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 and time delay . 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.
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 bywhere , , and . is the mass of the oscillator. is a periodic external force, and is the restoring force of the spring. and , respectively, denote a shape memory element of length and cross-section area. b, e, c, and are constants of the material. 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 aswhere , is denoted as delay, and is the delay position feedback parameter.
If is considered as the control parameter , equation (3) can be rewritten as
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 parameters and time delay 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 obtained
The 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.
2. Stability and B-T Bifurcation
In this section, we mainly establish the existence of the B-T bifurcation under some critical conditions.
Firstly, let ; then, system (4) can be equivalent to
Denoting the equilibrium of system (4) as , satisfies an algebraic equation as follows:where , , , and .
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)If , , and , then equation (6) has tree roots, and they are and (ii)If , , and , then equation (6) has tree roots, and they are and (iii)If , , , and , then equation (6) has five roots, and they are , , and (iv)If , then equation (6) has no real root(v)If and , then equation (6) has at least one positive real root(vi)If and and there exists such that , then equation (6) has at least two positive real roots(vii)If , , , , and and there exists such that , then equation (6) has two positive real roots
Proof. (i), (ii), and (iii) are easy to prove, and we do not show the process of proof.(iv)If , then and . Thus, we can obtain that equation (6) has no real root.(v)If and , then and , and we can obtain that equation (6) has at least one positive real root.(vi)If and and there exists such that , we can obtain and , and there exists and such that . Thus, equation (6) has at least two positive real roots.(vi)If , , , , and and there exists such that , we can obtain that , , , and have only one positive real root. Furthermore, from and , we get that equation (6) has two positive real roots. This completes the proof.Let and . Omitting the tilde, then system (5) can be rewritten aswhereThe characteristic equation of system (7) at the zero equilibrium isNext, 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 , then the following is obtained:(i)If and , is a single root of equation (9)(ii)If , and , is a double root of equation (9)(iii)If , all the roots of equation (9) have negative real parts except for the zero rootsHere, and .
Proof. Clearly, . By calculating, we can obtain the following result:It is easy to obtain if and , then . Thus, (i) holds.
If and , then and . Thus, (ii) holds.
If , , and , equation (9) has roots and . When , let be a root of equation (9); then, we haveLet ; then, equation (12) can be rewritten aswhere . If , it results in . 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 by choosing and as bifurcation parameters.
Taking , , and , system (5) can be rewritten aswhere and are perturbation parameters.
The phase space is chosen as the Banach space of the continuous mappings from [0,1] to. For , we define and . System (14) becomeswhere and are operators, given bywhere is the Dirac delta function, , and .From Lemma 1, equation (6) has a double-zero root, and all other eigenvalues have negative real parts. Let be the set of eigenvalues with zero real part; can be decomposed as , where is the generalized eigenspace associated with which has two zero eigenvalues and is the space adjoint with , and . Next, we defineand the bilinear form on is
Lemma 3 (see [24, 28, 29]). The bases and for and can be chosen such that , where , , , , and , , which satisfy(i).(ii).(iii).(iv).(v).(vi).
By calculating, we can obtainwhere and .
Let , where , namely,
From [30, 31], system (14) can be written aswhere
From [26], we can obtain the following result:where
The following normal form with versal unfolding on the center manifold can be obtained by some calculations:where , , , and . The detailed calculations can be found in Appendix.
By calculating, we can get
Thus, the map is regular, and system (14) is equivalent to the normal form (26), where .
Let , , and ; system (26) can be rewritten aswhere and .
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 and 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(2)System (28) undergoes a stable Hopf bifurcation on the set(3)System (28) undergoes a saddle homoclinic bifurcation on the set
(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 parameters , , , , , , and . Furthermore, based on results of Lemmas 1 and 2, we can calculate, , , , , , , , and . In Figure 1, the bifurcation diagrams of system (16) are composed of codimension-2 bifurcation point and three codimension-1 curves (saddle-node bifurcation curve, Hopf bifurcation curve, and saddle homoclinic bifurcation curve). When the parameters and change in different regions, system (16) will produce different dynamic properties.
To easily analyze the dynamics of system (7), we fix and only choose the value of to change. From Lemma 2, the B-T bifurcation point is obtained. Thus, we give some numerical examples as follows:(i)If setting , there exist a saddle and a stable focus in region (see Figure 1), as shown in Figures 2 and 3(ii)Fix ; Figures 4 and 5 show that a stable periodic solution occurs when bifurcation parameters pass through the Hopf bifurcation line H in region (see Figure 1)(iii)Fix ; Figures 6–8 show that a closed orbit exists through the homoclinic bifurcation line ③ (see Figure 1)
4. Conclusions
In this work, a shape memory alloy oscillator with delayed feedback has been analyzed. We mainly choose the two parameters and to investigate the B-T bifurcation of system (6). It is demonstrated that the feedback parameters and time delay 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 as the control parameter (see [22]). However, the periodic force 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.
Appendix
The following calculations of the norm forms of the equation are based on [25, 26, 32, 33].
Let . Then, system can be decomposed into the following form:where is the restriction of as an operator from to the Banach space . Employing Taylor expansion, system (A.1) becomeswhere denote the homogeneous polynomials of degree in . Then, we can obtain the following form:
Denote as the linear space of homogeneous polynomials and as the operator on with , where . can be decomposed as .
Then, can be written as
The base of can be chosen as follows:
The base of can be chosen as follows:
The base of can be chosen as follows:
Then, we can get the following result:where
Thus, the following normal form with versal unfolding on the center manifold can be obtained by some calculations:where , , , and .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
J. B. Wang carried out the study. L. F. Ma supervised the work and provided the support of funds. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. U1610253), the Key Research and Development Program of Shanxi Province (no. 201603D111004), and the Fund for Shanxi “1331 Project” Key Subjects Construction.