DOI: https://doi.org/10.36347/sjpms.2026.v13i03.003

Pages: 107-117

Chaotic dynamics is a core topic in nonlinear science that is defined by sensitive dependence on initial conditions and long-term aperiodic behavior with strange attractors as fractal geometric objects that realize these behaviors. Although chaotic dynamics has been investigated extensively, there continues to be an important lack in systematically connecting quantitative "chaoticity" of an attractor (for example, mixing efficiency, predictability horizon) to linear and nonlinear stability theory. The current research program plans to address this gap using a theoretical approach to studying the co-existence of local-instability (via Lyapunov exponents) and global-boundedness (via stability of equilibria, bifurcations), and a numerical experiments approach on well-known systems (for example Lorenz, Rössler, hyperchaotic model). Our major findings show that the performance of a chaotic system, in terms of its rate of entropy production and fractal dimension, is not simply a consequence of the value of its largest Lyapunov exponent but, more importantly, governed by the organization and stability of certain invariant sets that are often associated with chaotic systems (fixed points and periodic orbits). We showed conclusively that bifurcation parameters tune this chaotic performance directly. This leads us to conclude that stability theory supplies an essential "skeleton" of constraints that govern and bound what chaotic attractors can accomplish dynamically as a unified understanding, and one that holds promise for implications related to control, synchronization, and the design of applications in engineering and complex systems.