Bifurcation and Chaos in Time-Fractional Models of Thin-Film Ferroelectric Materials

The research explores the nonlinear dynamics of time-fractional thin-film ferroelectric materials based on a beta-fractional derivative to integrate an effect of polarization dependent on memory. The governing nonlinear system is studied analytically by the modified Kudryashov method, where explicit soliton-type solutions to the polarization field are generated which shows how the parameters of the fractional-order affect the propagation and stability of the waves. The system behavior to different parameters is characterized by a complex dynamical analysis of phase portraits, bifurcation diagram, Lyapunov exponents, and recurrence plots. The findings detect the periodic, quasi-periodic, and chaotic regimes transitions, as well as the development of the multi-stability and chaotic attractors. Specifically, bifurcation analysis has shown important parameter regimes controlling the changes between stable and unstable polarization states, which forms a direct connection between analytical and numerical dynamics. This work is innovative by the application of the beta-fractional derivative in the modeling of ferroelectric thin-films, which gives a superior representation of memory effects in comparison to the traditional fractional operators. Moreover, it is demonstrated that the modified Kudryashov approach results in efficient analytical solutions to nonlinear fractional equations and therefore it is beneficial compared to the current techniques. The occurrence of chaotic and multi-stable behavior is demonstrated to affect switching reliability, energy efficiency, and stability of ferroelectric-based sensors and non-volatile memory devices.