Bifurcation Analysis for the Generalized Nos´e–Hoover System

This study investigates the generalized Nos´e–Hoover system. The original version of the system is a chaotic system designed to represent the interaction between a harmonic oscillator and a heat bath maintained at a constant temperature. Despite its simplicity in just three dimensions, it exhibits complex and unusual dynamics. This investigation focuses on studying local bifurcations, including Saddle-Node and Hopf bifurcations, of the generalized Nos´e–Hoover system. In terms of cyclicity, the Lyapunov quantities technique is used to demonstrate that three periodic orbits can bifurcate from the Hopf point. This mathematical research contributes to understanding the equilibrium points, their stability and the dynamics of the nonlinear model when some of the parameters are varied.