Periodic orbits

t ln(|Λ_i(x(0), t)|) (1.1.6) so that |Λi(x(0), t)| ∼ e^λi(x(0))t. As we will see in section 1.1.3, chaotic systems also have the property of ergodicity, which means that the Lyapunov exponents will coincide with global values λi for almost all trajectories. The largest exponent λ dominates the local stretching for long times (for almost all trajectories).

1.1.2 Periodic orbits

A periodic orbit γ is a trajectory in the energy shell that repeats its motion after some timeT, so that

x(t) =x(t+T) (1.1.7)

holds for all times t. The minimum (positive) time T_γ for which this holds, is the period of the orbit. The orbit follows a closed loop in the energy shell returning to the same point after each period, and this provides us with a natural time scale to study motion near the periodic orbit. We take a point along the orbit and place a Poincar´e section orthogonal to the trajectory there. The system evolves in such a way that a local neighbourhood of the orbit is transported around the whole periodic orbit until it returns to the Poincar´e section. Because it passes along every point of the periodic orbit, the eigenvalues of the stability matrix of this transport, Mγ, do not depend on the starting point, and they are an invariant property of the orbit.

They provide the (f −1) pairs |Λγ,i|,|Λγ,i|⁻¹ of stretching and contracting factors of the orbit. Hence we can find the Lyapunov exponents λγ,i of the periodic orbit following|Λγ,i|= e^λγ,iTγ.

For systems with only 2 degrees of freedom, in the linearized approximation, the eigenvectors of Mγ define a pair of unstable and stable directions. After being transported around the periodic orbit once, a point with coordinates (u, s) in those directions, hits the Poincar´e section again at the point (u⁰, s⁰). These two points can be related to each other through the stability matrix as

u⁰ = Λγu, s⁰ = Λ⁻¹_γ s (1.1.8)

and it is clear that the points follow the hyperbolic motion associated with chaos.

Due to hyperbolicity, the two eigenvalues, which as we have seen are the inverse of each other, need to be real. If they are negative we obtain a reflection about the axes after each traversal of the periodic orbit and the intersection points switch from one half of the hyperbola to the other. The analysis above for the linearized approxima-tion can be extended to general dynamical systems where the stable and unstable manifolds are not straight lines but complicated curves. Due to the Birkhoff-Moser theorem (see Ozorio de Almeida, 1988, for example), we can make a change of coordi-nates to normal form coordicoordi-nates which lie along the stable and unstable manifolds.

In terms of these coordinates, the hyperbolic mapping from the Poincar´e section to itself still has the simple form

u⁰= Λ_γ[U(s, u)]u, s⁰ = Λ⁻¹_γ [U(s, u)]⁻¹s (1.1.9)

where the U is a measure of the non-linearity and tends to 1 as su→ 0, meaning that we recover true hyperbolae as we approach the axes.

When we consider systems with higher degrees of freedom (f > 2) in the lin-earized approximation, the eigenvalues ofMγ need no longer be real. We still have (f −1) pairs of eigenvalues, but if an eigenvalue Λ is complex, then it is part of a

loxodromic quartet because Λ^∗, _Λ¹ and _Λ¹∗ must also be eigenvalues.

For a particular loxodromic quartet j, we can write the four eigenvalues in the form e^{±λγ,jTγ±iφγ,j}, whereλγ,j >0. The eigenvectors are also complex and come in conjugate pairs, so to make the motion clear we split the quartet into a stable and unstable part. The eigenvalues e^{λγ,jTγ±iφγ,j} and one of their eigenvector pair give the unstable part, and their inverses the stable part. In the following we focus (as in Ozorio de Almeida, 1988) on the eigenvector associated to the unstable eigenvalue e^{λγ,jTγ+iφγ,j} with a positive sign in front of the phase φ_γ,j. Its real and imaginary part span the two-dimensional unstable plane. In this plane, after each iteration of the periodic orbit, all points increase their distance from the origin by a factor of

|Λγ,j|, as well as rotating clockwise by an angle ofφγ,j. The points then spiral out in this plane at the same time as they spiral inwards in the stable plane. We can define (non-invariant) stable and unstable directions that rotate (at the same rate as points in the planes) as we move around the periodic orbit (Turek et al., 2005).

In a system of coordinates along these directions, points would move along straight lines in each plane as they are transported around the periodic orbit, and we can separate the quartet into two pairs of stable and unstable directions. In fact, this rotation also happens when an eigenvalue is real and negative, but then the rotation is given by π.

Remaining in the linearized approximation, for each loxodromic quartet of eigen-values we have a stable and unstable plane, and for each real pair of eigeneigen-values a stable and unstable eigenvector. The stable manifold of the periodic orbit is the space spanned by all the stable planes and eigenvectors, while the unstable man-ifold is the (f −1) dimensional hyperplane spanned by the unstable equivalents.

Returning to general dynamics, the stable and unstable manifolds are no longer flat hyperplanes but complicated curved surfaces, but again we can rectify them using normal form coordinates. Then we can span this normal form space with (f −1) pairs of (possibly rotating) coordinates. The mapping from the Poincar´e section to itself for a point (u,s) in these coordinates has a simple hyperbolic form for each of its component parts (ui, si).