Transition From Stability to Chaos in High Performance Fiber Materials Using Predictive Modeling

ABSTRACT

This paper offers a thorough analytical and numerical analysis of nonlinear dynamics under the Ablowitz‐Kaup‐Newell‐Segur (AKNS) water wave equation, particularly in its possible applicability to advanced fiber materials. Utilizing an appropriate wave transformation, the governing nonlinear partial differential equation is simplified into an ordinary differential equation, which is solved exactly using the generalized Riccati equation mapping method (GREMM). The closed‐form solutions form the basis for a systematic study of the system's rich dynamical features such as equilibrium states, bifurcations, multi‐stability, and chaos. Bifurcation analysis identifies the key transitions, demonstrating the routes from stable periodic oscillations to irregular and chaotic states. Numerical simulations also confirm these transitions, demonstrating the system's evolution between regular, quasi‐periodic, and fully developed chaotic states. Illustrations like phase portraits, temporal evolution of series, Lyapunov exponents, Poincaré sections, and recurrence plots attest to the existence of chaos, fractal‐like self‐similarity, and huge sensitivity to initial conditions. Multi‐stability, which is the coexistence of more than one attractor under the same parameter conditions, highlights the tenuous balance of nonlinear responses in complex systems. These results are most directly applicable to high‐performance fiber materials, in which wave propagation, stress‐strain instabilities, and nonlinear dynamic responses have a profound impact on performance and reliability. The revealed bifurcation and chaos mechanisms shed light on how fiber composites, smart fibers, and adaptive materials could switch between different mechanical states under different loads, impacts, or environmental inputs. Mapping the entire bifurcation topology of the system provides a predictive platform for designing fibers with improved vibration damping, impact resistance, and energy absorption properties. In general, this work demonstrates the strength of linking analytical exact solutions with numerical computations in revealing the subtle interaction among nonlinearity, parameter variation, and dynamic behavior, providing insightful viewpoints toward designing and controlling next‐generation fiber materials and fiber‐reinforced structures. The novelty of this work lies in the integration of exact analytical solutions with advanced chaos diagnostics, providing deeper insight into the interplay between nonlinearity and system dynamics, and extending beyond existing studies by offering both a broader solution spectrum and a comprehensive dynamical characterization.