We begin with the classical Lorenz system, outlining the fundamental principle of its chaotic dynamics before introducing the mathematical and physical characterization of chaos: the intricate entanglement of stable and unstable manifolds within phase space, governed by the system’s local divergence and global boundedness. This defining feature arises from the hyperbolic eigenvalue structure of the Jacobian matrix at equilibrium.
We then turn to a recent development in the study of non-hyperbolic chaotic systems. Like the Lorenz system, these are first-order, three-dimensional, quadratic polynomial systems with deceptively simple expressions. Yet they differ profoundly in their equilibrium behaviour, exhibiting no equilibrium points, only stable equilibria, or even infinitely many equilibria. In such cases, traditional analytical tools, such as the manifold entanglement framework, lose validity.
Research into general non-hyperbolic chaotic systems is still in its early stages, leaving a wide landscape of compelling and unresolved problems for future exploration.