Proliferation of stability in phase and parameter spaces of nonlinear systems

Abstract

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 Hénon 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. In high-dimensional dynamical systems the possibility of controlling the dynamics through parametric changes is of great interest. Adding a time dependent parameter it is possible to control the dynamics of the paradigmatic Hénon map generating multiply Isoperiodic Stable Structures (ISSs) on the parameter space and consequently increasing the number of attractors on the phase space. Numerical simulations and analytical results explain the origin of new stable domains due to saddle-node bifurcations for specific parametric combinations. The distance between the multiplied ISSs can be controlled by the intensity of the time dependent parameter and general rules for the occurrence of proliferation are treated in details. We believe that the present study represents a significantly new insight on the use of alternating forces to control the dynamics of complex nonlinear systems modeled by twodimensional maps and can be extended to various applications ranging from physics, biology to engineering.

Optimizing thermally affected ratchet currents using periodic perturbations

“…They follow the rule for integers ω = m/k = 3/3, while other periods becomes m F = 3m. With these results we can conclude that the RM (2) follows the general rules for systems perturbed by a k-periodic external parameter [21,22]. We believe that this result corroborates with the general character inherent to the methodology applied here.…”

Optimal ratchet current for elastically interacting particles

“…Using strong coupling intensities in the system (1) we can observe that the domain of the parameter space with negative RCs is duplicated, as shown in Fig. 1(d) for Recently, similar findings were reported in time discrete dynamical systems by using external periodic perturbations to generate multiple attractors in phase space and ISSs in the parameter space [25,26]. Increasing the intensity of the external perturbation, the regular structures start to separate from each other and an effective enlargement of the available stable domain in the parameter space is obtained.…”

Controlling intermediate dynamics in a family of quadratic maps

“…The easy way of generating and moving shifted bifurcation diagrams by controlling the intermediate dynamics of composed maps is definitely the remarkable contribution of the present work. Consequences of shifted bifurcations in two dimensional systems were analyzed recently by multiplying isoperiodic stable structures in the parameter space of the Hénon map [16]. Future contributions intend to verify to which extend such multiplication of stable motion can be realized in the parameter space of classical [17,18] and quantum ratchets [19].…”