Get Chaos : a program collection for the PC; with many numerical PDF

By Hans Jürgen Korsch; Hansjörg Jodl; Timo Hartmann

ISBN-10: 3540748660

ISBN-13: 9783540748663

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41) where the (sufficiently well-behaved) functions f and g are chosen so that the mapping ρi+1 ρi =T . 42) θi+1 θi is still area-preserving, the perturbed map T can again be considered as a Poincar´e map generated by a Hamiltonian H0 + H1 . The KAM theorem then guarantees that the invariant circles of T having a sufficiently irrational value of α are slightly transformed invariant curves of T for small values of . 4 Special Topics 29 The Poincar´e-Birkhoff theorem [18] states that invariant circles with rational values α = r/s (r and s are coprime integer numbers) consisting entirely of fixed points of T are not completely destroyed.

84) ∗ ∗ is born. 85) f (x∗+ , r) < 1 , provided that r is close enough to r1 . This bifurcation of fixed points — a so-called ‘pitchfork bifurcation’ — is illustrated in Figs. 14. It is investigated numerically in Chap. 9 . For increasing values of the parameter r, the fixed points of f 2 can also lose their stability at r2 and bifurcate again into period-four orbits (fixed points of f 4 ), and so on. 4 Special Topics 37 Fig. 13. Pitchfork bifurcation: A stable period-one fixed point x∗ loses stability at a critical value of the parameter r = r1 , where the slope |f (x∗ )| is unity, and a ∗ and x∗ is born.

2. Billiard systems are particularly suitable as models for educational purposes, since billiard motion is easy to grasp. The numerical treatment, as opposed to other systems, does not require numerical integration of differential equations. This is an important advantage, because such computation is comparatively time-consuming, especially since chaotic phenomena are exhibited in the long-term time behavior of the orbits. α ϕ r P S 0 Fig. 1. Boundary curve r(ϕ) and coordinates of the billiard system.

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Chaos : a program collection for the PC; with many numerical experiments and a CD-ROM by Hans Jürgen Korsch; Hansjörg Jodl; Timo Hartmann


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