Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

Abstract

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

Detection and Analysis of Critical Dynamic Properties of Oligodendrocyte Differentiation

“…For dynamical systems, the first Lyapunov value is a well-established tool for making qualitative predictions about the long-term behavior of the system. To explain this, following [26], we calculate L 1 (for detailed discussion of L 1 , see appendix in [36] or also [27,28,[37][38][39][40][41][42]) close to the boundary of stability R = 0 of system (1). According to the Andronov-Hopf theory, the following relations are valid: (i) the sign of L 1 (at R = 0) defines the type (stable or instable) of equilibrium state; (ii) the type of equilibrium state (at R = 0) qualitatively defines the reconstruction of phase space.…”