Spectral Manifolds of Dynamical Transitions: From Linear Stability to Chaotic Attractors

This paper presents a comprehensive theoretical framework for understanding dynamical transitions in nonlinear systems, moving from states of linear stability to the emergence of complex chaotic attractors, through the lens of spectral manifolds. We integrate classical linear stability analysis, bifurcation theory, and advanced operator-theoretic approaches, particularly focusing on Spectral Submanifolds (SSMs) and the Koopman operator, to reveal the underlying spectral evolution during these transitions. The concept of spectral manifolds is developed as a trajectory or region in the complex spectral plane (or higher-dimensional spectral space) traced by the eigenvalues of linearized systems or spectral properties of global operators, as system parameters are varied. We demonstrate how the real and imaginary parts of eigenvalues of the Jacobian matrix characterize local stability, while their movement in the complex plane signals bifurcations such as Hopf and period-doubling. Beyond local analysis, we explore how global spectral properties, derived from Koopman and transfer operators, provide insights into the geometry of chaotic attractors and quantify their inherent unpredictability through Lyapunov exponents. The paper synthesizes these perspectives to offer a rigorous and unified methodology for diagnosing, characterizing, and potentially predicting critical transitions in diverse dynamical systems.