Bogdanov–Takens Bifurcation in a Shape Memory Alloy Oscillator with Delayed Feedback

Jinbin Wang,Lifeng Ma,J.-C Cortés

doi:10.1155/2020/9132501

Abstract

This work is focused on a shape memory alloy oscillator with delayed feedback. The main attention is to investigate the Bogdanov–Takens (B-T) bifurcation by choosing feedback parameters A 1,2 and time delay τ . The conditions for the occurrence of the B-T bifurcation are derived, and the versal unfolding of the norm forms near the B-T bifurcation point is obtained by using center manifold reduction and normal form. Moreover, it is demonstrated that the system also undergoes different codimension-1 bifurcations, such as saddle-node bifurcation, Hopf bifurcation, and saddle homoclinic bifurcation. Finally, some numerical simulations are given to verify the analytic results.

Highlights

Smart materials have been widely used in many fields such as aircraft manufacturing [1, 2], control field [3], energy [4, 5], and medical [6] due to their special properties. e discovery and application of shape memory alloys [7,8,9] is an important part of smart materials. e socalled shape memory alloy (SMA) [10] is a new type of smart material with special shape memory effect and pseudoelasticity, which can restore the previously defined shape when subjected to an appropriate thermomechanical loading process
SMA spring oscillators can exhibit rich dynamic behaviors based on their pseudo-elasticity, promoting the study of nonlinear dynamics and bifurcation of shape memory oscillators [11,12,13,14,15]
Savi et al [16] studied the nonlinear dynamics of shape memory alloy systems and established the constitutive model of the SMA

Summary

Introduction

Smart materials have been widely used in many fields such as aircraft manufacturing [1, 2], control field [3], energy [4, 5], and medical [6] due to their special properties. e discovery and application of shape memory alloys [7,8,9] is an important part of smart materials. e socalled shape memory alloy (SMA) [10] is a new type of smart material with special shape memory effect and pseudoelasticity, which can restore the previously defined shape when subjected to an appropriate thermomechanical loading process. Savi et al [16] studied the nonlinear dynamics of shape memory alloy systems and established the constitutive model of the SMA. De Paula et al [19] controlled a shape memory alloy two-bar truss by the delayed feedback method. In 2016, Yu et al [22] considered a typical dimensionless system of the SMA oscillator based on equation (1) as follows: x€ + α1x_ + α2x − α3x3 + α4x5 k cos(θt),. (2) e bifurcation diagram and topological classification of the trajectory of a universal unfolding are given (3) e second-order terms of the normal form on a center manifold of the SMA system are obtained e layout of this work is organized as follows: in Section 2, we, respectively, give conditions for the occurrence of the B-T bifurcation and mainly discuss the normal forms for the B-T bifurcation.

Stability and B-T Bifurcation

Numerical Simulation

Conclusions