Quantifying the Geometric Sensitivity of Attractor Basins: Power Law Dependence on Parameter Variations and Noise

We show that for many physical systems the dependence of attractor basin geometry on parameter variations and noise can be characterized by power laws. We introduce new invariants---the basin immunities---that quantify this dependence and we analyze their origin and properties. Results from extensive numerical experiments are presented; examples include the driven pendulum and the H\'enon map. Potential applications of basin immunities include quantifying the effect of parameter uncertainties and noise on the behavior of nonlinear devices, as well as improving parameter estimation algorithms.