδx′.

d

dt

(δxTΩδx′

)=

dδxT

dtΩδx′ + δxTΩ

dδx′

dt

=

(∂f

∂xδx

)TΩδx′ + δxTΩ

∂f

∂xδx′

= δxT

[(∂f

∂x

)TΩ + Ω

∂f

∂x

]δx′

= δxT

[(Ω∂2H

∂x2

)TΩ + ΩΩ

∂2H

∂x2

]δx′

= δxT

[(∂2H

∂x2

)TΩTΩ + ΩΩ

∂2H

∂x2

]δx′

= 0.

[linearization of f(x + δx, t), ΩΩ = −I, ΩT = −Ω, ∂2H/∂x2 symmetric].Symplectic condition =⇒ conservation of volume.

Definition. The differential symplectic area is a differential form of Poincare’sintegral invariant ∮

γ

p · dq =N∑i=1

∮γ

pidqi,

where γ is a closed path in (p, q) at constant t.

A generalization in extended 2N + 1 phase space (p, q, t) [Ott, fig. 7.2]:

Theorem (Poincare-Cartan integral theorem).∮Γ1

(p · dq −Hdt) =

∮Γ2

(p · dq −Hdt),

where Γ1 and Γ2 are paths around the ‘tube’ of trajectories.

5

Γ1 and Γ2 at constant times (dt = 0) =⇒ Poincare’s integral invariant.If H = H(p, q) and Γ1 and Γ2 are on this surface, we have

∮Hdt = 0 and∮

Γ1

p · dq =

∮Γ2

p · dq,

where Γ1 and Γ2 does not need to be at constant t.

3 Canonical changes of variables

Let us define a new set of phase-space variables X = (P,Q)T by using atransformation g : R2N → R2N , x 7→ X.

Definition. The transformation g is canonical, if it preserves the differentialsymplectic area;

δp · δq′ − δq · δp′ = δP · δQ′ − δQ · δP ′. (3)

Canonical transformations are typically performed by using a generatingfunction, e.g., S = S(P, q, t);

Q =∂S

∂P,

p =∂S

∂q,

which gives (P,Q) implicitly. The transformed Hamiltonian is K(P,Q, t) =H(p, q, t) +∂S/∂t. [Can be derived using the Hamilton’s principle; see Gold-stein.] In order to check the canonicity of the above one can substitute

δQ =∂2S

∂P 2δP +

∂2S

∂P∂qδq,

δp =∂2S

∂P∂qδP +

∂2S

∂q2δq,

into the condition (3) [homework].The new variables satisfy Hamilton’s equations: dX/dt = Ω(∂K/∂X)T .

Hence, the underlying symplectic structure of a given Hamiltonian systemis invariant, and can be represented using any suitable choice of canonicalphase-space variables.

6

4 Hamiltonian maps

Consider the map Mh : R2N → R2N :

Mh(x(t), t) = x(t+ h).

Its differential variation [linearize Mh(x+ δx, t)] is

∂Mh

∂xδx(t) = δx(t+ h).

Symplectic condition =⇒

δxT (t)Ωδx′(t) = δxT (t+ h)Ωδx′(t+ h)

=

(∂Mh

∂xδx(t)

)TΩ

(∂Mh

∂xδx′(t)

)= δxT (t)

(∂Mh

∂x

)TΩ

(∂Mh

∂x

)δx′(t)

=⇒ Ω =

(∂Mh

∂x

)TΩ

(∂Mh

∂x

)We say, ∂Mh/∂x is symplectic. Generally, matrix A is symplectic, if Ω =ATΩA. The product of symplectic matrices A and B is symplectic;

(AB)TΩ(AB) = BT (ATΩA)B = BTΩB = Ω.

Theorem (Poincare recurrence theorem). Examine a Hamiltonian H(p, q)with orbits bounded in a finite subset D ⊂ R2N . Let a ball R0 ∈ D haveradius ε > 0. In time, some of the orbits leaving R0 will return to it.

Proof. Evolve R0 with the volume-preserving Mh, obtaining subsequent re-gions R1, R2, . . .. Conclude that ∃Rr ∩ Rs 6= ∅ =⇒ Rr−s ∩ R0 6= ∅, r > s[Arnold, fig. 51].

Poincare maps

In general, a Poincare map gives an intersection of an orbit with a lower-dimensional subspace of the phase space, called Poincare section (or surfaceof section, SOS). For periodic orbits, plotting subsequent intersections typ-ically draws a figure which reflects deeper characteristics of the dynamicalsystem.

7

Definition. Poincare section for a Hamiltonian system H(p, q, t) where His τ -periodic in time:

• extended (2N + 1) -dimensional phase space (p, q, ξ), where ξ = t,

• additional equation dξ/dt = 1,

• let ξ′ = ξ mod τ ,

• set ξ′ = t0 ∈ [0, τ) as the 2N -dimensional SOS.

Because of the periodicity, Mτ (x, t0) = Mτ (x, t0 + kτ), k ∈ Z. Thesurface of section map xn+1 = M(xn) =Mτ (xn, t0) is symplectic.

Example (Standard map). The ‘kicked rotor’: bar of moment of inertia Iand length l, frictionless pivot [Ott, fig. 7.3]. Vertical impulse of strengthK/l at times t = 0, τ, 2τ, . . . Canonical variables (pθ, θ), Hamiltonian

H(pθ, θ, t) =p2θ

2I+K cos θ

∑n

δ(t− nτ),

where δ denotes the Dirac delta, and equations of motion (2)

dpθdt

= K sin θ∑

δ(t− nτ),

dθ

dt=pθI.

SOS: (pθ, θ) after each kick. Integration over t ∈ (nτ, (n+ 1)τ ] yields pn+1 −pn = K sin θn+1 and θn+1 − θn = pnτ/I. Setting τ/I = 1, we have

pn+1 = pn +K sin θn+1,

θn+1 = (θn + pn) mod 2π.

The mapping (N = 1) preserves differential areas, if

det(δxn, δx′n) = det [(∂xn+1/∂xn)(δxn, δx

′n)] ,

where xn = (pn, θn)T . Symplecticity can hence be verified by calculating

det

[∂pn+1/∂pn ∂pn+1/∂θn∂θn+1/∂pn ∂θn+1/∂θn

]= det

[1 +K cos θn+1 K cos θn+1

1 1

]= 1.

8

Definition. Poincare section for a Hamiltonian system H(p, q):

• motion in a (2N − 1)-dimensional energy surface H(p, q) = E,

• by choosing, e.g., q1 = 0, we determine a (2N − 2)-dimensional SOS.

For a unique SOS mapping xn+1 = M(xn), we often need an additionalconstraint, e.g., p1 > 0.

Example (Logarithmic potential). The potential Φ near the centre of a flat-tened galaxy can be modelled as

Φ(x, y) =1

2ln

(x2 +

y2

a2+ b2

), (4)

where a, b ∈ R are parameters [draw a picture]. For a star moving in thispotential, the Hamiltonian per unit mass is

H(px, py, x, y) =1

2

[p2x + p2

y + ln

(x2 +

y2

a2+ b2

)].

Equations of motion (2) become

px = − x

x2 + y2/a2 + b2,

py = − y

a2(x2 + y2/a2 + b2),

x = px,

y = py.

Choose y = 0 as the SOS. Now

H(px, py, x, 0) = E =⇒ p2y = 2E − p2

x − ln(x2 + b2

).

Hence, there are two points ±py corresponding to y = 0. We choose thatpy > 0 on the SOS. In this case, we cannot solve the equations of motion an-alytically, and points (px, x) on the SOS have to be determined by numericalintegration.

9

5 Integrable systems

Consider a Hamiltonian H(p, q). A constant of motion is a quantity thatdoes not change when the system evolves in time, e.g., H(p, q) = E. Ingeneral, for a function f(p(t), q(t)), we have

df

dt=∂f

∂pp+

∂f

∂qq =

∂f

∂q

∂H

∂p− ∂f

∂p

∂H

∂q.

If f(p, q) is a constant of motion, we can write [f,H] = 0, where the Poissonbracket for functions f and g is defined as

[f, g] :=∂f

∂q

∂g

∂p− ∂f

∂p

∂g

∂q.

Poisson bracket is anticommutative: [f, g] = −[g, f ].

Theorem (Liouville’s theorem on integrable systems). The Hamiltonian sys-tem H(p, q) is integrable, if it has N independent constants of motion fi(p, q),i = 1, . . . , N which are in involution, i.e., [fi, fj] = 0, ∀i, j.

[Proof: see, e.g., Arnold.] The independent constants of motion restrictan integrable system on an N -dimensional surface in the phase space. Be-cause the constants are in involution, any bounded orbit on this surface istopologically equivalent to an N -dimensional torus [Ott, fig. 7.4a].

Action-angle variables

The action-angle variables provide an explicit transformation between points(p, q) and points on the N -torus. Through a canonical change of variables,it is possible to transform to coordinates (P,Q) in which the new momentaPi(p, q), i = 1, . . . , N are constants of motion. In such a case dPi/dt =−∂H/∂Qi = 0 =⇒ H = H(P ).

A particularly convenient choice is

Pi = Ji :=1

2π

∮γi

p · dq,

where the paths γi wrap around each possible angle direction i = 1, . . . , Non the N -torus [Ott, fig. 7.4b]. The new variables Ji are actions. They areconstants of motion by the Poincare-Cartan integral theorem.

10

The corresponding conjugate coordinates Qi = θi are angles. Supposethat action-angle variables (J, θ) are obtained by the generating functionS(J, q);

θ =∂S

∂J,

p =∂S

∂q

By integrating the latter equation around γi, we examine the change in S;

∆iS =

∮γi

p · dq = 2πJi.

Then,

∆iθ =∂

∂J∆iS = 2π

∂

∂JJi,

or∆iθj = 2πδij,

where δij = 1, if i = j, and zero otherwise. Therefore, after one circuitaround γi, the angle θi increases by 2π, and the other angles return to theiroriginal values.

The new Hamiltonian is H = H(J), and

dJ

dt= 0,

dθ

dt=∂H

∂J=: ω(J).

A solution from initial values (J0, θ0) is J(t) = J0 and θ(t) = θ0 + ωt. Theconstants of motion ω are frequencies which can be interpreted as angularvelocities on the torus.

Definition. An orbit on the torus is quasiperiodic if there is no vector ofintegers m ∈ Zn such that

m · ω = 0.

A quasiperiodic orbit fills the surface of the torus as t→∞. On the otherhand, if ωi/ωj ∈ Q, ∀i, j, the orbit is periodic, and closes on itself. The set oftori with periodic orbits is dense in the phase space, but has a zero Lebesguemeasure. Hence, orbits on a randomly picked torus are quasiperiodic withprobability of one.

11

Example (harmonic oscillator). In general, for N = 1, we have

H(p, q) =p2

2m+ V (q),

where p, q ∈ R, and V (q) is a potential energy of a general form [Ott, fig.7.5]. We have

J =1

π

∫ q2

q1

2m[E − V (q)]1/2dq.

For the harmonic oscillator, V (q) = mω2hq

2/2, where ωh =√k/m is the

angular velocity of the oscillation, and k is the spring constant. From V (q) =E we have q2 = −q1 = [2E/(mω2

h)]1/2, and integration 1 yields J = E/ωh.

Thus,H(J) = ωhJ,

and ω(J) = ∂H/∂J = ωh, and is (in this rare case) independent of J . Theangle θ(t) = θ0 + ωht. In order to use the generating function S(J, q), wewrite H(p, q) = H(J), and substitute p = ∂S/∂q, obtaining

∂S

∂q=√

2m(ωhJ −mω2hq

2/2)

Solving for θ, and integrating 2, gives

θ =∂S

∂J= mωh

∫ [2m(ωhJ −mω2

hq2/2)

]−1/2dq

=

∫ (2J

mωh− q2

)−1/2

dq

= arcsin

(q

√mωh2J

),

and we have (p, q) as a function of (J, θ);

q =

√2J

mωhsin θ,

p =√

2Jmωh cos θ.

1∫(a2 − x2)1/2dx = x

2 (a2 − x2)1/2 + a2

2 arcsin xa

2∫

dx(a2−x2)1/2

= arcsin xa

12

The trajectory (p(θ), q(θ)) is an ellipse [Ott, fig. 7.6]. In general, the mapping(p, q) 7→ (J, θ) for N = 1 transforms a closed curve into a one-dimensionaltorus (a circle).

For N > 1, a general procedure for finding the action-angle variables(J, θ) involves solving the Hamilton-Jacobi equation

H

(∂S

∂q, q

)= E.

If we are lucky, the equation is solvable by separation of variables, and wehave N separation constants which are also constants of motion. The motionis typically quasiperiodic.

6 Perturbations and the KAM theorem

The phase space of integrable Hamiltonian systems is foliated by tori. Theseappear as nested closed curves on the SOS. The existence of N integrals ofmotion allows us to access the tori via action-angle variables.

We know how to solve the Hamilton-Jacobi equation (N > 1) for only ahandful of systems, e.g., harmonic oscillator, isochrone, and general Stackelpotentials. On the other hand, the logarithmic potential, for example, numer-ically demonstrates the existence of an additional constant of motion (besidesE) almost everywhere on the SOS [computer demo, logarithmic potential].

Motion on a torus is regular, and chaotic orbits are possible only in regionsof phase space where invariant tori do not exist.

In order to see how the integrable tori change under perturbation, westudy

H(p, q) = H0(p, q) + εH1(p, q),

where H0 is an integrable Hamiltonian, H1 is non-integrable, and ε ∈ R issmall.

We know that under small perturbations some Hamiltonian systems stayclose to integrable (Solar System), but on the other hand, some are globallychaotic (statistical mechanics).

Kolmogorov (1954), Arnold (1963), and Moser (1973) proved a result thatfor sufficiently small ε, most of the tori survive the perturbation (the KAMtheorem).

13

The actual proof is difficult [see, e.g., Arnold]. Some considerations follow.In action-angle variables of H0;

H(J, θ) = H0(J) + εH1(J, θ).

If there are tori in H, we have a new set of action-angle variables (J ′, θ′) forwhich

H(J, θ) = H ′(J ′),

and which are obtained by a generating function S(J ′, θ) such that

J =∂S

∂θ,

θ′ =∂S

∂J ′.

The corresponding Hamilton-Jacobi equation (HJE) is

H

(∂S

∂θ, θ

)= H ′(J ′).

We look for a solution in the form of a power series:

S = S0 + εS1 + ε2S2 + . . . .

We set S0 = J ′ · θ, because ε = 0 then corresponds to J = J ′, θ′ = θ.Substitution to HJE gives

H0

(J ′ + ε

∂S1

∂θ+ ε2∂S2

∂θ+ . . .

)+ εH1

(J ′ + ε

∂S1

∂θ+ . . . , θ

)= H ′(J ′).

By differentiating at J ′ with respect to the first order ε-terms we have

H0(J ′) + ε∂H0

∂J ′· ∂S1

∂θ+ εH1(J ′, θ) = H ′(J ′). (5)

Next, we write θ-dependent terms as Fourier series;

H1(J ′, θ) =∑m

H1,m(J ′) exp(im · θ),

S1(J ′, θ) =∑m

S1,m(J ′) exp(im · θ),

14

where H1, S1 are real-valued, and the coefficients H1,m, S1,m ∈ C, m ∈ ZN .By substituting into (5), and requiring that H ′ is a function of J ′ only, wehave

S1 = i∑m

H1,m(J ′)

m · ω0(J ′)exp(im · θ),

where ω0(J) = ∂H0(J)/∂J are the unperturbed frequencies. Similar termsS2, S3, . . . follow, if this technique is applied to higher orders of ε. Clearly, theresonant tori for which m · ω0 = 0 play a special role (‘the problem of smalldenominators’). The KAM theorem essentially proves that the series for Sconverges for ‘very nonresonant’ tori with a frequency vector ω satisfying

|m · ω| > K(ω)|m|−(N+1), ∀m ∈ ZN\0,

where K(ω) > 0, and |m| = |m1|+ |m2|+ . . .+ |mN |. The Lebesgue measureof the complement of this set, around resonant tori, goes to zero, as ε → 0.However, since the unperturbed resonant tori are dense in the phase space,whenever ε > 0, arbitrarily near each surviving torus there is a region ofdestroyed resonant tori which can host irregular motion.

7 The fate of resonant tori

Twist map (rn+1, φn+1) = M0(rn, φn) is a model for the SOS map of anintegrable Hamiltonian H0(J) with N = 2 [Ott, fig. 7.7];

rn+1 = rn,

φn+1 = [φn + 2πR(rn)] mod 2π,

where R(r) = ω1/ω2 is the ratio of frequencies for the torus r, and the SOS isdrawn at θ2 = constant. On a resonant torus at r = r, R(r) = j/k, j, k ∈ Zsuch that kω1− jω2 = 0, and k applications of the map return to the originalpoint;

Mk0 (r, φ) = (r, (φ+ 2πj) mod 2π) = (r, φ).

Hence, each point on r = r is a fixed point of Mk0 . We assume that R is

smooth and increasing. In such a case there are circles r− < r < r+ in sucha way such that under Mk

0 , the points on r− and r+ travel clockwise andcounterclockwise, respectively [Ott, fig. 7.8a].

15

When H0 is perturbed by the term εH1, we have a slightly changed mapMε;

rn+1 = rn + εg(rn, φn),

φn+1 = [φn + 2πR(rn) + εh(rn, φn)] mod 2π.

If ε is small enough, we still have the (distorted) circles r− and r+ where thepoints travel to opposite directions under Mk

ε . Thus, there must be a curverε(φ) in between where Mk

ε moves points only in radial direction [Ott, fig.7.8b].

When Mkε is applied to the points on rε(φ), we obtain another curve r′ε(φ)

[Ott, fig. 7.9]. Since Mkε is volume-preserving, the two curves intersect at a

finite and even number of points which are fixed points of Mkε . Since we

know the directions where the points travel under Mkε with respect to the

curves, we can identify half of the fixed points as elliptic and the other halfas hyperbolic. [Ott, fig. 7.10] This is the Poincare-Birkhoff theorem.

The number of elliptic (and hyperbolic) points is equal to k. An ellipticorbit on the SOS circles around the elliptic points. [computer demo, log-arithmic potential] The neighbourhood of an elliptic point can be furthermodelled by another perturbed twist mapping, resulting in a new set of el-liptic and hyperbolic points in a smaller scale. If this process is continued adinfinitum, a fractal-like structure of the phase space is revealed, with subse-quently smaller tori winding on bigger ones [Lichtenberg & Lieberman, fig.3.5].

A hyperbolic point lies at an intersection of an unstable and stable mani-fold, Wu and Ws, respectively. Consider a repeated SOS mapping (Mk

ε )n. Apoint on Ws approaches the hyperbolic point, as n→∞. The same happensto a point on Wu as n→ −∞.

The two manifolds Wu and Ws connect to another hyperbolic point in away that resembles the phase portrait of a simple pendulum. [Lichtenberg& Lieberman, fig. 3.3a] For the pendulum, Wu and Ws between the pointscoincide, forming a smooth separatrix. However, in the perturbed system,the Wu and Ws are generally different, and intersect at a homoclinic point.This point, mapped by (Mk

ε )n must also lie on both Wu and Ws. Hence,we conclude that there must be an infinite number of homoclinic points asn→ ±∞.

Since Mkε is area-preserving, Wu (Ws) must oscillate wildly as n → ∞

(n → −∞). Near a hyperbolic point, these oscillations overlap, and form a

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region where points from Wu and Ws are mixed (homoclinic tangle). [Licht-enberg & Lieberman, fig. 3.4b] Orbits in this region show extreme sensitivityto initial conditions, and hence, are chaotic [computer demo, logarithmicpotential].

8 Transition to global chaos

In two degrees of freedom, the chain of resonant islands and chaotic regionssurrounding them are confined between surviving KAM-tori. As the per-turbation strength ε increases, chaotic regions grow, and more KAM-tori aredestroyed. When all KAM-tori are destroyed, an can wander anywhere in theenergy surface, and the motion becomes ergodic [computer demo, standardmap].

For the standard map, the rotation number of the last surviving KAM-torus is the golden ratio which is the farthest away from any rational numberin the sense that its representation as a continued fraction is

1 +√

5

2=

1

1 +1

1 +1

1 + · · ·

.

For N = 2 the 2-tori can enclose volumes in the three dimensional energysurface H(p, q) = E. For N > 2 the situation is qualitatively different;e.g., for N = 3 the 3-tori cannot enclose 5-dimensional volumes much likelines cannot enclose 3-dimensional volumes. For any ε > 0, stochastic regionsform a web where orbits can, in principle, reach any part of the energy surfacethrough Arnold diffusion. However, Nekhoroshev theorem tells us that theArnold diffusion happens very slowly; at time scales exp(1/ε) or even slower.

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