Chaotic Threshold of a Nonlinear Zener Systems Based on the Melnikov Method

Fan, Shutong; Shen, Yongjun; Wen, Shaofang

doi:https://doi.org/10.1155/2022/1680496

Mathematical Problems in Engineering

On this page

Abstract Introduction Numerical Simulation Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1680496 | https://doi.org/10.1155/2022/1680496

Chaotic Threshold of a Nonlinear Zener Systems Based on the Melnikov Method

Shutong Fan,¹Yongjun Shen,^1,2and Shaofang Wen^2,3

Academic Editor: Nicholas Alexander

Received05 Feb 2022

Revised01 Mar 2022

Accepted08 Mar 2022

Published06 Apr 2022

Abstract

Regarding a nonlinear Zener model with a viscoelastic Maxwell element as the research object, the complicated dynamic behaviors such as homoclinic bifurcation and chaos under harmonic excitation are investigated. At first, the analytically necessary condition for chaos in the sense of Smale horseshoe is derived based on the Melnikov method. Then, the system parameters that meet the analytical condition and the main resonance condition are selected for the numerical simulation. From the bifurcation diagrams and the largest Lyapunov exponents, it is found that the homoclinic orbit breaks, and the system goes to chaos in a crisis way when the excitation amplitude passes the first threshold. The system enters another new chaotic state in the form of period-doubling bifurcation with the increase of the excitation amplitude. At last, the effects of nonlinear term, stiffness coefficient and damping coefficient of Maxwell element on the analytically necessary condition for chaos are analyzed, respectively, and the correctness of the analytical result is proved by numerical simulation. The research result shows that the critical excitation amplitude decreases with the increase of nonlinear term. In addition, the chaotic threshold increases first and then tends to remain unchanged with the raise of stiffness coefficient. The chaotic threshold increases first and then decreases with the enhancement of damping. These results provide a theoretical basis for the research of nonlinear viscoelastic system in the future.

1. Introduction

With the development of the society, viscoelastic materials have been developed rapidly in the field of vibration control and have shown superior benefits of reducing noise and vibration. As a result, it is widely applied in aviation [1, 2], machinery [3], civil engineering [4, 5], and many other fields, such as magnetorheological damper in automobile suspension [6], metal rubber in spacecraft vibration reduction [7], and air spring in high-speed train bogie [8]. To better analyze the characteristics of viscoelastic materials, a variety of simplified mechanical models were put forward. Among them, there are the two most classical models which are the Kelvin model and the Maxwell model, but each model has its drawbacks. The Kelvin model cannot reflect stress relaxation, and the Maxwell model cannot present the creep [9]. Therefore, scholars proposed the Zener model composed of a spring and a Maxwell element in parallel, which can not only be degenerate into the Kelvin model or the Maxwell model but also reflect the stress relaxation and creep characteristics of viscoelastic materials.

Specifically, in viscoelastic Zener system, there has been a lot of research studies on the vibration isolation performance and periodic solution of the model. Ruzicka [10] made a comprehensive analysis of its forced vibration response. Brennan et al. [11] studied the effects of stiffness and damping on the performance of the viscoelastic Zener system under the free or forced vibration. Wang et al. [12] presented the design method for the optimal damping of a three-parameter vibration isolator. Lou et al. [13] found that the Zener system has better impact resistance. After that, Brennan et al. [14–18] have conducted extensive research on nonlinear isolation systems. In [14], the vibration transmission characteristics of single degree-of-freedom passive vibration isolation systems with different nonlinear dampers were considered. In [15], the dynamic behaviors of a nonlinear two degree-of-freedom system are discussed and the interaction between the linear dynamics of the shaker and the nonlinear dynamics of the isolator is captured. In addition, in [16, 17], the method of nonlinear generation and propagation is found and the dynamic behaviors of the nonlinear system by using the method of dynamic stiffness are explained. To further study the performance of the Zener system, Shi et al. [18] established a four-parameter Zener model and found that the vibration isolation performance of the isolator has been improved in the high-frequency band. Wang et al. [19] and Lucas et al. [20] used nonlinear cubic stiffness elements to replace the spring in Maxwell element and linear mainspring to focus on the vibration isolation behavior, respectively, which improves the transmissibility of the Zener model damping isolator. Liu et al. [21] introduced geometric nonlinearity into the viscoelastic Zener model, which effectively suppresses the dynamic response in a resonant frequency band. Wei et al. [22] and Wang et al. [23] applied nonlinear damping to improve the viscoelastic Zener model, and they found that damped viscoelasticity to simulate the velocity correlation of the support is conducive to reflecting the structural response more realistically. Fan et al. [24] extended the multiscale method to approximately solve the nonlinear Zener model and discussed the influence of system parameters on the dynamic behavior and stability of the system. The studies of viscoelastic Zener system in the above literature only focus on the analytical solutions of linear and simple nonlinear resonance.

In recent years, many scholars still attach great importance to the research on complicated dynamic behaviors such as bifurcation and chaos of nonlinear systems. As one of the analytical methods for judging the chaotic threshold of nonlinear systems, the Melnikov method has been widely used since it was proposed. The Melnikov method [25, 26] can determine whether the system has a transverse zero point by measuring the distance between the stable and unstable manifolds of the hyperbolic fixed point in the nonlinear system and obtains the chaotic threshold in the sense of Smale horseshoe. Cao et al. [27] studied multistable and multiwell dynamic characteristics of a class of coupled typical geometric nonlinear systems and gave the analytical chaotic parameter conditions of homoclinic orbit and heteroclinic orbit based on the Melnikov method. Luo et al. [28] investigated the bifurcation and chaos of transverse vibration of viscoelastic radial transmission structure under nonlinear parametric excitation by the numerical method. For a Duffing oscillator with delayed displacement and velocity feedbacks under harmonic excitation, the necessary conditions for generating homoclinic orbit and heteroclinic orbit were, respectively, investigated by Shen et al. [29] and Wen et al. [30]. Li et al. [31] applied the Melnikov method to globally analyze the Duffing oscillator of the simultaneous primary and super-harmonic resonance. Moreover, Stanton et al. [32] and Li et al. [33, 34] studied the bistable and tristable piezoelectric energy harvesting system via the Melnikov method and concluded that electromechanical coupling factors affect its dynamic behavior. Their works obtained a qualitative research method for homoclinic bifurcation of energy harvesting system under harmonic excitation. The aforementioned literature mainly focuses on the homoclinic or heteroclinic bifurcation thresholds of the two-dimensional plane system.

At present, many researchers are interested in high-dimensional nonlinear systems to explore interesting dynamic behaviors. Yao et al. [35] discussed the multipulse global bifurcation and chaotic dynamics of nonlinear and nonplanar oscillation in the viscoelastic motion band of parametric excited state under resonance via the extended Melnikov method. Their works showed that the extended Melnikov method can be applied to the research of Shilnikov-type multipulse homoclinic bifurcation and chaotic dynamics of high-dimensional nonlinear systems. Davood et al. [36] considered the chaotic behavior of viscoelastic plates under the simultaneous action of subsonic flow and external excitation based on the extended Melnikov method. A detailed study on the dynamics of bistable impact oscillators with double rigid constraints by using the extended Melnikov method and dynamic experiments were conducted by Li et al. [37]. Zhang et al. [38] used the extended Melnikov method to investigate the multipulse jump, two-parameter co-oblique orbit, and chaotic dynamics of eccentric rotating ring truss antenna under parametric and external excitation. Furthermore, Wei et al. [39–44] discussed the hidden attractors contained in 3D, 4D, and 5D autonomous systems in detail, and their work further analyzed these phenomena produced by the system and put forward practical applications.

With regard to the nonlinear Zener model, the analysis of chaotic threshold is one of the focal points. The nonlinear Zener dynamical system with a Maxwell viscoelastic element adds a half degree of freedom, which in turn increases the complexity of the system. Most studies use the harmonic balance method to calculate and analyze the approximate analytical solution of the system, and few studies introduce nonlinear factors into the viscoelastic Zener model to analyze its dynamic behavior. Moreover, the Melnikov method used in most kinds of literature was to study the homoclinic or heteroclinic bifurcation thresholds of two-dimensional plane systems, and the application of the Melnikov method in this kind of nonlinear system proposed in this study is rarely reported. Based on the excellent characteristics of viscoelastic materials, the nonlinear viscoelastic Zener model is seen as the research object in the study. Note that the factors such as damping, viscoelastic Maxwell element, and external excitation are regarded as small disturbances in the plane integrable system to study the chaotic motion at resonance. In the analysis of chaotic threshold, it is found that the system enters the chaotic state in two different ways within a certain excitation amplitude range. The effects of system parameters on the necessary condition for chaos are further discussed by numerical simulation and analytical solution. It shows that the proper selection of system parameters can suppress the bifurcation and chaos of the system.

The study is arranged as follows. In Section 2, the Melnikov necessary condition for homoclinic bifurcation on the harmonic excitation condition is obtained. Section 3 presents the bifurcation curves, Poincare maps, and the largest Lyapunov exponents by the numerical method. Section 4 analyses the effects of nonlinear coefficients and stiffness and damping coefficients in Maxwell element under the necessary condition for chaos using analytical and numerical solutions, and the correctness of the analytical solution is proved. Conclusions are drawn in Section 5.

2. Melnikov Analysis of Nonlinear Viscoelastic System

This section investigates the nonlinear Zener system with linear damping, linear stiffness, and nonlinear terms, as shown in Figure 1, where m and k are the mass and linear stiffness coefficient of the system, respectively, α is the cubic nonlinear stiffness coefficient, c and k₁ are the damping coefficient and stiffness coefficient of Maxwell viscoelastic element, and F and ω are the amplitude and frequency of external excitation, respectively.

Obviously, this nonlinear Zener model is a 1.5 DOF system. According to Newton’s second law, the dynamic equation is written as [10, 24]

Here, considering the situation of k < 0 in system (1) and introducing the transformation,where is a small real parameter, and equation (1) can be simplified into the following form:

If , , and , the state equation of equation (2) is

In this study, assuming that the factors such as damping, viscoelastic Maxwell element, and external excitation are the small disturbances of equation (3) corresponding to Hamilton system, the undisturbed system is

Then, the homoclinic orbit through integral equation (4) is

Substituting equation (5) into the third equation of equation (3), we can obtain

Then, the calculation result of equation (6) iswhere is hypergeometric function and is polygamma function. To validate the analytic solution, (7) is numerically simulated by MATLAB, and Figure 2 demonstrates the consistency between the analytical and the simulation results. From the sophisticated form of the analytical solution of , the numerical solution is adopted in the following analysis.

Define the undisturbed system that does not contain :

Among them,

According to the Melnikov method, the Melnikov function is established aswhere denotes a wedge product. The definite integral of equation (9) can be divided into the following three parts:where . Since the analytical solution of is hard to be solved, the form of Simpson numerical integration is expressed approximately. Through the analysis of the above results, the necessary condition for homoclinic orbit cross-section intersection in equation (1) is

Substituting the original system parameters into equation (13), it could yieldthat is,

When inequality (15) holds, the homoclinic orbit of system (1) can cross intersect so that system (1) may enter chaos in the sense of Smale horseshoe. Notably, if the model in this study is simplified to a single degree-of-freedom system, the analytical result of the threshold of chaos is consistent with those in [25, 29], which indirectly proves the correctness of the analytical results in this study.

3. Numerical Simulation

By taking the system parameters , , , , , , and and substituting them into equation (13), the critical excitation amplitude for chaos is 0.133. In other words, when F > 0.133, the homoclinic orbit cross section of equation (1) may occur, and the greater excitation amplitude is than the threshold, the more likely chaos is to occur in the system.

In order to analyze the critical value of generating chaos more accurately, the trapezoidal method of free interpolation [45, 46] and the Lyapunov exponent are introduced to simulate equation (1) at different excitation amplitudes. The Lyapunov exponent, which is a reliable quantitative analysis method to judge whether the system, enters chaos by measuring the average change rate of separation or convergence of adjacent phase orbit traces and is calculated by the Jacobian method. Here, the initial value of F is 0 and the initial value is , and the simulation time t is set as . When F changes from 0 to 5 with the step size as 0.02, one can get the bifurcation diagram (Figure 3) and the largest Lyapunov exponent chart (Figure 4), respectively. From Figures 3 and 4, it can be found that there are two critical values, where the first critical value about 0.3 is the numerical threshold and the second critical excitation amplitude around 3 denotes the second occurrence from periodicity to chaos. Thus, the detailed bifurcation diagram and the variation of the largest Lyapunov exponent are shown in Figures 5–7, and the step sizes of Figures 5 and 7 are selected as 0.001 and 0.0005 individually.

Around the first critical value, i.e., when F is greater than 0.203, the system enters the chaotic state in a crisis way as shown in Figure 5, which is consistent with the sharp increase of the largest Lyapunov exponent of F at 0.3 in Figure 6. Figure 8 portrays the periodic orbit via taking F = 0.203 and the initial value (1.25715268823769, 0.111307241041528, 0.158363049145063, -0.119905053458755). Then, when F = 0.204 and the initial value (1.26119984719097, 0.115510861798501, 0.167496720727088, -0.121280433769164), the displacement time history, phase trajectory, and Poincare map are obtained as plotted in Figure 9. From Figures 8 and 9, it can be concluded that the homoclinic orbit breaks, and the system goes to chaos in a crisis way when the excitation amplitude increases from 0.203 to 0.204. Otherwise, at the second critical value, Figure 7 presents that, with the continuous increase of excitation, the system enters another chaos in the form of period-doubling bifurcation, where F = 3.2. At this point, the displacement time history, phase trajectory, and Poincare map are acquired by taking the initial value of the system as (2.2320514013694006, 0.23099234373530025, −1.85658853180 5132, −2.434406911433444), as shown in Figure 10.

(a)

(b)

(c)

(a)

(b)

(c)

4. Influence of System Parameters on Chaotic Motion

In this section, we are interested in the effects of nonlinear coefficient and stiffness and damping coefficients in Maxwell element under the necessary condition for chaos through analytical solution and numerical simulation. The numerical simulation uses ode23 t in MATLAB to calculate the nonlinear Zener system, that is, (1), and the unstable response in the calculation result is rounded off to ensure the stability of the solution.

The system parameters, , , , , , and , are selected to analyze the influence of nonlinear term α with the value range of 0 to 2. The relationship curves between chaotic threshold F and nonlinear coefficient α are obtained by MATLAB numerical simulation and Melnikov analytical method, respectively, in this study, as depicted in Figure 11. The numerical solution of the critical excitation amplitude is composed of small circles, and the analytical solution is consisted of asterisks. From the analysis of Figure 11, it can be found the chaotic threshold gradually decreases as the nonlinear coefficient increases. Through the comparison of analytical and numerical results, it can be observed that although there are quantitative differences between the two results, the overall trend is the same. This difference is reasonable because the Melnikov method is an approximate method to find chaotic motion. In short, the increase of nonlinear term can promote the bifurcation and chaos of the system.

Other two sets of system parameters are selected as , , , , , and , , , , , and , respectively, and the effects of stiffness coefficient and damping coefficient in viscoelastic Maxwell device are considered. At this time, takes 0 to 3 and takes 0 to 2, and the relationship curves between F and k₁ and the change trend of F and with c are shown in Figures 12 and 13, respectively. Figure 12 indicates that the tendencies of the numerical and analytical solutions are all increasing first and then remaining are unchanged with the increase of stiffness coefficient. It can be concluded from Figure 13 that the threshold F first increases and then decreases, and the error between the numerical solution and analytical solution increases with the increase of damping. Therefore, within a certain damping range, the proper selection of damping coefficient can inhibit the bifurcation and chaos of the nonlinear Zener system.

5. Conclusion

In this study, the chaotic dynamics of the nonlinear Zener model with a viscoelastic Maxwell device is investigated based on the Melnikov method, and the analytically necessary condition for chaos in the sense of Smale horseshoes is obtained. The system parameters satisfying the main resonance condition are selected and simulated by numerical iteration method. From the analysis of numerical simulation, it can be found that there are two different paths to chaos in a certain range of excitation amplitude, and these phenomena are confirmed by analyzing its dynamic response. At the first critical value, the system enters chaos in a crisis way. While at the second critical value, the system generates chaos in the way of period-doubling bifurcation. The effects of system parameters on the necessary condition for chaos are analyzed, and by comparing the threshold of chaos generated by the numerical and analytical solutions under different parameters, it is found that the trend of the analytical solution is consistent with the numerical iterative simulation, and the qualitative results are the same. It can be concluded that the chaotic threshold changes differently with the change of system parameters, which provides a basis for the research of this kind of the viscoelastic system. Above all, the chaotic motion of the system at resonance is studied in this study. The method of considering the factors such as damping, viscoelastic Maxwell element, and external excitation as the small disturbance of the system plane integrable system can be extended and applied to other similar nonlinear viscoelastic systems, which provides a new research idea for the dynamics of nonlinear viscoelastic systems.

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.

Acknowledgments

The authors are grateful to the support by the National Natural Science Foundation of China (12072206 and U1934201) and the Education department project of Hebei Province (ZD2020310).

References

X. D. He, X. C. Huang, and H. X. Hua, “Design of viscoelastic-dry friction damping ring for micro-vibration suppression of a flywheel,” Journal of Mechanical Engineering, vol. 56, no. 23, pp. 98–106, 2020.
View at: Google Scholar
H. C. Wu and Q. K. Han, “Dynamic characteristics analysis of aero engine casing structures with viscoe- lastic damping materials under elastic supports,” Journal of Aerospace Power, vol. 35, no. 7, pp. 1413–1424, 2020.
View at: Google Scholar
D. P. Zhang, D. Wu, and Y. J. Lei, “Vibration characteristics analysis of flexoelectric Timoshenko nanobeams in viscoelastic medium,” Journal of National University of Defense Technology, vol. 43, no. 6, pp. 8–16, 2021.
View at: Google Scholar
M. S. E. Nasab, M. M. Javidan, S. Chun, and J. Kim, “Experimental study on seismic retrofit of a RC frame using viscoelastic dampers,” Structures, vol. 34, no. 1, pp. 771–786, 2021.
View at: Publisher Site | Google Scholar
S. L. Jia, B. W. Wang, Q. Hu, and M. J. Chang, “Nonstationary seismic response analysis of energy dissipation system with general viscoelastic damping supported by braces,” Structural Engineers, vol. 37, no. 6, pp. 99–113, 2021.
View at: Google Scholar
H. You, Study on Fractional-Order Models and Application of Magneto-Rheological Damper in Vehicle Suspension, Shijiazhuang Tiedao University, Shijiazhuang, China, 2017.
Y. J. Chang, W. W. Tian, and E. L. Chen, “Dynamic model for the nonlinear hysteresis of metal rubber based on the fractional-order derivative,” Journal of Vibration and Shock, vol. 39, no. 14, pp. 233–241, 2020.
View at: Google Scholar
H. Zhu, J. Yang, Y. Zhang, and X. Feng, “A novel air spring dynamic model with pneumatic thermodynamics, effective friction and viscoelastic damping,” Journal of Sound and Vibration, vol. 408, no. 1, pp. 87–104, 2017.
View at: Publisher Site | Google Scholar
T. Q. Yang, W. B. Luo, and P. Xu, Viscoelastic Theory and Application, Science Press, Beijing, China, pp. 13–24, 2004.
E. Ruzicka, Forced Vibration in Systems with Elastically Supported Dampers, Massachusetts Institute of Technology, Cambridge, MA, USA, 1957.
M. J. Brennan, A. Carrella, and T. Waters, “On the dynamic behaviour of a mass supported by a parallel combination of a spring and an elastically connected damper,” Journal of Sound and Vibration, vol. 309, no. 3, pp. 823–837, 2008.
View at: Publisher Site | Google Scholar
C. Wang, J. Y. Sun, and Z. Y. Zhang, “Design and experiment of a three-parameter isolation system with optimal damping,” Journal of Mechanical Engineering, vol. 51, no. 15, pp. 90–96, 2015.
View at: Publisher Site | Google Scholar
J. J. Lou, S. J. Zhu, and S. M. Tang, “Study on effect of Ruzicka isolator on shock isolation,” Noise and Vibration Control, vol. 31, no. 4, pp. 37–41, 2011.
View at: Google Scholar
B. Tang and M. J. Brennan, “A comparison of two nonlinear damping mechanisms in a vibration isolator,” Journal of Sound and Vibration, vol. 332, no. 3, pp. 510–520, 2013.
View at: Publisher Site | Google Scholar
G. Gatti and M. J. Brennan, “On the effects of system parameters on the response of a harmonically excited system consisting of weakly coupled nonlinear and linear oscillators,” Journal of Sound and Vibration, vol. 330, no. 18-19, pp. 4538–4550, 2011.
View at: Publisher Site | Google Scholar
J. C. Carranza, M. J. Brennan, and B. Tang, “Sources and propagation of nonlinearity in a vibration isolator with geometrically nonlinear damping,” Journal of Vibration and Acoustics, vol. 138, no. 2, pp. 1–6, 2016.
View at: Publisher Site | Google Scholar
B. Tang, M. J. Brennan, and G. Gatti, “Use of the dynamic stiffness method to interpret experimental data from a nonlinear system,” Journal of Sound and Vibration, vol. 421, no. 1, pp. 91–110, 2018.
View at: Publisher Site | Google Scholar
W.-k. Shi, C. Qian, Z.-y. Chen, Y. Cao, and H.-h. Zhang, “Modeling and dynamic properties of a four-parameter zener model vibration isolator,” Shock and Vibration, vol. 2016, no. 1, pp. 1–16, 2016.
View at: Publisher Site | Google Scholar
X. Wang, H. Yao, and G. Zheng, “Enhancing the isolation performance by a nonlinear secondary spring in the Zener model,” Nonlinear Dynamics, vol. 87, no. 4, pp. 2483–2495, 2017.
View at: Publisher Site | Google Scholar
H. S. Lucas, J. Paulo, and G. Paupitz G, “On the dynamic behavior of the Zener model with nonlinear stiffness for harmonic vibration isolation,” Mechanical Systems and Signal Processing, vol. 112, no. 1, pp. 343–358, 2018.
View at: Google Scholar
H. P. Liu, D. S. Shen, and P. P. Zhao, “Dynamic performance of a three-parameter isolator with nonlinear characteristic,” Journal of Vibration Engineering, vol. 34, no. 3, pp. 490–498, 2021.
View at: Google Scholar
W. Wei, Y. Yuan, and P. Tan, “A rate-dependent constitutive model of high damping rubber bearings considering the influence of compressive load and its effect on the seismic response,” China Civil Engineering Journal, vol. 53, no. 2, pp. 23–32, 2020.
View at: Google Scholar
J. L. Wang, S. Jin, and Y. Kang, “Comparative structural response of bilinear model and rate-dependent model of high damping rubber bearing,” Journal of Civil Engineering and Management, vol. 38, no. 2, pp. 77–83+90, 2021.
View at: Google Scholar
S. T. Fan and Y. J. Shen, “Extension of multi-scale method and its application to nonlinear viscoelastic system,” Chinese Journal of Theoretical and Applied Mechanics, vol. 54, no. 2, pp. 495–502, 2022.
View at: Google Scholar
J. Guckenheimer J and P. J. Holmes, Nonliner Oscillations, Dynamical Systems and Bifurcations of Vector fields, Springer, New York, NY, USA, 1983.
J. B. Li and F. J. Chen, Chaos, Melnikov Method and New Development, Science Press, Beijing, China, 2012.
Q. Cao, R. Tian, and Y. Han, “Investigations of nonlinear dynamic for SD oscillator,” Journal of Shijiazhuang Tiedao University, vol. 23, no. 1, pp. 32–37, 2010.
View at: Google Scholar
Y. Luo and H. Ding, “Bifurcation and chaos in nonlinear parametric resonance of viscoelastic transporting structures,” Scientia Sinica, vol. 43, no. 2, pp. 199–206, 2013.
View at: Publisher Site | Google Scholar
Y. J. Shen, S. F. Wen, and S. P. Yang, “Analytical threshold for chaos in a Duffing oscillator with delayed feedbacks,” International Journal of Non-linear Mechanics, vol. 98, no. 1, pp. 1–8, 2018.
View at: Publisher Site | Google Scholar
S.-F. Wen, J.-F. Chen, and S.-Q. Guo, “Heteroclinic bifurcation behaviors of a Duffing oscillator with delayed feedback,” Shock and Vibration, vol. 2018, no. 1, Article ID 7213606, 12 pages, 2018.
View at: Publisher Site | Google Scholar
H. Li, Y. J. Shen, and S. P. Yang, “Simultaneous primary and super-harmonic resonance of Duffing oscillator,” Acta Physica Sinica, vol. 70, no. 4, pp. 119–128, 2021.
View at: Google Scholar
S. C. Stanton, B. P. Mann, and B. A. M. Owens, “Melnikov theoretic methods for characterizing the dynamics of the bistable piezoelectric inertial generator in complex spectral environments,” Physica D: Nonlinear Phenomena, vol. 241, no. 6, pp. 711–720, 2012.
View at: Publisher Site | Google Scholar
H. T. Li and W. Y. Qin, “Bifurcation and chaos threshold of bistable piezoelectric vibration energy harvesting systems,” Applied Mathematics and Mechanics, vol. 35, no. 6, pp. 652–662, 2014.
View at: Google Scholar
H. T. Li, H. Ding, L. Q. Chen, and W. Y. Qin, “Homoclinic bifurcations and chaos thresholds of tristable piezoelectric vibration energy harvesting systems,” Applied Mathematics and Mechanics, vol. 41, no. 12, pp. 1311–1322, 2020.
View at: Google Scholar
M. Yao, W. Zhang, and J. W. Zu, “Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt,” Journal of Sound and Vibration, vol. 331, no. 11, pp. 2624–2653, 2012.
View at: Publisher Site | Google Scholar
Y. Davood and N. Hamed, “Chaos prediction in nonlinear viscoelastic plates subjected to subsonic flow and external load using extended Melnikov’s method,” Nonlinear Dynamics, vol. 84, no. 3, pp. 1163–1179, 2016.
View at: Google Scholar
S. Li, H. Wu, X. Zhou, T. Wang, and W. Zhang, “Theoretical and experimental studies of global dynamics for a class of bistable nonlinear impact oscillators with bilateral rigid constraints,” International Journal of Non-linear Mechanics, vol. 133, no. 1, Article ID 103720, 2021.
View at: Publisher Site | Google Scholar
W. Zhang, Y. Zheng, T. Liu, and X. Y. Guo, “Multi-pulse jumping double-parameter chaotic dynamics of eccentric rotating ring truss antenna under combined parametric and external excitations,” Nonlinear Dynamics, vol. 98, no. 1, pp. 761–800, 2019.
View at: Publisher Site | Google Scholar
Z. Wei, I. Moroz, J. C. Sprott, A. Akgul, and W. Zhang, “Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 27, no. 3, Article ID 033101, 2017.
View at: Publisher Site | Google Scholar
Z. We, I. Moroz I, and J. C. Sprott, “Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo,” International Journal of Bifurcation and Chaos, vol. 27, no. 2, p. 1730008, 2017.
View at: Publisher Site | Google Scholar
Z. Wei, W. Zhang, and M. Yao, “On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system,” Nonlinear Dynamics, vol. 82, no. 3, pp. 1251–1258, 2015.
View at: Publisher Site | Google Scholar
W. Zhang and Z. Wei, “Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system,” Nonlinear Dynamics, vol. 82, no. 1, pp. 131–141, 2015.
View at: Google Scholar
Z. Wei, W. Zhang, and Z. Wang, “Hidden attractors and dynamical behaviors in an extended rikitake system,” International Journal of Bifurcation and Chaos, vol. 25, no. 2, Article ID 1550028, 2015.
View at: Publisher Site | Google Scholar
Z. Wei and W. Zhang, “Hidden hyperchaotic attractors in a modified lorenz-stenflo system with only one stable equilibrium,” International Journal of Bifurcation and Chaos, vol. 24, no. 10, Article ID 1450127, 2014.
View at: Publisher Site | Google Scholar
M. E. Hosea and L. F. Shampine, “Analysis and implementation of TR-BDF2,” Applied Numerical Mathematics, vol. 20, no. 1, pp. 21–37, 1996.
View at: Publisher Site | Google Scholar
L. F. Shampine, M. W. Reichelt, and J. A. Kierzenka, “Solving index-1 DAEs in MATLAB and simulink,” SIAM Review, vol. 41, no. 3, pp. 538–552, 1999.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Shutong Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies