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The role of unstable periodic orbits was already fully appreciated by Poincar´e [8, 9], who noted that hidden in the apparent chaos is a rigid skeleton, a tree of cycles (periodic orbits) of increasing lengths and selfsimilar structure, and suggested that the cycles should be the key to chaotic dynamics. Periodic orbits have been at core of much of the mathematical work on the theory of the classical and quantum dynamical systems ever since. We refer the reader to the reprint selection [10] for an overview of some of that literature.

8 on p. 54 days and gobbles up terabits of memory, you are using brain-damaged high level software. Try writing a few lines of your own Runge-Kuta code in some mundane everyday language. 9 on p. 54 methods, this in not the time or place to expand on them; how you learn them is your business. 10 on p. 54 And if you have developed some nice routines for solving problems in this text or can point another students to some, let us know. 11 on p. 55 fast track: chapter 3, p. 5 Inﬁnite-dimensional ﬂows Flows described by partial diﬀerential equations are considered inﬁnite dimensional because if one writes them down as a set of ordinary diﬀerential equations (ODE) then one needs an inﬁnity of the ordinary kind to represent the dynamics of one equation of the partial kind (PDE).

By considering all possible trajectories, we obtain the vector ∂t f 0 (x) at any point x ∈ M and deﬁne a vector ﬁeld v(x) = ∂f 0 (x) . tex 4apr2002 on p. 52 38 CHAPTER 2. 7), together with a short trajectory segment. The vectors are drawn superimposed over the conﬁguration coordinates (x(t), y(t)) of phase space M, but they belong to a diﬀerent space, the tangent bundle T M. (b) The ﬂow lines. Each “comet” represents the same time interval of a trajectory, starting at the tail and ending at the head.