Dynamics of strain bifurcations in magnetostrictive ribbon

Abstract : The rich chaotic dynamics exhibited by sinusoidally driven nonlinear oscillators are ubiquitous to a large number of systems such as turbulence in fluid systems, chemical oscillators, cardiac tissues etc. Manipulated poperties of chaos such as control of chaos realized in driven nonlinear oscillators, are also reported in several disciplines, for instance control of output of laser system, enhanced performance of permanent magnet synchronous motor, control of cardiac arrhythmias etc. Natural to multistate weakly driven oscillators in the presence of ubiquitous noise is the stochastic resonance, another generic noise induced cooperative phenomenon that enhances the signal to noise ratio. However, it is rare to find such a broad spectrum of dynamical features in a single system. Surprisingly, physical realization of several of these features was reported by Vohra and his group, almost twenty five years ago. In their study of strain bifurcations in magnetostrictive ribbons subjected to the combined influence of sinusoidal (ac) and dc magnetic fields, the authors report an extraordinarily large number of dynamical features such as (a) quasiperioidic (QP) route to chaos when the amplitude of the ac magnetic field hac, was increased in the presence of an additional dc magnetic field hdc, (b) period doubling (PD) route to chaos when hdc was increased keeping the ac field fixed, (c) suppression and shift of period doubling bifurcation point and induced subcritical bifurcation under small amplitude near resonant conditions, (d) control of chaos, specifically, suppression and induced chaos with the application of a near resonant perturbation to one of the subharmonics, and (e) stochastic resonance. However, to the best of our knowledge, modeling such a rich dynamics exhibited by a single system in terms of relevant strain and magnetic order parameters has remained a challenge. We develop a coupled nonlinear oscillator model [1] involving magnetization and strain to explain several experimentally observed dynamical features exhibited by forced magnetostrictive ribbon. Our starting point is to write down the relevant free energies and then to derive a coupled set of partial differential equations for strain and magnetization. These equation can be further reduced to one space dimension since samples are long ribbons. Using the dominant mode of vibration, the equations are further reduced to a coupled set of ordinary differential equations for strain and magnetization. We show that the model recovers the observed period doubling route to chaos as function of the dc field for a fixed ac field and quasiperiodic route to chaos as a function of the ac field keeping the dc field constant [1]. The model also predicts induced and suppressed chaos under the influence of an additional small amplitude near resonant ac field. Our analysis suggests rich dynamics in coupled order parameter systems like magnetomartensites and magnetoelectric materials [2,3]. The general nature of these equations suggest much richer dynamics in ferromagnetic martensites samples that posses even stronger elastic and magnetic nonlinearities and also in magnetoelectric materials. The model also explains some old results on internal friction studies of martensites. Interestingly, these equations also support mixed mode oscillations.