^{1,a)}and G. Ananthakrishna

^{1,b)}

### Abstract

We develop a unified model to explain the dynamics of driven one dimensional ribbon for materials with strain and magnetic order parameters. We show that the model equations in their most general form explain several results on driven magnetostrictive metallic glass ribbons such as the period doubling route to chaos as a function of a dc magnetic field in the presence of a sinusoidal field, the quasiperiodic route to chaos as a function of the sinusoidal field for a fixed dc field, and induced and suppressed chaos in the presence of an additional low amplitude near resonant sinusoidal field. We also investigate the influence of a low amplitude near resonant field on the period doubling route. The model equations also exhibit symmetry restoring crisis with an exponent close to unity. The model can be adopted to explain certain results on magnetoelastic beam and martensitic ribbon under sinusoidal driving conditions. In the latter case, we find interesting dynamics of a periodic one orbit switching between two equivalent wells as a function of an ac magnetic field that eventually makes a direct transition to chaos under resonant driving condition. The model is also applicable to magnetomartensites and materials with two order parameters.

Modeling dynamics of real physical systems is generally difficult. Such a task can be challenging when a single experimental system exhibits a broad spectrum of dynamical features which is not usually found in a single system. One such system was experimentally realized almost two decades ago in studies on driven magnetostrictive metallic glass ribbons. These investigations show that the system exhibits the period doubling (PD) route to chaos as a function of a dc magnetic field for a fixed sinusoidal magnetic field and the quasiperiodic (QP) route to chaos as a function of the sinusoidal magnetic field keeping the dc field fixed. The authors reported several other dynamical features that include suppressed and induced chaos in the presence of an additional small amplitude near resonant sinusoidal field, and stochastic resonance. Here, we design a model with strain and magnetization as order parameters (OPs) and show that the model not only captures the first three features but also exhibits rich dynamics. The modelequations are sufficiently general that they can be adopted to explain the dynamical features of driven magnetoelastic beam and driven martensitic ribbon. We also find interesting switching dynamics in the case of forced martensitic ribbon hitherto not reported.

G.A. acknowledges the support from Indian National Science Academy for the Senior Scientist position. This work was supported BRNS Grant No: 2007/36/62-BRNS/2564.

I. INTRODUCTION

II. A UNIFIED MODEL

III. DRIVEN MAGNETOSTRICTIVE METALLIC GLASS RIBBON

A. Driven magnetoelastic beam

IV. DRIVEN MARTENSITES

V. DISCUSSION

### Key Topics

- Chaos
- 56.0
- Bifurcations
- 24.0
- Free energy
- 24.0
- Alternating current power transmission
- 20.0
- Amorphous metals
- 19.0

## Figures

(a) Period doubling bifurcation as a function of *h _{dc} * for , and

*p*= 0.32. (b) The corresponding largest Lyapunov exponent for the same set of parameters.

(a) Period doubling bifurcation as a function of *h _{dc} * for , and

*p*= 0.32. (b) The corresponding largest Lyapunov exponent for the same set of parameters.

(a) Quasiperiodic orbit in the space for keeping , and *p* = 0.32. (b) The corresponding Poincaré map in the plane. (c) Poincaré map in the plane for . (d) Largest Lyapunov exponent as a function of *h _{ac} *. (e) Poincaré map for a chaotic orbit in the plane for , and (f) Poincaré map for a chaotic orbit in the plane for .

(a) Quasiperiodic orbit in the space for keeping , and *p* = 0.32. (b) The corresponding Poincaré map in the plane. (c) Poincaré map in the plane for . (d) Largest Lyapunov exponent as a function of *h _{ac} *. (e) Poincaré map for a chaotic orbit in the plane for , and (f) Poincaré map for a chaotic orbit in the plane for .

(a) Bifurcation diagram as a function of *h _{dc} * for , and

*p*= 0.32. (b) Bifurcation diagram as a function of

*h*for the same set of parameters for and that clearly shows delayed chaos. Note that the period two orbit appears to switch by a small amount intermittently. (c) Bifurcation diagram as a function of

_{dc}*h*for the same set of parameters for and that clearly shows induced chaos. (d) Suppressed and induced chaos for and as a function of

_{dc}*h*.

_{r}(a) Bifurcation diagram as a function of *h _{dc} * for , and

*p*= 0.32. (b) Bifurcation diagram as a function of

*h*for the same set of parameters for and that clearly shows delayed chaos. Note that the period two orbit appears to switch by a small amount intermittently. (c) Bifurcation diagram as a function of

_{dc}*h*for the same set of parameters for and that clearly shows induced chaos. (d) Suppressed and induced chaos for and as a function of

_{dc}*h*.

_{r}(a) Bifurcation diagram as a function of *h _{dc} * for , and . (b) Bifurcation diagram as a function of

*h*for , and .

_{dc}(a) Bifurcation diagram as a function of *h _{dc} * for , and . (b) Bifurcation diagram as a function of

*h*for , and .

_{dc}(a) Symmetry restoring crisis during period doubling bifurcation as a function of *h _{ac} *. The parameter values are the same as used in Fig. 1 , except that we keep and and vary

*h*. Other parameter values are , and

_{ac}*p*= 0.32. (b) Power law dependence of the mean residence time in the Pre-crisis attractor.

(a) Symmetry restoring crisis during period doubling bifurcation as a function of *h _{ac} *. The parameter values are the same as used in Fig. 1 , except that we keep and and vary

*h*. Other parameter values are , and

_{ac}*p*= 0.32. (b) Power law dependence of the mean residence time in the Pre-crisis attractor.

(a) Phase plot in the plane for , and . (b) The corresponding Poincaré plot.

(a) Phase plot in the plane for , and . (b) The corresponding Poincaré plot.

(a) Resonance curve given by Eq. (14) for , and . Also shown is the numerically obtained resonance curve that shows several resonances for Ω less than unity. (b) A typical amplitude plot for .

(a) Resonance curve given by Eq. (14) for , and . Also shown is the numerically obtained resonance curve that shows several resonances for Ω less than unity. (b) A typical amplitude plot for .

(a) Bifurcation diagram for (b) A typical Poincaré map in the chaotic regime for .

(a) Bifurcation diagram for (b) A typical Poincaré map in the chaotic regime for .

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