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

Authors: Jinbin Wang; Lifeng Ma
Journal: Complexity
 (2020)

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## 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 parametersA1,2 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.


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## Body

## 1. 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. The discovery and application of shape memory alloys [7–9] is an important part of smart materials. The so-called shape memory alloy (SMA) [10] is a new type of smart material with special shape memory effect and pseudo-elasticity, 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 [11–15]. 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.The governing equation of motion of a shape memory oscillator [20, 21] is given by(1)mx¨+cx˙+Kx,T=Fcosωt,where q¯=qA/L, b¯=bA/L3, and e¯=eA/L5. m is the mass of the oscillator. Fcosωt is a periodic external force, and Kx,T=q¯T−Tmx−b¯x3+e¯x5 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. TM 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:(2)x¨+α1x˙+α2x−α3x3+α4x5=kcosθt,and they added a time-delayed feedback to control equation (2), and equation (2) can be rewritten as(3)x¨+α1x˙+α2x−α3x3+α4x5=kcosθt+A1xτ+A2xτ3,where xτ=xt−τ, τ is denoted as delay, and A1,2 is the delay position feedback parameter.Ifkcosθt is considered as the control parameter δ, equation (3) can be rewritten as(4)x¨+α1x˙+α2x−α3x3+α4x5=δ+A1xτ+A2xτ3.They used the normal form theory (NFT) and center manifold theorem (CMT) to calculate the conditions of the Hopf bifurcation and stability of equation (4).The 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 [24–28].Motivated by the above works, we consider system (4) and investigate the B-T bifurcation under some critical conditions. The main contributions of this paper are as follows:(1)
The feedback parametersA1,2 and time delay τ are selected to analyze their impact on codimension-2 bifurcations of system (4)(2)
The bifurcation diagram and topological classification of the trajectory of a universal unfolding are given(3)
The second-order terms of the normal form on a center manifold of the SMA system are obtainedThe layout of this work is organized as follows: in Section2, 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.


## 2. Stability and B-T Bifurcation

In this section, we mainly establish the existence of the B-T bifurcation under some critical conditions.Firstly, letx˙=y; then, system (4) can be equivalent to(5)x˙=y,y˙=−α1y−α2x+α3x3−α4x5+δ+A1xτ+A2xτ3.Denoting the equilibrium of system (4) as E0=x0,0, x0 satisfies an algebraic equation as follows:(6)gx=e1x5+e2x3+e3x+e4=0,where e1=α4, e2=−α3+A2, e3=α2−A1, and e4=−δ.Next, we discuss the existence conditions of the root of equation (6).Lemma 1.
For the roots of equation (6), the following results hold:(i)
Ife1<0, e3>0, and e4=0, then equation (6) has tree roots, and they are x1=0 and x2,3=±−e2−e22−4e1e3/2e1(ii)
Ife1>0, e3<0, and e4=0, then equation (6) has tree roots, and they are x1=0 and x2,3=±−e2−e22−4e1e3/2e1(iii)
Ife2/e1<0, e3/e1>0, △=e22−4e1e3>0, and e4=0, then equation (6) has five roots, and they are x1=0, x2,3=±−e2−e22−4e1e3/2e1, and x4,5=±−e2−e22−4e1e3/2e1(iv)
Ifei>0i=1,2,3,4, then equation (6) has no real root(v)
Ife1>0 and e4<0, then equation (6) has at least one positive real root(vi)
Ife1>0 and e4>0 and there exists ξ>0 such that gξ<0, then equation (6) has at least two positive real roots(vii)
Ifx>0, e1>0, e2>0, e3<0, and e4>0 and there exists ξ>0 such that gξ<0, then equation (6) has two positive real rootsProof.
(i), (ii), and (iii) are easy to prove, and we do not show the process of proof.(iv)
Ifei>0i=1,2,3,4, then g0=e4>0 and g′x>0. Thus, we can obtain that equation (6) has no real root.(v)
Ife1>0 and e4<0, then g0=e4<0 and limx⟶+∞gx=+∞, and we can obtain that equation (6) has at least one positive real root.(vi)
Ife1>0 and e4>0 and there exists ξ>0 such that gξ<0, we can obtain g0=e4>0 and limx⟶+∞g′x=+∞, and there exists x1∈0,ξ and x2∈ξ,+∞ such that gx1=gx2=0. Thus, equation (6) has at least two positive real roots.(vi)
Ifx>0, e1>0, e2>0, e3<0, and e4>0 and there exists ξ>0 such that gξ<0, we can obtain that g′0=e3<0, g″x=20e1x3+6e2x>0, limx⟶+∞g′x=+∞, and g′x=0 have only one positive real root. Furthermore, from g0=e4>0 and limx⟶+∞gx=+∞, we get that equation (6) has two positive real roots. This completes the proof.
Letx˜=x−x0 and y˜=y. Omitting the tilde, then system (5) can be rewritten as(7)x˙=y,y˙=−α1y+c1x+c2x2+c3x3+c4x4+c5x5+d1xτ+d2xτ2+d3xτ3,where(8)c1=−α2+3α3x0+5α4x04,c2=3α3x0−10α4x03,c3=α3+10α4x02,c4=−5α4x0,c5=−α4,d1=A1+3x02A2,d2=3A2x0,d3=A2.
The characteristic equation of system (7) at the zero equilibrium E0=0,0T is(9)Fλ,τ=λ2+α1λ−c1−d1e−λτ=0.
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.
Ifd0<0, then the following is obtained:(i)
Ifd1=d0 and τ≠τ0, λ=0 is a single root of equation (9)(ii)
Ifλ=0,d1=d0, and τ=τ0, λ=0 is a double root of equation (9)(iii)
Ifd1=d0, all the roots of equation (9) have negative real parts except for the zero roots
Here,τ0=−α1/d0 and d0=−c1.Proof.
Clearly,F0,τ=d1+c1=0. By calculating, we can obtain the following result:(10)∂Fλ,τ∂λ=2λ+α1+d0τe−λτ,∂2Fλ,τ∂λ2=2−d0τ2e−λτ.
It is easy to obtain ifλ=0 and τ≠τ0, then ∂Fλ,τ/∂λ|λ=0,τ≠τ0≠0. Thus, (i) holds.
Ifλ=0 and τ=τ0, then ∂Fλ,τ/∂λ|λ=0,τ=τ0=0 and ∂2Fλ,τ/∂λ2|λ=0,τ=τ0=2−α12/d0>0. Thus, (ii) holds.
Ifτ=0, α1>0, and d1=d0, equation (9) has roots λ1=0 and λ2=−α1<0. When τ≠0, let λ=iωω>0 be a root of equation (9); then, we have(11)−ω2+d0−d0cosωτ=0,α1ω+d0sinωτ=0.(12)ω4+α12−2d0ω2=0.
Letω2=t>0; then, equation (12) can be rewritten as(13)t2+pt=0,where p=α12−2d0. If d0<0, it results in p>0. Clearly, equation (13) has no positive roots. Thus, (iii) holds. This completes the proof.
Next, we will investigate the B-T bifurcation of system (7) near d0,τ0 by choosing d1 and τas bifurcation parameters.
Takingt=t/τ, d1=d0+μ1, and τ=τ0+μ2, system (5) can be rewritten as(14)x˙=τ0+μ2y,y˙=τ0+μ2−α1y+c1x+c2x2+c3x3+c4x4+c5x5+d0+μ1xt−1+d2x2t−1+d3x3t−1,where μ1 and μ2 are perturbation parameters.
The phase spaceC=C0,1;R2 is chosen as the Banach space of the continuous mappings from [0,1] toR2. For ∀x∈C, we define xtt+θ and x=sup−1<θ<0xθ. System (14) becomes(15)x˙t=Ext+Fxt,where E and F are operators, given by(16)Eφ=φ˙,DE=φ∈C10,1;R2:φ˙0=∫−10dηθφ−θ,dηθ=A0δθ+B0δθ+1dθ,Fφ=0,θ∈−1,0,L1μxt+Gxt,μ,θ=0,where δθ is the Dirac delta function, A0=0τ0c1τ0−α1τ0, and B0=00d0τ00.(17)L1μxt=μ2y0−μ2α1y0+c1μ2x0+d0μ2x−1+τ0μ1x−1,Gxt,μ=0c2τ0x20+c3τ0x30+c4τ0x40+c5τ0x50+τ0d2x2−1+τ0d3x3−1.
From Lemma1, equation (6) has a double-zero root, and all other eigenvalues have negative real parts. Let Λ be the set of eigenvalues with zero real part; C can be decomposed as C=A⊕B, where A is the generalized eigenspace associated with Λ which has two zero eigenvalues and A∗ is the space adjoint with A, and B=φ∈C:φ,ψ=0,∀φ∈A∗. Next, we define(18)E∗ψ=−ψ˙,DE∗=ψ∈C10,1;R2∗:ψ˙0=−∫−10dηθψ−θ,and the bilinear form on C×C∗ is(19)ψ,φ=ψ0φ0−∫−10∫0θψξ−θdηθφξdξ.Lemma 3 (see [24, 28, 29]).
The basesΦ and Ψ for A and A∗ can be chosen such that Ψ,Φ=I,Φ˙=Φ0100,andΨ˙=−0100Ψ, where Φθ=φ1θ,φ2θθ∈−1,0, ψs=colψ1s,ψ2ss∈0,1, φ1θ=φ10∈Rn\0, φ2θ=φ20+φ10θ,φ20∈Rn, and ψ2s=ψ20∈Rn∗\0, ψ1s=ψ10−sψ20,ψ10∈Rn∗, which satisfy(i)
A0+B0φ10=0.(ii)
A0+B0φ20=B0+Iφ10.(iii)
ψ20A0+B0=0.(iv)
ψ10A0+B0=ψ20B0+I.(v)
ψ20φ20−1/2ψ20B0φ10+ψ20B0φ20=1.(vi)
ψ10φ20−1/2ψ10B0φ10+ψ10B0φ20+1/6ψ20B0φ10−1/2ψ20B0φ20=0.By calculating, we can obtain(20)Φθ=1b∗+θ01τ0,Ψs=−sα1d∗−sd∗−d∗α1τ0α1d∗d∗,where b∗=α12τ02+6+3α1τ0/6α1τ0+3α12τ02 and d∗=2τ0/2+α1τ0.LetX=ΦZ+W, where Z∈R2andW∈B, namely,(21)x1θ=Z1+b∗+θZ2+w1θ,x2θ=1τ0Z2+w2θ,x10=Z1+b∗Z2+w10,x20=1τ0Z2+w20,x1−1=Z1+b∗−1Z2+w1−1.From [30, 31], system (14) can be written as(22)Z1.=Z2−d∗a1τ0F22,Z2.=a1d∗F21+d∗F22,where(23)F21=μ2τ0Z2,F22=c2τ0+τ0d2Z12+2c2τ0b+2τ0d2b∗−1Z1Z2+c2τ0b∗2+τ0d2b∗−12Z22+τ0μ1Z1+τ0b∗−1μ1Z2+−α1τ0+c1x∗−d0μ2Z2.From [26], we can obtain the following result:(24)f2Z,0,μ1=h1Z12+h2Z1Z2+h3Z22+h4μ1Z1+h5μ2Z1+h6μ1Z2+h7μ2Z2f1Z12+f2Z1Z2+f3Z22+f4μ1Z1+f5μ2Z1+f6μ1Z2+f7μ2Z2,where(25)h1=−d∗α1c2+d2,h2=−d∗α12c2b∗+2d2b∗−1,h3=−d∗α1c2b∗2+d2b∗−12,h4=−d∗α1,h5=0,h6=−d∗α1b∗−1,h7=−d∗τ0α1−α1τ0−d0,f1=d∗τ0c2+d2,f2=d∗τ02c2b∗+2d2b∗−1,f3=d∗τ0c2b∗2+d2b∗−12,f4=d∗τ0,f5=0,f6=d∗τ0b∗−1,f7=−d0d∗.The following normal form with versal unfolding on the center manifold can be obtained by some calculations:(26)Z1.=Z2,Z2.=λ1Z1+λ2Z2+η1Z12+η2Z1Z2,where λ1=d∗τ0μ1, λ2=τ0b∗−1d∗μ1−d0d∗μ2, η1=d∗τ0c2+d2, and η2=−2d∗/α1c2+d2+d∗τ02c2b∗+2d2b∗−1. The detailed calculations can be found in Appendix.By calculating, we can get(27)∂λ1,λ2∂μ1,μ2|μ=0=−d0d∗2τ0≠0.Thus, the mapμ1,μ2⟶λ1,λ2 is regular, and system (14) is equivalent to the normal form (26), where η1·η2>0.LetZ1⟶η1/η22Z1−λ1η22/2η12, Z2⟶−η12/η23Z2, and t⟶−η2/η1t; system (26) can be rewritten as(28)Z1.=Z2,Z2.=v1+v2Z2+Z12−Z1Z2,where v1=−λ12η24/4η14 and v2=−λ2η2/η1−λ1η22/2η12.System (28) as a universal unfolding [30, 31] with codimension-2 has been well studied. We can acquire the complete bifurcation diagram and topological classification of the trajectory of system (28), and the bifurcation diagram of system (28) on the perturbation parameter v1 and v2 planes is shown in Figure 1 [17, 21]. Furthermore, we give a concise form to list the conclusion as follows:(1)
System (28) undergoes a saddle-node bifurcation on the set(29)SN=v1,v2:v1=0,v2≠0=μ1,μ2:μ1=0,−τ0b∗−1d∗μ1−d0d∗μ2η2η1−d∗τ0μ1η222η12≠0.(2)
System (28) undergoes a stable Hopf bifurcation on the set(30)H=v1,v2:v1=−v22,v2>0=μ1,μ2:d∗τ0μ12η244η14=−τ0b∗−1d∗μ1−d0d∗μ2η2η1+d∗τ0μ1η222η122.(3)
System (28) undergoes a saddle homoclinic bifurcation on the set(31)T=v1,v2:v1=−4925v22=μ1,μ2:d∗τ0μ12η244η14=−4925τ0b∗−1d∗μ1−d0d∗μ2η2η1+d∗τ0μ1η222η122.Figure 1
The bifurcation diagram (a) and phase portraits (b) of system (15) on the perturbation parameter v1 and v2 planes.
(a)(b)


## 3. 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 parametersα1=1, α2=0.1, α3=1, α4=1, δ=0.8, A1=−1, and A2=−1. Furthermore, based on results of Lemmas 1 and 2, we can calculatex0=0.634, τ0=0.344, d0=−2.902, c2=−0.646, c3=5.02, c4=−3.17, c5=−1, d2=−1.902, and d3=−1. In Figure 1, the bifurcation diagrams of system (16) are composed of codimension-2 bifurcation point v1,v2=0,0 and three codimension-1 curves (saddle-node bifurcation curve, Hopf bifurcation curve, and saddle homoclinic bifurcation curve). When the parameters v1 and v2 change in different regions, system (16) will produce different dynamic properties.To easily analyze the dynamics of system (7), we fix μ1=−0.001 and only choose the value of μ2 to change. From Lemma 2, the B-T bifurcation point d0,τ0=−2.902,0.344 is obtained. Thus, we give some numerical examples as follows:(i)
If settingμ1,μ2=−0.001,0.006, there exist a saddle and a stable focus in region A2 (see Figure 1), as shown in Figures 2 and 3(ii)
Fixμ1,μ2=−0.001,0.026; Figures 4 and 5 show that a stable periodic solution occurs when bifurcation parameters pass through the Hopf bifurcation line H in region A3 (see Figure 1)(iii)
Fixμ1,μ2=−0.001,0.1664; Figures 6–8 show that a closed orbit exists through the homoclinic bifurcation line T∈③ (see Figure 1)Figure 2
Waveform plot of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.006, τ=0.35, tspan=0,500, and x0,y0=0.1,0.2.Figure 3
State trajectory of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.006, τ=0.35, tspan=0,500, and x0,y0=0.1,0.2.Figure 4
Waveform plot of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.026, τ=0.37, tspan=0,300, and x0,y0=0.1,0.2.Figure 5
Phase portraits of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.026, τ=0.37, tspan=0,300, and x0,y0=0.1,0.2.Figure 6
Waveform plot of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.1664, τ=0.5104, tspan=0,100, and x0,y0=0.1,0.2.Figure 7
Waveform plot of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.1664, τ=0.5104, tspan=0,100, and x0,y0=0.1,0.2.Figure 8
Phase portraits of system (7) near d0,τ0=−2.902,0.344 with parameters μ1,μ2=−0.001,0.1664, τ=0.5104, tspan=0,100, and x0,y0=0.1,0.2.


## 4. Conclusions

In this work, a shape memory alloy oscillator with delayed feedback has been analyzed. We mainly choose the two parametersd1=A1+3x02A2 and τ to investigate the B-T bifurcation of system (6). It is demonstrated that the feedback parameters A1,2 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 kcosθt as the control parameter δ (see [22]). However, the periodic force kcosθt has an important effect on the vibration and memory characteristics of the SMA system. Therefore, further discussion and analysis of the SMA system will be our future work.



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