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

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.


Introduction
In recent years, 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, thus 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. Fu and Lu [17] investigated the nonlinear dynamics and vibration damping of dry friction oscillators with SMA restraints. Costa et al. [18] applied the extended time-delayed feedback approach to investigate the chaos control of an SMA two-bar truss. de Paula et al. [19] controlled a shape memory alloy two-bar truss by the delayed feedback method. e governing equation of motion of a shape memory oscillator [20,21] is given by where q � (qA/L), b � (bA/L 3 ), and e � (eA/L 5 ). m is the mass of the oscillator. F cos(ωt) is a periodic external force, and K(x, T) � q(T − T m )x − bx 3 + ex 5 is the restoring force of the spring. L and A, respectively, denote a shape memory element of length and cross-section area. b, e, c, and q are constants of the material. T M corresponds to the temperature where the martensitic phase is stable. In 2016, Yu et al. [22] considered a typical dimensionless system of the SMA oscillator based on equation (1) as follows: and they added a time-delayed feedback to control equation (2), and equation (2) can be rewritten as where x τ � x(t − τ), τ is denoted as delay, and A 1,2 is the delay position feedback parameter. If k cos(θt) is considered as the control parameter δ, equation (3) can be rewritten as ey used the normal form theory (NFT) and center manifold theorem (CMT) to calculate the conditions of the Hopf bifurcation and stability of equation (4). e deep insight of the system dynamics is helpful to understand the nonlinear dynamics of shape memory alloy systems. However, many studies on time-delay systems have focused on analyzing the bifurcations of codimension-1, such as Hopf bifurcation [23]. Actually, the time-delay system may have more complicated dynamics when two separate parameters or many parameters are changed simultaneously. B-T bifurcation, which is a typical codimension-2 bifurcation, is studied in [24][25][26][27][28].
Motivated by the above works, we consider system (4) and investigate the B-T bifurcation under some critical conditions. e main contributions of this paper are as follows: (1) e feedback parameters A 1,2 and time delay τ are selected to analyze their impact on codimension-2 bifurcations of system (4) (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. In Section 3, some numerical simulations are implemented to validate the above analysis. We give some conclusions in Section 4, respectively.

Stability and B-T Bifurcation
In this section, we mainly establish the existence of the B-T bifurcation under some critical conditions. Firstly, let _ x � y; then, system (4) can be equivalent to Denoting the equilibrium of system (4) as E 0 � (x 0 , 0), x 0 satisfies an algebraic equation as follows: where e 1 � α 4 , e 2 � − (α 3 + A 2 ), e 3 � α 2 − A 1 , and e 4 � − δ. Next, we discuss the existence conditions of the root of equation (6).
Let x � x − x 0 and y � y. Omitting the tilde, then system (5) can be rewritten as 2 Complexity where e characteristic equation of system (7) at the zero Next, we give the conditions for the existence of the B-T bifurcation and investigate the dynamical classification near the B-T bifurcation point. □ Lemma 2. If d 0 < 0, then the following is obtained: all the roots of equation (9) have negative real parts except for the zero roots Proof. Clearly, F(0, τ) � d 1 + c 1 � 0. By calculating, we can obtain the following result: It is easy to obtain if λ � 0 and τ ≠ τ 0 , then (9) has roots λ 1 � 0 and λ 2 � − α 1 < 0. When τ ≠ 0, let λ � iω(ω > 0) be a root of equation (9); then, we have Let ω 2 � t > 0; then, equation (12) can be rewritten as where p � α 2 1 − 2d 0 . If d 0 < 0, it results in p > 0. Clearly, equation (13) has no positive roots. us, (iii) holds. is completes the proof.
Next, we will investigate the B-T bifurcation of system (7) near (d 0 , τ 0 ) by choosing d 1 and τas bifurcation parameters.
Let X � ΦZ + W, where Z ∈ R 2 and W ∈ B, namely, From [30,31], system (14) can be written as From [26], we can obtain the following result: where e following normal form with versal unfolding on the center manifold can be obtained by some calculations: where 1)). e detailed calculations can be found in Appendix.

Numerical Simulation
In this section, we use the dde23 method in MATLAB and show some numerical simulations to illustrate the analysis results given in the previous sections.
In order to easily verify the obtained results, we choose parameters  Figure 1, the bifurcation diagrams of system (16) are composed of codimension-2 bifurcation point (v 1 , v 2 ) � (0, 0) and three codimension-1 curves (saddle-node bifurcation curve, Hopf bifurcation curve, and saddle homoclinic bifurcation curve). When the parameters v 1 and v 2 change in different regions, system (16) will produce different dynamic properties.
To easily analyze the dynamics of system (7) Figure 1)

Conclusions
In this work, a shape memory alloy oscillator with delayed feedback has been analyzed. We mainly choose the two parameters d 1 � A 1 + 3x 2 0 A 2 and τ to investigate the B-T bifurcation of system (6). It is demonstrated that the feedback parameters A 1,2 and time delay τ have an important influence on the shape memory alloy oscillator. As the two parameters of the SMA oscillator change, the conditions for the occurrence of B-T bifurcation and some phase portraits and bifurcation diagrams are given. By using the CMT and NFT of functional differential equations, we investigate some typical codimension-1 bifurcations such as saddle-node bifurcation, Hopf bifurcation, and saddle homoclinic bifurcation. Some numerical simulations further verify the obtained analytic results.
In our paper, second-order terms of the normal form on a center manifold are given, but the higher order is not investigated. System (2) or (3) is only discussed by considering k cos(θt) as the control parameter δ (see [22]).  However, the periodic force k cos(θt) has an important effect on the vibration and memory characteristics of the SMA system. erefore, further discussion and analysis of the SMA system will be our future work.