On the study and application of limit cycles of a kind of piecewise smooth equation. (English) Zbl 1439.34024
Summary: We study the piecewise smooth equation of the form: \[ \frac{dx}{dt}=S(t,x)= \begin{cases} S_1(t,x)=a_1(t)x^m+b(t), & \quad \text{if} \quad x\ge{0}, \\ S_2(t,x)=a_2(t)x^m+b(t), & \quad \text{if} \quad x<0, \end{cases}\] where \((t,x)\in [0,2\pi ]\times{\mathbb{R}}, m\in{{\mathbb{Z}}^+}\) and \(a_1(t), a_2(t), b(t)\) are \(2\pi \)-periodic smooth functions. A solution of the equation satisfying \(x(0)=x(2\pi )\) is called a periodic solution. Moreover, such solution is called a limit cycle if and only if it is isolated. We obtain that the maximum number of limit cycles for this equation is 1 (resp. 2) if \((-1)^ma_1(t)\cdot a_2(t)<0\) (resp. \((-1)^ma_1(t)\cdot a_2(t)>0)\). In this study we pay more attention to the examples in which the equation has limit cycle(s) crossing the separation straight line \(x=0\). In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve \(x^2+y^2=1\).
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
Keywords:
limit cycles; upper bound; piecewise smooth equation; planar piecewise smooth systems with separation curve being \({\mathbb{S}}^1\)References:
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