Hénon map chaotic system critical points analysis and classification for the dynamic control of brain stimulation

Abstract

Brain activities demonstrate chaotic behaviors which can be simulated by the outputs of chaotic systems such as Hénon map. Brain stimulation has been used to control the brain dynamics to treat certain neurological disorders such as Parkinson's disease and Epilepsy. However, these clinical practices are based on empirical trials and lack theoretical support. Moreover, the long-term effect of brain stimulation to the brain neural network remains unknown. Therefore, it is critical to review the recent clinical practices and bring up sound theoretical study for brain stimulation. This paper presents the analysis and classification of critical points of Hénon map chaotic system based on different system parameters. The dynamical control of Hénon map is based on five characteristics of chaotic systems, namely bifurcation, Lyapunov exponent, critical point, Jacobian matrix, and the eigenvalues of the Jacobian matrix evaluated at the critical point. The critical points are classified into eight types based on system parameters to evaluate the system behavior (stable or unstable), which can then be used for generating stimulation for dynamical system control. These analysis methods can be used for various chaotic systems in general.