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By Rudolf Dvorak

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The dynamics of the dissipative standard map is foliated by attracting periodic orbits, invariant quasi-periodic attractors and strange attractors. They can be seen as analogous to the periodic, quasi-periodic and chaotic orbits of the classical standard map, respectively. 7 Phase portraits of the dissipative standard map. (a) coexistence of a periodic and quasiperiodic attractor with corresponding basin of attraction. (b) strange attractor for f (x) D sin(x) C sin(3x) (see text). 37 38 3 Numerical and Analytical Tools strange attractors have a fractal noninteger dimension.

24) We introduce the following quantities for the connecting position vector of particles ν and μ q μ ν its norm r ν μ which stands for the distance between the masses ν and μ respectively qN ν μ ) for their mutual velocity qN ν μ . 2 Lie-Series Numerical Integration As next step we perform a series expansion which leads to "1 # X (τ D) n τD q ν (τ) D e q ν (0) D q ν (0) n! nD0 Ä Ã Â n (τ)2 2 (τ)3 3 (τ) D 1 C τD C q ν (0) D C D C CO Dn 2! 3! n! n 2 N0 At the core of the integration algorithm lies the computation of the Lie series operator, that is the calculation of the orders 2) D n q(0).

2. For every problem one needs to find out the most effective scheme for the recurrence formulas. 3. For complicated right hand sides of the differential equations the finding of recurrence formulae for D becomes more and more complicated. All in all, for most problems in celestial mechanics it is an efficient tool, well designed to deal with close encounters between celestial bodies, for example nearEarth asteroids, or comets suffering from transformed orbits due to close encounters to Jupiter.

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