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Analytical Approximation of the Dissipative Standard Map

Published online by Cambridge University Press:  12 April 2016

Alessandra Celletti
Affiliation:
Dipartimento di Matematica, Università di L’Aquila, 1-67010 L’Aquila (Italy)
Gabriella Della Penna
Affiliation:
Observatoire de Nice, BP 229, F-06304 Nice Cedex 4 (France)
Claude Froeschlé
Affiliation:
Observatoire de Nice, BP 229, F-06304 Nice Cedex 4 (France)

Extract

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We investigate the dynamics of a dissipative standard mapping defined by the equations

where yR, xT and ε is a real parameter, we refer to 0 < α < 1 as the “dissipative parameter” and to ψ as the “dissipative coefficient” (ε = α = 0 provides an integrable mapping). Notice that the dynamics is contractive, since the jacobian of the above mapping equals to 1 − α. In particular, we want to compare (see Celletti et al., 1997) the solutions associated to the conservative map (i.e., α = 0) with that related to (1) (α ≠ 0). For simplicity, we consider the case when α = ε2 and construct explicit approximate solutions to the conservative and dissipative systems, using a suitable parametrization like in (Celletti and Chierchia, 1988).

Type
Extended Abstracts
Copyright
Copyright © Kluwer 1999

References

Celletti, A., Chierchia, L.: 1988, Construction of analytic KAM surfaces and effective stability bounds, Commun, in Math. Physics, 118, 119 CrossRefGoogle Scholar
Celletti, A., Della Penna, G., Froeschlé, C.: 1997, Analytic approximation of the solution of the dissipative standard map, International Journal of Bifurcation and Chaos, in press.CrossRefGoogle Scholar