Random parameter-switching synthesis of a class of hyperbolic attractors.

The parameter perturbation methods (the most known being the OGY method) apply small wisely chosen swift kicks to the system once per cycle, to maintain it near the desired unstable periodic orbit. Thus, one can consider that a new attractor is finally generated. Another class of methods which allow the attractors born, imply small perturbations of the state variable [see, e.g., J. Güémez and M. A. Matías, Phys. Lett. A 181, 29 (1993)]. Whatever technique is utilized, generating any targeted attractor starting from a set of two or more of any kind of attractors (stable or not) of a considered dissipative continuous-time system cannot be achieved with these techniques. This kind of attractor synthesis [introduced in M.-F. Danca, W. K. S. Tang, and G. Chen, Appl. Math. Comput. 201, 650 (2008) and proved analytically in Y. Mao, W. K. S. Tang, and M.-F. Danca, Appl. Math. Comput. (submitted)] which starts from a set of given attractors, allows us, via periodic parameter-switching, to generate any of the set of all possible attractors of a class of continuous-time dissipative dynamical systems, depending linearly on the control parameter. In this paper we extend this technique proving empirically that even random manners for switching can be utilized for this purpose. These parameter-switches schemes are very easy to implement and require only the mathematical model of the underlying dynamical system, a convergent numerical method to integrate the system, and the bifurcation diagram to choose specific attractors. Relatively large parameter switches are admitted. As a main result, these switching algorithms (deterministic or random) offer a new perspective on the set of all attractors of a class of dissipative continuous-time dynamical systems.