Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractionalorder delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system.


Introduction
As an important branch of mathematics, fractional calculus had been studied for more than 300 years.In recent years, it had attracted more attention in a lot of research fields, such as physics, chemistry, mechanics, biology, electromagnetics, materials science, and control engineering [1][2][3][4][5][6][7][8][9][10][11].This was due to the fact that many features in time and space could be explained by the fractional-order calculus model, such as memory and nonlocality.At present, the forms of fractional-order differential systems can be classified into two categories.The first one is to simply add fractional-order derivative term into the original integerorder system, so as to establish a fractional-order system.For example, Shen et al. [12][13][14][15][16] studied several linear and nonlinear fractional-order oscillators by the averaging method or incremental harmonic balance method and found that the fractional-order derivatives had both damping and stiffness effects on the dynamical response in those oscillators.Chen et al. [17,18] studied the response of some nonlinear fractional-order oscillator under Gaussian white noise excitation.Yang et al. [19,20] investigated the stochastic response of nonlinear system with Caputo-type fractional derivative subject to Gaussian white noise.Xu et al. [21] proposed a new technique to deal with strongly nonlinear stochastic systems with fractional derivative damping and random harmonic excitation.The other one is that the classic integer-order derivatives in dynamical system are directly extended to the fractional-order ones, so that a fractionalorder differential system in state space is obtained.This kind of systems include fractional-order Lorenz, van der Pol, and Duffing system, and one could study the stability region, bifurcations, chaos, and its control [22].For example, Li and Peng [23] found that chaos existed in Chen's system with a fractional order by utilizing the fractional calculus techniques.Ahmed et al. [24] proposed some Routh-Hurwitz stability conditions for some fractional-order systems.Li and Wu [25] studied the chaotic behaviors and the Hopf bifurcation in a new fractional-order hyperchaotic system based on the Lorenz system.Čermák and Nechvátal [26] discussed the stability conditions and chaotic behavior of the Lorenz system involving the Caputo fractional derivative with order between 0 and 1.Time delay is more common and inevitable in dynamical and control systems, and it could lead to the instability 2 Shock and Vibration of the dynamical system and the damage to the control performance [27][28][29].At present, some researches had been done on fractional-order and time delay systems.For example, Deng et al. [30] studied the stability of n-dimensional linear fractional differential equation with time delays.Shi and Wang [31] presented the BIBO stability criterion of a fractional-order delayed system.Babakhani et al. [9] studied the existence of solutions at the neighborhood of equilibrium for fractional-order delayed differential equations and the Hopf bifurcations.
In recent years, the research on fractional-order van der Pol oscillator had attracted more and more attention of the scholars.For example, Guo et al. [32,33] studied the steadystate solution of the fractional van der Pol system with time delay via residue harmonic balance technique.Liu et al. [34] analyzed the asymptotic behaviors of the steady-state responses of a fractional van der Pol oscillator by homotopy analysis method and memory-free principle.Xie and Lin [35] investigated the asymptotic solution of the van der Pol oscillator with small fractional damping by using the method of two-scale expansion.Shen et al. [36] obtained the approximately analytical solution for the limit cycle of van der Pol oscillator with two kinds of fractional-order derivatives and analyzed its properties about amplitude and frequency.Wen et al. [37] investigated the influence of fractional-order delayed control on parameter-excited vibration for Mathieu-Duffing oscillator based on the switch of stability.
Different from the aforementioned references, the primary resonance of van der Pol oscillator under fractionalorder delayed feedback and forced excitation is analytically studied by averaging method in this paper.Particularly the effects of all the parameters in fractional-order delayed feedback on the primary resonance of the van der Pol system are studied, and the calculation process of the averaging method in fractional-order system is simplified.The paper is organized as follows.In Section 2 the approximate solution for primary resonance of van der Pol oscillator under fractional-order delayed feedback is obtained.The equivalent stiffness and damping coefficients denoted by the feedback gain, fractional order, time delay, and so on are defined.In Section 3 the stability condition of the steadystate solution is obtained.Then, the results of approximate analytical solution and numerical integration are compared by numerical simulation in Section 4.Moreover, the effects of the parameters in the fractional-order delayed feedback on the amplitude-frequency equation are also given in this section.Finally, the main conclusions are made in Section 5.
Introduce the following transformations: Equation ( 1) becomes where  0 is natural frequency and  is a small positive dimensionless parameter.We focus on the primary resonance by averaging method [12][13][14][15]37], which means  ≈  0 .Hence, one could introduce  2 =  2 0 +  to illustrate the approximation degree, where  is the detuning factor.
Then, (4) can be written as Letting  =  + , the solution of ( 5) can be assumed as where the amplitude  and the phase  are slow-varying functions of time .Substituting (6a), (6b), and (6c) into (5), one could obtain where Applying the averaging method to (7a) and (7b) in time interval [0, ], one could obtain In the above equation, the time  is selected as is an aperiodic one.One could obtain the simplified forms of the first part in (9a) and (9b) In order to calculate the second part in (9a) and (9b), one could use the formula in literature [3] Then, it yields Hence, After a similar calculation, one could obtain This calculation process is simpler than that in the literature [37].Combining (10a) and (10b) with (13a) and (13b), one could obtain Substituting the parameters with the original ones, (14a) and (14b) could be written as Thus, one could get the approximate analytical solution of the system.Reorganizing (15a) and (15b), it yields where are defined, respectively, as the equivalent damping coefficient and the equivalent stiffness coefficient.From (16a), (16b), (17a), and (17b), we could conclude that the feedback gain  1 , fractional-order , and time delay  have important effects on   () and   ().Since the feedback gain  1 is linearly related to   () and   (), it affects the response amplitude and resonance frequency in van der Pol Shock and Vibration oscillator simultaneously.When fractional-order  ̸ = 0, the fractional-order delayed feedback has the functions of both delayed displacement feedback and delayed velocity feedback.When  → 0, fractional-order delayed feedback is almost equivalent to delayed displacement feedback.However, it is almost the same as delayed velocity feedback when  → 1.Moreover, we could find that the amplitude and resonance frequency are affected periodically with the change of .

Amplitude-Frequency Equation and Stability Condition of the Approximate Solution
Now we study the steady-state solution, which is more important and meaningful in vibration control.By putting ȧ = 0 and θ = 0 in (16a) and (16b), we could obtain where  0 and  0 are the amplitude and phase of the steadystate solution, respectively.Eliminating  0 from (18a) and (18b), the amplitudefrequency equation is obtained as follows: For simplicity, one could define Then, another equivalent form of ( 19) can be written as From ( 24), we could see that there may be one or three steadystate solutions in the case of primary resonance.Next, we study the stability of the steady-state solution.Letting  =  0 + Δ and  =  0 + Δ and linearizing (16a) and (16b) at ( 0 ,  0 ), it yields Combined with (18a) and (18b), one could eliminate  0 from (22a) and (22b) and get the characteristic equation as follows: that is, where Based on the Hurwitz criterion, one could obtain the necessary and sufficient conditions for the stability of the steadystate solution as follows: Substituting ( 25) into ( 26), one could obtain the stability conditions as (27b)

Comparison between Approximate Analytical and Numerical Solution.
In order to verify the correctness and precision of the approximate analytical solution, the numerical results of (1) are presented to compare the differences between the approximate analytical solutions and the numerical solutions.An illustrative example system is studied herein as defined by system parameters:  1 = 1,  = 4,  = 10,  1 = −0.25, = 0.5, and  = 1.4.
Here we select time delay  = 3, 2.25, and 1.5, respectively, so that one could obtain three different response modes of amplitude-frequency curve shown in Figure 1, where the solid line is for the stable solution and the dot line is for the unstable one.
Next the numerical formula [22] is adopted as where   = ℎ ( = 1, 2, 3, . ..) is the time sample points, ℎ is step length, and    is the fractional binomial coefficient with the iterative relationship as Letting  =  × ℎ, where  is natural number, one could obtain [37] Based on ( 28)-( 30), one could get the numerical iterative algorithm of (1) as follows: where is velocity, and Here we select ℎ = 0.001, and the total computation time is generally 300 s.Omitting the frontal 80% response, we take the maximum value of the posterior 20% response as the steady-state amplitude of the numerical results.As a comparison, the amplitude-frequency curve by numerical integration is also shown in Figure 1 denoted by small circles.From Figure 1, it could be found that the approximate analytical solutions agree very well with the numerical results and achieve satisfactory precision in all the three response modes.the amplitude-frequency curves shown in Figure 2 when the fractional feedback gain  1 varies.From Figure 2, one could find that the response amplitude decreases when the fractional feedback gain  1 decreases gradually, which means the equivalent damping of the system increases with the decrease of  1 .Moreover, the resonance frequency will increase along with the decrease of the fractional feedback gain  1 , which is because the equivalent stiffness coefficient becomes also larger.In this process, the amplitude-frequency curve of the system is shifted to the left.From Figure 2(a), one could find that the topology structure of amplitude-frequency curve is even changed due to the variation of  1 .It can be seen that the decrease of fractional-order feedback gain  1 leads to the increase of the resonance frequency (i.e., natural frequency) of the system, and the amplitude-frequency curve is shown to be shifted and its topology structure is also changed.

Effects of the Fractional
Next we select the system parameters  1 = 1,  = 4,  = 10,  1 = −0.25, and  = 3.When the fractionalorder  changes, one could obtain the amplitude-frequency curves shown in Figure 3.It could be found that the larger the fractional-order , the larger the maximum amplitude.The reason is that the equivalent linear damping coefficient will decrease along with the increase of fractional-order .Moreover, the resonance frequency will be larger along with the increase of the fractional-order , which is due to the fact that the equivalent stiffness coefficient becomes also larger.In this process, the amplitude-frequency curve of the system is shifted to the left.From Figure 3(a), it could be found that the topology structure of the amplitude-frequency curve will be changed due to the variation of .
However, the above analysis is only applicable to the case of  = 3.When time delay  takes another value, the effects of the fractional-order  on this system may vary, which is due to the fact that the fractional-order  is coupling with time delay  as trigonometric function in fractional-order delayed feedback.
Finally, the system parameters are selected as  1 = 1,  = 4,  = 10,  1 = −0.25, = 0.5, and  = 1.4.The amplitude-frequency curves are shown in Figure 4 when time delay  takes some different values.From the observation of Figure 4(a), one could find that the response amplitude of the system increases gradually when time delay  is increased from 0.5.At the same time, the system resonance frequency decreases, the amplitude-frequency curve of the system shifts from right to left, and its topology structure changes.From Figure 4(b), we could see that the response amplitude of the system begins to decrease gradually when time delay  continues to increase.Moreover, the resonance frequency increases, and the system amplitude-frequency curve continues to shift to the left.As shown in Figure 4(c), it could be found that the response amplitude of the system begins to increase gradually again as time delay  continues to increase.Meanwhile, the resonance frequency decreases and the amplitude-frequency curve of the system shifts to the right gradually.It is easy to see that the amplitudefrequency curve of Figure 4(d) varies with time delay  as shown in Figure 4(a), which is due to the equivalent damping and equivalent stiffness contain trigonometric functions in (17a) and (17b).One could obtain  ≈ 2/ 0 = 3.9738.Therefore, time delay  periodically affects the amplitudefrequency curve of the system (see Figure 5).

Conclusions
The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation by averaging method is studied, and the approximate analytical solution is obtained.The steady-state solutions and stability conditions are investigated.The effects of the fractional feedback, the fractional order, and time delay on the solution are analyzed, which are characterized by the equivalent damping coefficient and equivalent stiffness coefficient.Moreover, it is found that the changes of fractional-order delayed feedback parameters may change the amplitude and topology structure of the amplitude-frequency curve.These results have important influence on the dynamical behavior and could be of great significance to the optimization and control of a similar system.

5 Figure 1 :
Figure1: Comparisons of the amplitude-frequency curves by the approximate analytical solution with that by numerical integration in three response modes.

Figure 2 :
Figure 2: Effects of the fractional feedback gain  1 on the amplitude-frequency curves in three response modes ( = 0.5).