By P. Civitanovic

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Identify y with x and integrate x over the whole phase space. The result is an expression for tr Lt as a sum over neighborhoods of prime cycles p and their repetitions tr Lt = Tp p ∞ r=1 δ(t − rTp ) det 1 − Mrp . 10. After the rth return to a Poincar´e section, the initial tube Mp has been stretched out along the expanding eigendirections, with the overlap with the initial volume given by 1/ det 1 − Mrp → 1/|Λp |, the same weight we obtained heuristically in sect. 1. 13) is disquieting in the way reminiscent of the Poisson resummation formulas of Fourier analysis; the left-hand side is the smooth eigenvalue sum tr eAt = esα t , while the right-hand side equals zero everywhere except for the set t = rTp .

Simon) 0 Y(t) 0 -5 -10 -10 -5 10 5 X(t) The plane is still rotating with constant angular velocity, but trajectories are now spiraling out. In general, any flow in the plane either escapes, falls into an attracting equilibrium point, or converges to a limit cycle - richer dynamics requires at least one more dimension. In order to prevent the trajectory from escaping to ∞, kick it into 3rd dimension when x reaches some value c by adding z˙ = b + z(x − c) , c > 0. 13) Now z shoots upwards exponentially, z ≃ e(x−c)t .

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.

### Chaos. Classical and Quantum by P. Civitanovic

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