Lennard-Jones hyperelastic oscillators: Observing strange attractors, Feigenbaum cascades, and chaos

It is demonstrated that the widely used Lennard-Jones (LJ) potential in the mechanics of cross-linked polymers-and an oscillator based on it-can give rise to several notable phenomena: (i) The emergence of subharmonic and superharmonic oscillations across a broad range of driving force amplitudes; (ii) the presence of exponentially decaying amplitudes in the discrete part of the amplitude spectrum, associated with superharmonic components; (iii) the manifestation of multi-periodic, quasi-periodic, and chaotic regimes, depending on the amplitude of the driving force; (iv) the appearance of Feigenbaum cascades at transition zones between multi-periodic and chaotic behavior; and (v) the formation of strange attractors in the corresponding Poincaré sections, indicative of chaotic dynamics. The analysis is based on solving an autonomous system of three coupled first-order equations using the Adams-Bashforth-Moulton solver, which is well-suited for stiff dynamical systems. These findings offer deeper insight into the vibrational performance of seismic and vibration absorbers constructed from rubber-like materials modelled by LJ potentials.