Proliferation of stability in phase and parameter spaces of nonlinear systems

In this work we show how the composition of maps allows us to multiply, enlarge and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using H\'enon maps with distinct parameters we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.