Global Dynamics and Bifurcations of an Oscillator with Symmetric Irrational Nonlinearities

Abstract

This study’s objective is an irrationally nonlinear oscillating system, whose bifurcations and consequent multi-stability under the circumstances of single potential well and double potential wells are investigated in detail to further reveal the mechanism of the transition of resonance and its utilization. First, static bifurcations of its nondimensional system are discussed. It is found that variations of two structural parameters can induce different numbers and natures of potential wells. Next, the cases of mono-potential wells and double wells are explored. The forms and stabilities of the resonant responses within each potential well and the inter-well resonant responses are discussed via different theoretical methods. The results show that the natural frequencies and trends of frequency responses in the cases of mono- and double-potential wells are totally different; as a result of the saddle-node bifurcations of resonant solutions, raising the excitation level or frequency can lead to the coexistence of bistable responses within each well and cause an inter-well periodic response. Moreover, in addition to verifying the accuracy of the theoretical prediction, numerical results considering the disturbance of initial conditions are presented to detect complicated dynamical behaviors such as jump between coexisting resonant responses, intra-well period-two responses and chaos. The results herein provide a theoretical foundation for designing and utilizing the multi-stable behaviors of irrationally nonlinear oscillators.

Comparisons for Global Dynamics of a Geometrically Nonlinear Oscillator among Single-, Double- and Quadruple-Well Configurations

“…The geometric configuration of the springs causes irrational nonlinearity in the dynamic systems, hence causing difficulties in solving the analytical solutions of these systems. Theoretical and numerical approaches have been employed to investigate their nonlinear characteristics, including static bifurcations [20][21][22], approximative solutions [23][24][25], stochastic bifurcations [26] and global bifurcation analysis [27][28][29]. The SD oscillator subjected to harmonic excitation was found to possess the stiffness-hardening characteristics for a smooth parameter greater than one [23] but exhibited bistable potential wells and quad-stable responses for smooth parameters less than one [22,24].…”