An Iteration Scheme Suitable for Solving Limit Cycles of Nonsmooth Dynamical Systems

The Mickens iteration method (MIM) is modified to solve self-excited systems containing nonsmooth nonlinearities and/or nonlinear damping terms. If the MIM is implemented routinely, the unknown frequency and amplitude of limit cycle (LC) would couple to each other in complicated nonlinear algebraic equations at each iteration. It is cumbersome to solve these algebraic equations, especially for nonsmooth systems. In the modified procedures, the unknown frequency is substituted by the determined value obtained at the previous iteration. By this means, the frequency is decoupled from the nonlinear terms. Numerical examples show that the LCs obtained by the modified MIM agree well with numerical results. The presented method is very suitable for solving self-excited systems, especially those with nonlinear damping and nonsmooth nonlinearities.

In principle, the approximations can be obtained to any desired accuracy by the MIM as long as the iteration proceeds.In the MIM, algebraic equations are introduced at each iteration to eliminate the so-called secular terms.In the applications in conservative oscillators, the algebraic equations are linear.Nonlinear damping and nonsmooth terms appear widely in many dynamical systems [14,15].As for the oscillators with nonlinear damping terms, however, very complicated nonlinear algebraic equations have to be solved at each iteration [16].Moreover, the algebraic equations cannot be deduced for systems with nonsmooth nonlinearities.It is necessary and worthwhile, therefore, to propose some approaches to simplify the MIM.This paper will present a modified iteration algorithm by decoupling the unknown frequency from nonlinear terms.

A Modified MIM
Consider a self-excited oscillator ..

𝑥 +𝑓 (𝑥
where the superscript denotes the differentiation with respect to time  and (, ẋ ) is a nonlinear term with damping terms.Assume that system (1) has at least one limit cycle (LC) solution.Since the LC frequency and amplitude are independent of initial conditions, they should be considered as unknowns to be determined at every iteration.Denote the angular frequency as  and introduce the transformation as  = ; thus, we rewrite (1) as subject to the following initial conditions: where  is the unknown LC amplitude and the superscript denotes the differentiation with respect to .Note that  will be approximated as a series {  } by eliminating the secular terms at each iteration stage.In order to obtain the LC, the MIM [16] can be given as with the initial conditions being rewritten at each iteration as Note that the coefficient of the first harmonic in  −1 remains still to be an unknown, that is,  −1 .This unknown will couple with the unknown frequency,  −1 , which will result in a coupled nonlinear term (i.e.,  −1   −1 ) in the right side of (4).If higher powers of   exist, these terms will lead to very complicated functions in  −1 and  −1 .Different from conservative systems,  2  −1 can no longer be considered as an independent unknown.In order to simplify the MIM, therefore, a modified scheme is proposed as As  = 1, we choose  −1 =  0 .In the th iteration,  −2 is a given constant that is obtained at the ( − 1)th iteration.The square of the unknown frequency, that is,  2 −1 , can be treated as an independent parameter, because  −1 appears only in According to the initial conditions, the starting iteration solution can be chosen as It is obvious that as long as the series {  ,  = 1, 2, . ..} and {  ,  = 1, 2, . ..} are convergent, they must converge to the exact solutions.The right-hand side of ( 6) can be expressed by Fourier series as where the harmonic coefficients Here, () is a positive integer denoting the order of the highest harmonic.Approximations  −1 and  −1 are determined by eliminating the secular terms, that is, letting These equations can be solved analytically if  2 −1  −1 is considered as an independent unknown.They can also be numerically solved by Newton-Raphson method.The latter is employed in this study.
Different from the existing procedures [16], as  increases,  2  −1 is always an independent unknown in the modified MIM.Moreover, unknown  −1 does not couple with nonlinear terms.It simplifies the MIM to a large extent, as shown later.

Numerical Examples
Example 1 (system with nonlinear damping terms).The van der Pol equation is chosen to illustrate the previous procedures more clearly: ..
According to the modified MIM, the corresponding iteration scheme is given as Introducing a new time variable  =  −1  at each iteration stage, we rewrite (11) as where the superscript denotes the derivative with respect to .The iteration algorithm begins with an initial solution Then, we obtain the governing equations in  1 () as Equating the coefficients of cos  and sin  to zeros results into which yields that  0 = 1 and  0 = 2. Substituting them into (16), we have Considering initial conditions (14), we can obtain where  1 is to be determined at the next iteration stage.According to iterative scheme (11), the equation in  2 () is deduced as Equate the coefficients of cos  and sin  to zeros: By solving (19) numerically, we can determine  1 and  2 1 .Here, we obtain the second-order approximation and expand it as According to [17], the Lindstedt-Poincare (LP) method provides the second-and forth-order approximate frequency  LP 2 = 1− when compared with  LP 4 .
Figure 1 shows the comparison of the phase planes between iteration solutions (  ) and numerical result.Rapid convergence of   to the numerical result can be observed.Note that all numerical solutions are obtained by the fourthorder Runge-Kutta (RK) integration method.When || > 1, the iteration procedure presented by Chen and Liu [16] does not converge.This is probably the difference between the starting function ( 0 =  0 cos ) and the exact solution is too large.The modified MIM is still effective for || ≤ 1.5.As Figure 2 shows, the LC solution with  = 1.5 obtained by the presented method is in excellent agreement with numerical one.It is necessary to point out that the presented method is able to track unstable LCs, whereas the RK method is not.Also plotted in Figure 2 are the results provided by the LP method [17].The iteration results are much more precise than the 2nd and 4th-order LP approximations.
It is necessary to point out that the presented method is able to track unstable LCs, whereas the RK method is not.Figure 3 shows an unstable LC of the van der Pol equation with  = −1 obtained by the presented method.As shown, the RK begins at the LC; however, the solution curve converges to the equilibrium.In order to further demonstrate the merit of the modified MIM when applied to problems with nonlinear damping terms, we consider the following self-excited system [18]: The nonlinear term contains high powers of ẋ , that is, ( ẋ ) = ( ẋ ) 3 .If the original MIM is employed, the algebraic equations governing  will become very complicated.Therefore, it is necessary to employ the modified approach.Figure 4 indicates that the approximations obtained by the presented method converge rapidly to the numerical solution as  increased.
Example 2 (system with nonsmooth nonlinearity).The modified MVIM is further applied to nonsmooth dynamical system expressed as ..
Here, (, ẋ ) is a nonlinear damping term, and () is a nonsmooth function.If substituting the  −1 into (), on account of  −1 contained unknown quantities ( −1 ), so () can not be expanded as Fourier progression by numerical integration.To this end, ( 9) cannot be deduced by eliminating The LC solutions of system (10) with  = 1.5 obtained by the modified MIM, RK method, and LP method, respectively.The iteration solution is denoted as IS and the th-order LP approximation as LP. the secular terms.Likewise, we present the following iteration scheme: In this scheme, ( −2 ) can be expanded as a Fourier series since  −2 has been determined at the previous iteration.Let us consider a van der Pol type oscillator with a nonsmooth function as  with Figure 5 shows the LC of system ( 14) with  = 1 and  = 0.5.The 5th-order approximations obtained by the presented The LC can also be obtained very accurate, as Figure 6 shows.

Conclusions
The Mickens iteration method (MIM) has been modified, so that it is suitable for solving LC solutions of self-excited systems with nonsmooth and/or damping nonlinearities.Different from the routinely-used MIM, the modified method decouples the unknown frequency from nonlinear terms.This modification simplifies the MIM significantly.Numerical examples show the feasibility and validity of the presented method, which implies that it could be applicable to more nonlinear dynamical systems, especially those with nonlinear damping terms and nonsmooth nonlinearities.

Figure 1 :
Figure1: The LC solutions of system(10) with  = 1 obtained by the modified MIM and RK method, respectively.The th-order solution (  ) is represented by IS.

Figure 2 :
Figure2: The LC solutions of system(10) with  = 1.5 obtained by the modified MIM, RK method, and LP method, respectively.The iteration solution is denoted as IS and the th-order LP approximation as LP.

Figure 3 :
Figure 3: Comparison of the LC solution of system (10) with  = −1, provided by the modified MIM, and by RK method, respectively.

Figure 4 :Figure 5 :
Figure 4: The LC solutions of system (21) with  = 1 obtained by the modified MIM and RK method, respectively.The th-order solution (  ) is represented by IS.

Table 1 :
Comparison of the second-order frequency obtained by IS and LP method with the forth-order approximation obtained by LP method, when  = 1.