Abstract
In this paper, the bifurcation and chaotic motion of a piecewise Duffing oscillator with delayed displacement feedback under harmonic excitation are studied. Based on the Melnikov method, the necessary critical conditions for the chaotic motion in the system are obtained, and the chaos threshold curve is obtained by calculation and numerical simulation. The accuracy of the analytical result is proved by some typical numerical simulation results, including the local bifurcation diagrams, phase portraits, Poincaré maps, and the largest Lyapunov exponents. The effects of excitation frequency and time delay of the displacement feedback are analytically discussed. It could be found that the critical excitation amplitude will increase obviously with the increase of the excitation frequency, and under the selection of certain parameters, the critical excitation amplitude takes the time delay of 0.58 as the inflection point, which decreases at first and then increases.
1. Introduction
Nonsmooth systems, as a part of nonlinear systems, exist in many mechanical systems, such as vehicle suspension systems and buffers. Compared with smooth systems, nonsmooth systems usually exhibit completely different vibration characteristics. At present, several dynamic behaviors and mechanisms of nonsmooth systems are not very clear, which may have several adverse effects on the dynamic characteristics of the systems. Therefore, exploring the complex dynamics of nonsmooth dynamic systems and their bifurcation and chaos phenomena is very important.
In recent years, many scholars at home and abroad have shown close attention to the motion characteristics and theoretical systems of nonsmooth dynamic systems, and various nonsmooth dynamic models have been established. Hu [1, 2] found that the nonsmoothness of the vector field of the continuous piecewise linear system destroys the second-order differentiability of the Poincaré map at the fixed point, which is the root cause of the singularity of the system. Luo [3] analyzed various unstable and stable periodic motion patterns of piecewise linear systems under periodic excitation. Zhang et al. [4] conducted an in-depth study on the nonsmooth system caused by the coupling of different scales in the frequency domain and revealed the mechanism of its complex oscillation. Liu [5] combined the theoretical results of classical smooth systems with the dynamic theory of nonsmooth systems that had been preliminarily formed, studied nonsmooth systems in practical engineering, and explored the effective methods for the stability analysis of nonsmooth dynamic systems and the deeper dynamic law. He [6] applied a numerical algorithm to analyze the dynamic problem of a piecewise linear system and studied the dynamic behavior of tensegrity structures under simple harmonic load, including stable periodic motion, quasiperiodic motion, chaos, and bifurcation, which proved that this method had high accuracy and efficiency. Ma [7] studied the Melnikov method and homoclinic chaos control of the global dynamics of a planar nonsmooth oscillator and obtained sufficient conditions of chaos control. Numerical simulation with MATLAB further proved the effectiveness of the homoclinic chaos suppression method. In addition, Wang et al. [8, 9], Wang et al. [10]and Ye and Zhang [11] have also studied the chaos control and chaotic motion characteristics of nonlinear systems and nonlinear piecewise systems in recent two years.
Time delay generally exists in dynamic systems. In the past, when dealing with nonlinear dynamic systems, time delay was often ignored to solve problems. With the increasingly refined requirements for control systems and the in-depth study of time-delay systems, time-delay systems have demonstrated a self-evolutionary trend, which further shows that the evolution of time-delay systems does not exist in isolation.
In recent years, many academic teams in China have carried out systematic research on time-delay dynamic systems and made various research advances, including the dynamics and control of stochastic systems with time delay [12–19], time-delay feedback control design [20], and experiments [21–23]. Jia and Cai [24] studied the synchronization of fractional-order time delay chaotic systems and used methods such as state observers to synchronize the time delay chaotic systems. Qu et al. [25] took the 1/4 car semiactive suspension as the research object, considered the time delay and nonlinear damping factors, and established the suspension vertical vibration system model. In the study, the harmonic expansion of the system dynamic equation was carried out with the incremental harmonic balance method, the steady-state response of the system was obtained by iterative calculation, and the correctness of the incremental harmonic balance method was verified by the Runge–Kutta method. For a class of discrete systems with state-varying time delays and uncertain disturbances, Wu et al. [26] constructed a predictive model using linear sliding mode functions and designed a new reference trajectory in combination with double power functions. According to predictive control theory, the performance index function was defined, and the stability of the system was proven.
The effect of time delay has been paid attention to. Even a small amount of time delay will affect the stability and dynamic performance of the entire dynamic system, resulting in complex dynamic behavior [27–31], such as bifurcation and chaos [32]. A deep study of multiple parameters such as time delay and the change trend when hybrid vibration occurs can better ensure the stability of the nonlinear system under time delay and avoid serious damage to the good performance of the system due to time delay.
Discussing bifurcation and chaos under different parameters of the nonlinear dynamic equation under the time-delay state can help understand the influence of each parameter on the nonlinear time-delay system to verify the influence of the parameters on the system further. In this paper, the Melnikov method is used to study the necessary conditions for the occurrence of chaos in a piecewise Duffing oscillator with time delay displacement feedback under simple harmonic excitation. This paper is organized as follows: in the first section, the Melnikov function is established based on the Melnikov method, and the necessary conditions for chaos in the piecewise Duffing oscillator with time delay displacement feedback are obtained. In the second section, the correctness of the analytical solution is verified by comparing the numerical simulation results with the analytical results. In the third section, the influence of excitation frequency and time delay on the chaotic behavior of the system is analyzed. Finally, the conclusions obtained are summarized and analyzed.
2. Necessary Conditions for the Chaos of the Duffing Oscillator
To effectively control the influence of the chaotic behavior of the piecewise nonlinear constrained system on the stability of the system, the vibration displacement of the system is used as the control quantity. Linear time-delay displacement feedback control is introduced, and based on this, the time delay displacement feedback control model of the piecewise Duffing oscillator is established.where is the linear stiffness coefficient, is the viscous damping coefficient, is the excitation amplitude, ω is the excitation frequency, is the feedback gain, and is the amount of time delay.
is the piecewise nonlinear spring force, which is expressed as
is the time delay displacement feedback controller.
2.1. Homoclinic Orbit of the Piecewise Duffing Oscillator
Equation (1) can be transformed into an equation of state as follows:
The Hamiltonian function of the system (3) is
When , that is, when system damping does not exist, the formed homoclinic orbit satisfies the differential equation as follows:
Assuming , can be obtained, and
After definite integral calculation and simplification by the above equation, the parameter equations of two homoclinic orbits can be obtained at the saddle point of the system as follows:where “” of represents the positive axis of the homoclinic orbit, “” represents the negative axis, “” of is the upper half axis of the homoclinic orbit, and “” represents the lower half axis.
2.2. Necessary Conditions for Determining Chaos by Melnikov Function
The basic idea of Melnikov’s method is to reduce the dynamic system to a Poincaré map on the plane and study whether the map has the mathematical conditions for transversal homoclinic or heteroclinic orbits to determine the criterion of chaotic phenomena in the sense of the Smale horseshoe [33].
According to the Melnikov process, the Melnikov function of the homoclinic orbit of the system can be defined aswhere “” represents the wedge product of a 2D vector.
In the above equation, suppose t0 is the initial moment of any real number, and and are
The Melnikov function can be obtained as follows:
The results of equation (10) are as follows:
2.3. Chaos Prediction
The definite integral of equation (11) can be divided into three parts, namely, , , and , which can obtain
The calculation results of , and are as follows:
Substituting equations (13), (14), and (15) into (11), one could have
Finally, the necessary conditions for chaos in the sense of Smale’s horseshoe are derived as
To simplify the above expression, let and . Then,
3. Numerical Simulation
To verify the correctness of the analytical analysis in this paper, the numerical simulation results are compared with the analytical results, and the basic parameters of the system are selected as = 30, = 20, = 10, c = 0.5, and μ = 0.001. When τ = 1.2 and ω = 3.9, the partial bifurcation diagram of the system is shown in Figure 1, which clearly exhibits the evolution of the system into chaos after period-doubling bifurcation:
When F = 37.15, the system motion state is a single-period motion, and the system phase diagram and the Poincaré section are shown in Figures 2(a) and 2(b). At this time, the phase diagram of the system is a closed curve, and the Poincaré section is a point. The system is a single-period motion.
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With the increase of the excitation amplitude, when excitation amplitude F = 62.80, the system phase diagram and Poincaré section are shown in Figures 3(a) and 3(b). The phase diagram consists of two closed curves, and the Poincaré section diagram consists of two isolated points. Evidently, as excitation amplitude increases, the system bifurcates and moves from period 1 to period 2.
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As excitation amplitude continues to increase, when excitation amplitude F = 73.50 N, the phase diagram is shown in Figure 4(a), and the Poincaré section is shown in Figure 4(b). The trajectories of the phase diagram of the system are entangled with each other, and the Poncaré section diagram is chaotic. They all reflect that the system is in a state of chaotic motion.
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The above details show that with the continuous increase of excitation amplitude, the system gradually enters chaotic motion from periodic motion through period-doubling bifurcation. When excitation amplitude F = 74.30, the system produces chaotic motion, and when selecting the excitation amplitude value of 70–77, the maximum Lyapunov index diagram corresponding to the partial bifurcation diagram in Figure 1 is shown in Figure 5, which clearly illustrates that when external excitation amplitude value is 74.3 (point P in the figure), the system enters a state of chaotic motion. In the differential equation with time delay, the calculation of the largest Lyapunov exponent is based on the transformation from the circumferential equation with time delay to the infinite-dimensional ordinary differential equation or iterative mapping:
When , the infinite-dimensional delay differential equation is expressed as
Because the system is infinite-dimensional, when in the continuous range, an infinite number of initial conditions are needed, and the system can be approximated by an infinite-dimensional ODE system. For nonlinear functions, it is necessary to approximate the continuous evolution of X by the following form of iterative mapping with discrete (but small) time steps:
In the above equation, , represents the dimension of the iterated map. When , the eigenvalues of the flow in the vicinity of the stable equilibrium can be obtained and expressed by Lambert function [34], then the largest Lyapunov exponent can be obtained [35]. Some detailed numerical calculation formats and some typical examples can be listed in references [35, 36].
Based on the analytical equation (17) obtained by the previous analysis, the necessary condition for the system to produce chaos under the same conditions is the excitation amplitude F = 74.06 can be concluded. Comparing the numerical solution with the analytical solution, the difference is 0.24.
To verify the correctness of the analytical results further, the relationship curve between critical excitation amplitude F and excitation frequency ω for chaos in the piecewise Duffing system, determined by equation (17), is shown as the solid line in Figure 6. At the same time, excitation frequency ω is selected as 3.1, 3.3, 3.4, 3.5, 3.6, 3.7, and 3.9, and the critical excitation amplitude of chaos obtained by numerical simulation F is shown in Figure 6. A comparison reveals that the same as in the situation when excitation frequency is 3.9 or above, a certain error exists between the analytical solution and the numerical solution because the result obtained by the Melnikov function is a first-order approximation; thus, a quantitative difference exists between the two. However, the analytical solution qualitatively agrees with the numerical solution, and the error between them is within a reasonable range.
4. Influence of Excitation Frequency and Time Delay on the Chaotic Motion of the Piecewise Duffing System
This section mainly studies the influence of excitation frequency ω and time delay on necessary conditions of chaos in the Smale horseshoe sense, that is, the threshold of the system from periodic motion to chaotic motion.
First, the influence of the external excitation frequency on the chaotic motion of the system is analyzed. The solid line graph in Figure 6 is the relationship between the critical excitation amplitude and the excitation frequency of chaos generated by the system obtained by analytical result (17). With the increase of external excitation frequency , the critical excitation amplitude of chaos also increases remarkably, with a noticeable growth rate.
The basic parameters remain unchanged. When selecting ω = 3.5, τ = 1.2, further simulation analysis verifies this conclusion. Figure 7 is a partial cross diagram of the system. Figures 8–10 are the single-period motion, two-period motion, and chaotic motion of the system, respectively. Combining Figures 7–10 shows that when excitation amplitude F = 53, chaotic motion occurs. Compared with the critical excitation amplitude value F = 74.3 obtained when ω = 3.9, the higher the excitation frequency is, the greater the critical excitation amplitude value for chaos in the system, which means that under the same conditions, increasing the excitation frequency can restrain the chaotic motion of the system to a certain extent.
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Next, the influence of time delay on the chaotic motion of the system is analyzed. According to equation (17), the relationship between critical excitation amplitude and time delay when the piecewise Duffing system produces chaos is obtained as shown in Figure 11, represented by the solid line. The small circle in the figure is still the result of a numerical simulation.
Figure 11 shows that when , the corresponding critical excitation amplitude when the system appears chaotic motion gradually decreases as increases; when , the corresponding excitation amplitude when the system appears in chaotic motion gradually increases as increases; when , the excitation amplitude increases more slowly as increases. Equation (14) shows that the critical excitation amplitude for chaos in the system is mainly affected by the analysis results and , and q1 and depend on the linear stiffness and time delay of the system.
If the linear stiffness of the system is constant, the change of and only depends on the time delay of the system. When time delay τ is small, critical excitation amplitude is mainly affected by . At this time, with the increase of τ, gradually becomes smaller. Therefore, when time delay is small, excitation amplitude decreases with the increase of time delay; when τ increases to a certain value, the critical amplitude of system chaos is gradually affected by , gradually increases with the increase of , and its increasing effect on the critical value is greater than the decreasing effect of . Thus, as τ increases, system critical excitation amplitude also gradually increases; when time delay τ continues to increase to a certain extent, and approach a constant value. Hence, the system critical excitation amplitude gradually tends to be constant with the increase of the time delay τ.
A set of parameters is still selected to verify this conclusion. The basic parameters remain unchanged; taking , , the partial bifurcation diagram of the piecewise Duffing system is obtained by numerical simulation, as shown in Figure 12. From Figure 12 and the corresponding phase diagram and Poincaré diagram under different excitation amplitudes (as shown in Figures 13–15), chaos occurs in the system when excitation amplitude is concluded. This result is compared with the case of and , and it accords with the relationship between the time delay τ obtained by analysis and the excitation amplitude F corresponding to the occurrence of chaos. The author selects other points for verification, and the results all agree with the relationship curve between the excitation amplitude and time delay shown in Figure 7. Thus, the above conclusion is correct. When time delay is in the range of , the generation of chaotic motion can be delayed by reducing time delay. When time delay is in , the generation of chaotic phenomena in the system can be delayed by increasing time delay.
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5. Conclusions
In this paper, the Melnikov method is used to study the chaotic motion of the piecewise Duffing oscillator with time delay displacement feedback, the necessary conditions for chaotic analysis in the sense of Smale horseshoe are studied, and the analytical results obtained. The influence of excitation frequency and time delay on the critical excitation amplitude of the piecewise Duffing system is analyzed, and numerical verification is carried out.
The results show that as the excitation frequency increases, the critical excitation amplitude increases when chaotic motion occurs, and the increase is apparent. By taking different excitation frequencies, the numerical solution is compared with the analytical solution. The findings reveal that the overall trend is consistent, and the qualitative results are the same. The necessary condition for the Melnikov method to analyze the chaotic system is the first-order approximate result. Therefore, a certain difference between the numerical result and the analytical solution is acceptable, and the excitation frequency is appropriately increased to increase the excitation amplitude of chaotic phenomena and delay the occurrence of chaos. Time delay also affects the critical excitation amplitude of the chaotic motion of the system. Under the conditions of this paper, when the time delay is less than 0.58, the chaotic critical excitation amplitude of the Duffing system decreases gradually as the time delay increases. In this range, the generation of chaos can be suppressed by reducing the time delay. When the time delay is greater than 0.58 and less than 1.2, the critical excitation amplitude of chaos of the piecewise Duffing system increases with the increase of time delay. Therefore, within this range, the generation of chaos in the system can be delayed by increasing the time delay to prevent the chaotic behaviors of the system better.
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
Project supported by the National Natural Science Foundation of China (no. 11872256) and the Department of Education of Hebei Province to cultivate the innovative ability of graduate students (no. YC2021042).