Discrete sine-gordon model with hysteretic links

Abstract

The article considers the collective dynamics of a set of nonlinear pendulums with hysteresis coupling between individual elements — the so-called discrete hysteretic sine-Gordon model. Hysteresis couplings can be formalized using the Bouc — Wen model, which is a convenient tool for modelling the phenomenon of hysteresis in mechanical systems. The article presents the simulation results for the evolution of localized oscillatory modes (breathers) obtained using the interactive Simulink environment integrated with MATLAB. Phase portraits and power spectrum density demonstrated the regularisation and filtering role of the hysteresis elements (using the terms of the Bouc — Wen model). Analysis of the dynamics of the localized natural frequencies showed that in the presence of hysteresis coupling, the asymptotic behaviour corresponds to the limiting cycle. We also analysed the resonance properties of the discrete hysteresis sine-Gordon system in the case of harmonic excitation of one of the pendulums. A bistable oscillatory mode was simulated using the frequency “scan” method. The method was used to calculate the amplitude-frequency characteristic and determine the frequency interval corresponding to unstable oscillatory modes. The obtained results demonstrated the efficiency of the hysteresis blocks as filtering and regulating elements of the complex nonlinear oscillatory system considered in the study.