Adaptation to the edge of chaos in continuous time dynamical systems

Abstract

We present a study involving the effect of self-adjusting feedback on the evolution of continuous time dynamical systems. This self-adjusting feedback results in the convergence (adaptation) of the system's dynamics to the edge of chaos. In particular, the delay-differential equations describing the Mackey-Glass system are studied. However, as a prelude, adaptation in the Rössler system is explored. For both these cases, the control parameter is taken to be a slowly varying function of one of the dynamical variables of the system. This slowly varying feedback is computed using a low-pass filter. Numerical simulations show that the parameter values depart from the chaotic regime and adapt to the edge of chaos in both systems. This is verified by repeating the simulations for numerous randomly chosen initial conditions. The final parameter values are observed to be distributed densely around the periodic windows at the edge of chaos.