Discovering parameter values by measuring self-similar structures in the phase-space of dissipative systems with constant Jacobian

¹ Departamento de Física, Universidade do Estado de Santa Catarina 89223-100 Joinville, Brazil
² Departamento de Física, Universidade Federal do Paraná - 81531-990 Curitiba, Brazil
³ Instituto de Física, Universidade Federal do Rio Grande do Sul 91501-970 Porto Alegre, Brazil
⁴ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa 1749-016 Lisboa, Portugal
⁵ Institut für Computer Anwendungen, Universität Stuttgart Pfaffenwaldring 27, D-70569 Stuttgart, Germany.

Corresponding authors: dfi2pcr@dcc.fej.udesc.br mbeims@fisica.ufpr.br jgallas@if.ufrgs.br

Received: 27 September 1999
Accepted: 10 January 2000

Abstract

This paper shows that dissipative dynamical systems with constant Jacobian allow one to discover the numerical values of physical parameters under which the system is operating. This is done by performing measurements on self-similar (fractal) structures of the phase-space. Parameter recovery is illustrated explicitly for the Ikeda laser ring-cavity map and the Hénon map. The first model involves transcendental equations of motion that can be solved only numerically. Analytical results are obtained for the second model. In both cases the macroscopic dissipation rate of the dynamical system is recovered from the speed at which fractal “fingers” making up basins of attraction accumulate towards basin boundaries.