Finding attractors of continuous-time systems by parameter switching

08 March 2012


There are many different interactions in nature and systems could evolve according to more than one dynamics for short periods of time. Therefore, it is reasonable to think that the evolution of some natural processes could be explained by the alternation of different dynamics for relatively short periods of time. RIST's Marius F. Danca has found that if the control parameter p, of a continuous-time nonlinear system belonging to a large class of systems, is switched within a set of chosen values in a deterministic or even random manner, while the underlying model is numerically integrated, the obtained attractor is a numerical approximation of one of the existing attractors of the considered system. The numerically obtained trajectory is extremely similar to the trajectory obtained for p replaced with the averaged value of the switched parameters. The algorithm could be considered as an excellent alternative for control and anticontrol of chaos for continuous-time systems. Moreover, this kind of attractor synthesis resembles Parrondo's philosophy in a winning game: ''losing + losing = winning''. Thus, switching the control parameter within a set of values which generate e.g. chaotic attractors, one can obtain a stable periodic motion which, in Parondian terms this means: "chaos+chaos=order" (a kind of control like algorithm). Also one can have anticontrol: "order+order=chaos" or other possible combinations. The work leading to these results has been performed in the last four years, and has been documented through more than ten papers in nonlinear science journals. A review is presented in: Marius-F. Danca, M. Romera, G. Pastor, and F. Montoya, Finding Attractors of Continuous-Time Systems by Parameter Switching, Nonlinear Dynamics (2011), doi:10.1007/s11071-011-0172-6.