Recovering parameters from self-similar structures in phase space of dissipative systems with constant Jacobian

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