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By P. Civitanovic

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5 5. 8 6. 9 • Averaging, numerical 1. 3 2. 2 • Averaging, periodic orbits 1. 2 2. 5 3. 3 4. 1 5. 1 6. 3 7. 7 Solutions for some of the problems are given in appendix N. Often going through a solution is more instructive than reading the chapter that problem is supposed to illustrate. 8. A GUIDE TO THE LITERATURE 27 R´ esum´ e The goal of this text is an exposition of the best of all possible theories of deterministic chaos, and the strategy is: 1) count, 2) weigh, 3) add up. In a chaotic system any open ball of initial conditions, no matter how small, will spread over the entire accessible phase space.

Chapter 12 For a deterministic system almost every point has a unique future, so trajectories cannot intersect. ) for which a trajectory is not defined. We may think such sets a nuisance, but it is quite the contrary - they will enable us to partition phase space, so that the dynamics can be better understood. Locally, the phase space M looks like Rd , meaning that d numbers are sufficient to determine what will happen next. Globally, it may be a more complicated manifold formed by patching together several pieces of Rd , forming a torus, a cylinder, or some other geometric object.

Atmospheric Phys. 20, 130 (1963). 2] G. Duffing, Erzwungene Schwingungen bei ver¨ anderlicher Eigenfrequenz (Vieweg. Braunschweig 1918). 3] O. R¨ossler, Phys. Lett. 57A, 397 (1976). org/wiki/Rossler map. 5] J. Peinke, J. E. R¨ossler, and R. Stoop, Encounter with Chaos.

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