## Study on Periodic Solutions of a Class of Continuous and Discontinuous Piece-Wise Linear Systems

Yang Jing1, Ke Changcheng1, Wei Zhouchao,1,2

 基金资助: 国家自然科学基金.  11772306浙江省自然科学基金.  LY20A020001

 Fund supported: the NSFC.  11772306the NSF of Zhejiang Province.  LY20A020001

Abstract

In recent years, the study of non-smooth systems has become a hot spot, and the qualitative analysis of piece-wise linear systems has become an indispensable research problem. In this paper, a transformed Michelson differential system is studied, and the existence of periodic solutions for continuous and discontinuous piece-wise linear systems is proved by means of average theory.

Keywords： Continuous piece-wise linear differential system ; Discontinuous piece-wise linear differential system ; Periodic solution ; Average theory

Yang Jing, Ke Changcheng, Wei Zhouchao. Study on Periodic Solutions of a Class of Continuous and Discontinuous Piece-Wise Linear Systems. Acta Mathematica Scientia[J], 2021, 41(4): 1053-1065 doi:

## 2 主要结论

$\begin{eqnarray} \left\{\begin{array}{ll} \dot{x_1} = y, \\ \dot{x_2} = z, \\ { } \dot{x_3} = c^2-y-\frac{x^2}{2}, \end{array}\right. \end{eqnarray}$

2016年, Llibre[18]对系统(2.1)做变量替换: $(x, y, z, c)\rightarrow (2{\mu}x_1, 2{\mu}x_2, 2{\mu}x_3, 2{\mu}p)$, 其中${\mu}>0$充分小, 且$p>0$. 然后做变换$x_1^2\rightarrow |x_1|$, 则得到新的分段线性系统(2.2), 并研究了该系统的周期解问题

$\begin{eqnarray} \left\{\begin{array}{ll} \dot{x_1} = x_2, \\ \dot{x_2} = x_3, \\ \dot{x_3} = -x_2+\mu(2p^2-|x_1|). \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{ll} \dot{x_1} = x_2+\mu(2p^2-|x_1|), \\ \dot{x_2} = x_3+\mu(2p^2-|x_1|), \\ \dot{x_3} = -x_2+\mu(2p^2-|x_1|), \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{ll} \dot{x_1} = x_2+\mu(2p^2-|x_1|-{\rm sign}(x_1)), \\ \dot{x_2} = x_3+\mu(2p^2-|x_1|-{\rm sign}(x_1)), \\ \dot{x_3} = -x_2+\mu(2p^2-|x_1|-{\rm sign}(x_1)). \end{array}\right. \end{eqnarray}$

$a\in(-1, 0)$, $a$由函数$g(a) = 0$决定,

### 3.1 连续分段线性微分系统

$\begin{eqnarray} \dot{x} = F_0(t, x)+\mu{F_1(t, x)}+{\mu}^{2}I(t, x, \mu), \end{eqnarray}$

$\begin{eqnarray} \dot{x}(t) = F(t, x) = \left\{\begin{array}{ll} F^1(t, x), (t, x)\in\Sigma^+, \\ F^2(t, x), (t, x)\in\Sigma^-, \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} \dot{x}(t) = F(t, x) = \gamma_+(t, x)F^1(t, x)+\gamma_-(t, x)F^2(t, x), \end{eqnarray}$

$\begin{eqnarray} \dot{x} = F_0(t, x)+\mu{F_1(t, x)}+{\mu}^{2}I(t, x, \mu), \end{eqnarray}$

$F_i^1\in{C^1}, i = 0, 1, I^1, I^2$是连续函数, 且第二个变量满足Lipschitz条件, 且对变量$t\in{{\Bbb R}}$, 这些函数都是周期为T的周期函数. 对$z\in{D}, \mu>0$充分小, 令$x(t, z, \mu)$表示系统(3.6)的解, 且$x(0, z, \mu) = z.$

$\begin{eqnarray} F(z) = \int_0^T{{N(t, z)^{-1}}F_1(t, x(t, z, 0))}{\rm d}t, \end{eqnarray}$

(1) 存在一个开有界子集$C\in{D}$, 对充分小的$\mu,$每一个从$C$出发的轨道都只在其交点处达到不连续点集.

(2) 对$\alpha\in{C}$, 有$F(\alpha) = 0$, 则存在$\alpha$的邻域$U\in{C}$, 对所有$z\in\overline{U}\backslash{\{\alpha\}}$, $F(z)\neq0$, $\rm{det}(D_zF(\alpha))\neq0.$

## 4 证明定理2.1

$\begin{eqnarray} \left\{\begin{array}{ll} \dot{x_1} = \rho\sin{\xi}+\mu(2p^2-|x_1|), \\ \dot{\rho} = \mu(2p^2-|x_1|)(\sin{\xi}+\cos{\xi}), \\ { } \dot{\xi} = 1-\frac{\mu}{\rho}(\sin{\xi}-\cos{\xi})(2p^2-|x_1|). \end{array}\right. \end{eqnarray}$

$\xi$为新的自变量, 微分系统(4.1)变为

$\begin{eqnarray} \left\{\begin{array}{ll} { } \frac{{\rm d}x_1}{{\rm d}\xi} = x_1' = \rho\sin{\xi}+\mu(\sin{\xi}^{2}-\sin{\xi}\cos{\xi}+1)(2p^2-|x_1|)+o(\mu^{2}), \\ { } \frac{{\rm d}\rho}{{\rm d}\xi} = \rho' = \mu(\sin{\xi}+\cos{\xi})(2p^2-|x_1|)+O(\mu^{2}), \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{ll} x_1' = \rho\sin{\xi}, \\ \rho' = 0. \end{array}\right. \end{eqnarray}$

$C^0$, 且满足Lipschitz条件, 则系统(4.1)满足定理3.1的假设条件. 由定理3.1, 我们要计算平均函数

$N(0)$是单位矩阵. 因此

$\begin{eqnarray} F(x_1^0, \rho^{0}) & = &\int_0^{2\pi}(2p^2-|x_1^0+\rho^{0}(1-\cos{\xi})|) \left(\begin{array}{cc} 1&\cos{\xi}-1\\ 0&1 \end{array}\right) \left(\begin{array}{cc} \sin{\xi}^{2}-\sin{\xi}\cos{\xi}+1\\ \sin{\xi}+\cos{\xi} \end{array}\right){\rm d}\xi{}\\ & = &(f_1(x_1^0, \rho^{0}), f_2(x_1^0, \rho^{0})), \end{eqnarray}$

$\frac{x_1^0+\rho^{0}}{\rho^{0}} = {-1}$时, 除了$\xi = \pi$时有$G(\xi) = 0$, 对所有的$\xi $$x_1^0+\rho^{0}-\rho^{0}\cos{\xi}<0. 情况2 当 x_1^0>0 , 即 \frac{x_1^0+\rho^{0}}{\rho^{0}}>1 时, 对所有的 \xi$$ x_1^0+\rho^{0}-\rho^{0}\cos{\xi}>0$;

$\frac{x_1^0+\rho^{0}}{\rho^{0}} = 1$时, 除了$\xi = 0, 2\pi$时有$G(\xi) = 0$, 对所有的$\xi $$x_1^0+\rho^{0}-\rho^{0}\cos{\xi}>0. 情况3 当 -2\rho^{0}<x_1^0<0 , 即 |\frac{x_1^0+\rho^{0}}{\rho^{0}}|<1 (i) x_1^0+\rho^{0}-\rho^{0}\cos{\xi}<0, \xi\in(-\arccos\frac{x_1^0+\rho^{0}}{\rho^{0}}, \arccos\frac{x_1^0+\rho^{0}}{\rho^{0}}); (ii) x_1^0+\rho^{0}-\rho^{0}\cos{\xi}>0, \xi\in(\arccos\frac{x_1^0+\rho^{0}}{\rho^{0}}, 2\pi-\arccos\frac{x_1^0+\rho^{0}}{\rho^{0}}). 针对情况1, 平均函数为 计算得平均函数有唯一零点 (x_1^0, \rho^{0}) = (-2p^2, 0). 因此在情况1下平均理论不提供周期解. 针对情况2, 平均函数为 计算得平均函数有唯一零点 (x_1^0, \rho^{0}) = (2p^2, 0). 与情况1相同, 情况2下平均理论不提供周期解. 针对情况3, 平均函数为 其中 进一步计算得到 为了计算 f_1(x_1^0, \rho^{0}) = f_2(x_1^0, \rho^{0}) = 0 的零点, 需做变换 a = \frac{x_1^0+\rho^{0}}{\rho^{0}} , -1<a<1 , 则有 f_2(x_1^0, \rho^{0}) = \pi{\rho^{0}}+2\rho^{0}a\sqrt{1-a^2}-2\rho^{0}\arccos{a} = 0. 由于 \rho^{0}>0 , 解得 a = 0 , 则有 x_1^0 = -\rho^{0}. 代入 f_1(x_1^0, \rho^{0}) = 0 , 解得 x_1^0 = -\pi{p^2} , \rho^{0} = \pi{p^2}. 计算 (f_1, f_2)$$ (x_1^0, \rho^{0}) = (-\pi{p^2}, \pi{p^2})$的雅可比矩阵, 有

$\begin{eqnarray} \rho^{0} = \frac{-4\sqrt{1-a^{2}}}{\pi+2a\sqrt{1-a^{2}}-2\arccos{a}}, \; \; x_1^0 = a\rho^{0}-\rho^{0}, \end{eqnarray}$

$x_1^0, \rho^{0}$由(5.5)式给出.

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Andronov A , Vitt A , Khaikin S . Theory of Oscillations. Oxford: Pergamon Press, 1996

Bernardo M Di , Budd C , Champneys A R , et al.

Piece-wise smooth dynamical systems: theory and applications

Appl Math Sci, 2007, 163 (4): 1072- 1118

Simpson D J W .

Bifurcations in piece-wise smooth continuous systems

J Nonlinear Sci, 2010, 69 (1): 23- 36

Makarenkov O , Lamb J S W .

Dynamics and bifurcations of nonsmooth systems: A survey

Physica D, 2012, 241 (22): 1826- 1844

Carvalho B D , Fernando M L .

More than three limit cycles in discontinuous piece-wise linear differential systems with two zones in the plane

Int J Bifurcat Chaos, 2014, 24 (4): 1450056

Euzebio R D , Llibre J .

On the number of limit cycles in discontinuous piece-wise linear differential systems with two pieces separated by a straight line

J Integral Equ Appl, 2015, 424 (1): 475- 486

Llibre J , Teixeira M A .

Piece-wise linear differential systems without equilibria produce limit cycles

Nonlinear Dyn, 2017, 88 (1): 157- 164

Wang J F , Huang C X , Huang L H .

Discontinuity-induced limit cycles in a general planar piece-wise linear system of saddle-focus type

Nonlinear Anal-Hybri, 2019, 33, 162- 178

Li S M , Llibre J .

On the limit cycles of planar discontinue piece-wise linear differential systems with a unique equilibrium

Discrete Cont Dyn-B, 2019, 24 (11): 5885- 5901

Fatou P .

Sur le mouvement d'un systeme soumis a des forces a courte periode

Bull Soc Math, 1928, 56, 98- 139

Bogoliubov N , Krylov N .

Application of methods of nonlinear mechanics in the theory of stationary oscillations

Časo Pěst Mate Fysiky, 1935, 64 (5): 107- 115

Hale J K .

On the method of averaging

IRE Trans CT, 1960, 7 (4): 517- 519

Hlanay A .

On the method of averaging for differential equations with retarded argument

J Integral Equ Appl, 1966, 14 (1): 70- 76

Sethna P R , Meyer K R , Bajaj A K .

On the method of averaging, integral manifolds and systems with symmetry

Siam J Appl Math, 1985, 45 (1): 343- 359

Lehman B , Bass R M .

Extensions of averaging theory for power electronic systems

IEEE Trans Power Electr, 1996, 11 (4): 542- 553

Llibre J .

Averaging theory and limit cycles for quadratic systems

Rad Mat, 2002, 3 (2): 215- 228

Llibre J , Novaes D , Teixeira M A .

Higher order averaging theory for finding periodic solutions via Brouwer degree

Nonlinearity, 2014, 27 (1): 563- 583

Llibre J , Oliveira R , Rodrigues C A B .

On the periodic solutions of the Michelson continuous and discontinuous piece-wise linear differential system

Comp Appl Math, 2018, 37 (2): 1550- 1561

Michelson D .

Steady solutions for the Kuramoto-Sivashinsky equation

Physics D, 1986, 19 (1): 89- 111

Wilczak D .

The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof

Found Comput Math, 2006, 6 (4): 495- 535

Llibre J .

The Michelson system is neither global analytic, nor Darboux integrable

Physics D, 2010, 239 (8): 414- 419

Llibre J , Makhlouf A .

Zero-Hopf bifurcation in the generalized Michelson system

Chaos, 2015, 89, 228- 231

Verhulst F . Nonlinear Differential Equations and Dynamical Systems. Berlin: Springer-Verlag, 1996

Francoise B A , Francoise J P , Llibre J .

Periodic solutions of nonlinear periodic differential systems with a small parameter

Comm Pure Appl Anal, 2007, 6 (1): 103- 111

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