Type II Intermittency Route to Hidden Attractor in Modified Lorenz System

The location and basins of attraction of attractors involved in nonlinear smooth dynamical systems play a crucial role in understanding the dynamics of these systems across a wide range of parameter values. Nonlinear dynamical systems, described by autonomous nonlinear coupled ordinary differential equations with three or more variables, often exhibit chaotic dynamics within specific parameter ranges. In the literature, three distinct routes to chaos are well known: period doubling, crisis, and intermittency. Recent research in the field of nonlinear dynamical systems focuses on the characterization and identification of hidden attractors. The primary objective of this study is to investigate the dynamics and the successive local and global bifurcations that give rise to chaotic behavior in a Modified Lorenz System (MLS). Our findings reveal that chaos emerges via the type II intermittency route. Furthermore, we identify the chaotic attractor as a hidden attractor, with its existence corroborated by the dissipative nature of the model beyond the subcritical Hopf bifurcation. Key contributions of this research include the derivation of analytical criteria for a degenerate pitchfork bifurcation, the numerical calculation of the first Lyapunov exponent, and a semi-analytic proof of the homoclinic bifurcation. These results enhance our understanding of the complex dynamics within the MLS and contribute to the broader field of nonlinear dynamics and chaos theory.