Computational Issues in Nonlinear Dynamical Systems: Creating Bifurcation Diagrams of Lyapunov Exponents Using a Novel Adaptive Approach

Nonlinear dynamical systems with oscillatory (periodic, quasi-periodic and chaotic) responses are analyzed in this paper through the method of Lyapunov exponents. The main goal is to present 1D (one varying parameter) and 2D (two varying parameters) bifurcation diagrams of Lyapunov exponents created via a new adaptive approach, for selected two systems, describing an electric arc system and a model of calcium oscillatory phenomenon. First, the well-known Lyapunov exponents approach is discussed, and the orthogonalization procedure of the Parker-Chua method is modified to deal with a possibility of existence of co-linear vectors of Jacobian matrix (singularity of the matrix) in the orthogonalization algorithm. The co-linearity of vectors may lead to an extremely high matrix conditional coefficient, and, as a consequence, a failure of the procedure of orthogonalization in the Lyapunov exponents computation. Thus, a new adaptive approach is proposed to avoid the co-linearity issue. Four corrections of the parameters involved in the algorithm are proposed in the new adaptive approach. The advantage of using the newly design adaptive method to improve the procedure of the Lyapunov exponents computation is illustrated through a series of numerical examples. Visual results in the forms of 1D and 2D bifurcation diagrams are presented for an electric arc system and calcium oscillatory model.