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The courses taught so far start out with the introductory chapters on qualitative dynamics, symbolic dynamics and flows, and then continue in different directions: Deterministic chaos. Chaotic averaging, evolution operators, trace formulas, zeta functions, cycle expansions, Lyapunov exponents, billiards, transport coefficients, thermodynamic formalism, period doubling, renormalization operators. Spatiotemporal dynamical systems. Partial differential equations for dissipative systems, weak amplitude expansions, normal forms, symmetries and bifurcations, pseudospectral methods, spatiotemporal chaos.

9) can be read off the formula, a zero zp = |Λp |1/np for each term in the product. ” Dead wrong! 9) used? We start by computing the lengths and eigenvalues of the shortest cycles. 5. CHAOS FOR CYCLISTS chapter 14 17 assume that the numerics is under control, and that all short cycles up to given length have been found. In our pinball example this can be done by elementary geometrical optics. It is very important not to miss any short cycles, as the calculation is as accurate as the shortest cycle dropped – including cycles longer than the shortest omitted does not improve the accuracy (unless exponentially many more cycles are included).

17). The excitations thus transferred are dissipated by the strongly damped short wavelengths, and a “chaotic equilibrium” can emerge. The √ very short wavelengths |k| 1/ ν will remain small for all times, but √ the intermediate wavelengths of order |k| ∼ 1/ ν will play an important role in maintaining the dynamical equilibrium. 17). 17), care has to be exercised to ensure that no modes essential to the dynamics are chopped away. In practice one does this by repeating the same calculation at different truncation cutoffs N , and making sure that inclusion of additional modes has no effect within the accuracy desired.

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