Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 361hetero– and homo–clinic points, whose orbits converge to one periodic orbitin the past and to another (or the same) in the future. He showed that havingintersected once, the invariant manifolds must intersect infinitely often.Moreover the existence of one heteroclinic orbit implies the existence of aninfinity of others.The distance between the stable and unstable manifolds can be quantifiedby Melnikov’s integral. This leads to a technique for proving thenon–existence of integrals for a slightly perturbed, integrable Hamiltonian.For integrable systems, nearby orbits separate linearly in time. However,dynamical systems can have exponentially separating orbits. Let δx be atangent vector at the phase–space point x and δx t be the evolved vectorfollowing the orbit of x. Then, recall that the average rate of exponentiationof δx t is the Lyapunov exponent λ (see, e.g., [Chen and Dong (1998)])λ(x, δx) =lim 1/t ln |δx t|.t−→∞If λ is nonzero, then the predictions one can make will be valid for a timeonly logarithmic in the precision. Therefore, although deterministic in principle,a system need not be predictable in practice.A concrete example of the complexity of behavior of typical Hamiltoniansystems is provided by the ‘horseshoe’, a type of invariant set found nearhomoclinic orbits. Its points can be labelled by doubly infinite sequences of0’s and 1’s corresponding to which half of a horseshoe shaped set the orbitis in at successive times. For every sequence, no matter how complicated,there is an orbit which has that symbol sequence. This implies, e.g., that asimple pendulum in a sufficiently strongly modulated time–periodic gravitationalfield has an initial condition such that the pendulum will turn overonce each period when there is 1 in the sequence and not if there is a 0 forany sequence of 0’s and 1’s.3.12.3.3 Hamilton–Poisson MechanicsNow, instead of using symplectic structures arising in Hamiltonian mechanics,we propose the more general Poisson manifold (g ∗ , {F, G}). Here g ∗ isa chosen Lie algebra with a (±) Lie–Poisson bracket {F, G} ± (µ)) and carriesan abstract Poisson evolution equation F ˙ = {F, H}. This approachis well–defined in both the finite– and the infinite–dimensional case. Itis equivalent to the strong symplectic approach when this exists and offersa viable formulation for Poisson manifolds which are not symplectic(for technical details, see see [Weinstein (1990); Abraham et al. (1988);