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

Authors: Q. X. Liu; Y. M. Chen; J. K. Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

Recent years have witnessed the wide applications of iteration techniques, such as the Mickens iteration method (MIM) [1–6] and the variational iteration method [7–10]. In order to improve the efficiency, Lim and Wu [11], Marinca and Herisanu [12], and Hu [13] modified the MIM, respectively.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.


## 2. A Modified MIM

Consider a self-excited oscillator(1)x..+f(x,x.)=0,
where the superscript denotes the differentiation with respect to time t and f(x,x.) 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 ω and introduce the transformation as τ=ωt; thus, we rewrite (1) as
(2)ω2x′′+ω2x=ω2x-f(x,ωx′)
subject to the following initial conditions:
(3)x(0)=α,x′(0)=0,
where α is the unknown LC amplitude and the superscript denotes the differentiation with respect to τ. Note that α will be approximated as a series {αk} by eliminating the secular terms at each iteration stage. In order to obtain the LC, the MIM [16] can be given as
(4)ωk-12(xk′′+xk)=ωk-12xk-1-f(xk-1,ωk-1xk-1′),000000000000000000000000000000k=1,2,…,
with the initial conditions being rewritten at each iteration  as
(5)xk(0)=αk,xk′(0)=0.
Note that the coefficient of the first harmonic in xk-1 remains still to be an unknown, that is, αk-1. This unknown will couple with the unknown frequency, ωk-1, which will result in a coupled nonlinear term (i.e., ωk-1xk-1′) in the right side of (4). If higher powers of x′ exist, these terms will lead to very complicated functions in ωk-1 and αk-1. Different from conservative systems, ωk-12 can no longer be considered as an independent unknown. In order to simplify the MIM, therefore, a modified scheme is proposed as
(6)ωk-12(xk′′+xk)=ωk-12xk-1-f(xk-1,ωk-2xk-1′),000000000000000000000000000000k=1,2,….
As k=1, we choose ω-1=ω0. In the kth iteration, ωk-2 is a given constant that is obtained at the (k-1)th iteration. The square of the unknown frequency, that is, ωk-12, can be treated as an independent parameter, because ωk-1 appears only in ωk-12xk-1. According to the initial conditions, the starting iteration solution can be chosen as
(7)x0(τ)=α0cosτ.
It is obvious that as long as the series {ωk,k=1,2,…} and {xk,k=1,2,…} are convergent, they must converge to the exact solutions. The right-hand side of (6) can be expressed by Fourier series as
(8)ωk-12xk-1-f(xk-1,ωk-2xk-1′)=∑i=1φ(k)[ck-1,i(ωk-12,αk-1)cos(iτ)000000000.00+sk-1,i(ωk-12,αk-1)sin(iτ)],
where the harmonic coefficients ck-1,i(ωk-12,αk-1) and sk-1,i(ωk-12,αk-1) are functions in ωk-12 and αk-1. Here, φ(k) is a positive integer denoting the order of the highest harmonic. Approximations ωk-1 and αk-1 are determined by eliminating the secular terms, that is, letting
(9)ck-1,1(ωk-12,αk-1)=0,sk-1,1(ωk-12,αk-1)=0,0000000000000000000000000000000k=1,2,….
These equations can be solved analytically if ωk-12αk-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 k increases, ωk-12 is always an independent unknown in the modified MIM. Moreover, unknown ωk-1 does not couple with nonlinear terms. It simplifies the MIM to a large extent, as shown later.


## 3. Numerical Examples

Example 1 (system with nonlinear damping terms).
The van der Pol equation is chosen to illustrate the previous procedures more clearly:(10)x..+x+ε(x2-1)x.=0,
where ε is a given constant. As known, (10) has a stable LC solution when ε>0 while an unstable one when ε<0.
According to the modified MIM, the corresponding iteration scheme is given as(11)x..k+ωk-12xk=ωk-12xk-1-xk-1-ε(xk-12-1)x.k,000000000000000000000000000000k=1,2,….
Introducing a new time variable τ=ωk-1t at each iteration stage, we rewrite (11) as
(12)ωk-12(xk′′+xk)=ωk-12xk-1-xk-1-εωk-1(xk-12-1)xk-1′,
where the superscript denotes the derivative with respect to τ. The iteration algorithm begins with an initial solution
(13)x0(τ)=α0cosτ.
Then, we obtain the governing equations in x1(τ) as
(14)ω02(x1′′+x1)=(ω02α0-α0)cosτ-(εα0ω0-εα03ω04)sinτ+εα03ω04sin3τ,x1(0)=α1,x1′(0)=0.
Equating the coefficients of cosτ and sinτ to zeros results into
(15)ω02α0-α0=0,εα0ω0-εα03ω04=0
which yields that ω0=1 and α0=2. Substituting them into (16), we have
(16)x1′′+x1=2εsin3τ.
Considering initial conditions (14), we can obtain
(17)x1=α1cosτ+3εsinτ4-εsin3τ4,
where α1 is to be determined at the next iteration stage.
According to iterative scheme (11), the equation in x2(τ) is deduced as
(18)ω12(x2′′+x2)=∑i=19[c1,i(ω12,α1)cos(iω1t)000000+s1,i(ω12,α1)sin(iω1t)].
Equate the coefficients of cosτ and sinτ to zeros:
(19)3ω124-15α164-α134-34=0,α1ω12-α1-α138+75128=0.
By solving (19) numerically, we can determine α1 and ω12. Here, we obtain the second-order approximation and expand it as
(20)ω2IS=1-ε216+29ε42048+o(ε4).
According to [17], the Lindstedt-Poincare (LP) method provides the second- and forth-order approximate frequency ω2LP=1-ε2/16 and ω4LP=1-ε2/16+17ε4/3072, respectively. The attained approximation agrees well with the 4th-order LP solution. Table 1 indicates that ω2IS is more accurate than  ω2LP when compared with ω4LP.Table 1
Comparison of the second-order frequency obtained by IS and LP method with the forth-order approximation obtained by LP method, whenε=1.
ε
ω
2
IS
ω
2
LP
ω
4
LP
|
ω
2
IS
-
ω
4
LP
|
|
ω
2
LP
-
ω
4
LP
|
1
0.944799584
0.93750000
0.943033854
1.76
E
-
3
5.53
E
-
3
0.5
0.984820946
0.98437500
0.984720866
1.00
E
-
4
3.46
E
-
4
0.25
0.996121460
0.99609375
0.996115367
6.09
E
-
6
2.16
E
-
5Figure1 shows the comparison of the phase planes between iteration solutions (xk) and numerical result. Rapid convergence of xk to the numerical result can be observed. Note that all numerical solutions are obtained by the fourth-order Runge-Kutta (RK) integration method. When |ε|>1, the iteration procedure presented by Chen and Liu [16] does not converge. This is probably the difference between the starting function (x0=α0cost) and the exact solution is too large. The modified MIM is still effective for |ε|≤1.5. As Figure 2 shows, the LC solution with ε=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.Figure 1
The LC solutions of system (10) with ε=1 obtained by the modified MIM and RK method, respectively. The kth-order solution (xk) is represented by ISk.Figure 2
The LC solutions of system (10) with ε=1.5 obtained by the modified MIM, RK method, and LP method, respectively. The iteration solution is denoted as IS and the kth-order LP approximation as LPk.Also plotted in Figure2 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. Figure3 shows an unstable LC of the van der Pol equation with ε=-1 obtained by the presented method. As shown, the RK begins at the LC; however, the solution curve converges to the equilibrium.Figure 3
Comparison of the LC solution of system (10) with ε=-1, provided by the modified MIM, and by RK method, respectively.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]:
(21)x..+x+ε[(x.)2-1]x.+f(x.)=0.
The nonlinear term contains high powers of x., that is, f(x.)=(x.)3. If the original MIM is employed, the algebraic equations governing ω 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 k increased.Figure 4
The LC solutions of system (21) with ε=1 obtained by the modified MIM and RK method, respectively. The kth-order solution (xk) is represented by ISk.Example 2 (system with nonsmooth nonlinearity).
The modified MVIM is further applied to nonsmooth dynamical system expressed as(22)x..+f(x,x.)+ηg(x)=0.
Here, f(x,x.) is a nonlinear damping term, and g(x) is a nonsmooth function. If substituting the xk-1 into g(x), on account of xk-1 contained unknown quantities (αk-1), so g(x) can not be expanded as Fourier progression by numerical integration. To this end, (9) cannot be deduced by eliminating the secular terms. Likewise, we present the following iteration scheme:
(23)ωk-12(x¨k+xk)=ωk-12xk-1+f(xk-1,ωk-2x˙k-1)+ηg(xk-2).
In this scheme, g(xk-2) can be expanded as a Fourier series since αk-2 has been determined at the previous iteration.
Let us consider a van der Pol type oscillator with a nonsmooth function as(24)x¨+x+ε(1-x2)x˙+ηg(x)=0
with
(25)g(x)={x-1x≥10-1<x<1x+1x≤-1.
Figure 5 shows the LC of system (14) with ε=1 and η=0.5. The 5th-order approximations obtained by the presented method agree well with the numerical solution when the nonsmooth term is given as
(26)g(x)=sgn(x)={1,x>00,x=0-1,x<0.
The LC can also be obtained very accurate, as Figure 6 shows.Figure 5
LC solutions of system (24) with (25) (ε=1, η=0.5) provided by the modified MIM and by RK method, respectively.Figure 6
LC solutions of system (24) with (26) (ε=1, η=0.5) provided by the modified MIM and by RK method, respectively.


## 4. 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.



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