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arXiv · 1808.01553

Limit cycles appearing from perturbations of cubic piecewise smooth center with double invariant real straight lines

Jihua Yang·Liqin Zhao
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Abstract

This paper investigates the exact number of limit cycles given by the averaging theory of first order for the piecewise smooth integrable non-Hamiltonian system \begin{eqnarray*} (\dot{x},\ \dot{y})=\begin{cases} (-y(x+a)^2+\varepsilon f^+(x,y),\ x(x+a)^2+\varepsilon g^+(x,y)),\ \ x\geq0,\\ (-y(x+b)^2+\varepsilon f^-(x,y),\ x(x+b)^2+\varepsilon g^-(x,y)),\ ~ \, x<0,\\ \end{cases}\end{eqnarray*} where $ab\neq 0$, $0<|\varepsilon|\ll 1$, and $f^\pm(x,y)$ and $g^\pm(x,y)$ are polynomials of degree $n$. It is proved that the exact number of limit cycles emerging from the period annulus surrounding the origin is linear depending on $n$ and it is at least twice the associated estimation of smooth systems.

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Jihua Yang, Liqin Zhao. 2018-08-05. Limit cycles appearing from perturbations of cubic piecewise smooth center with double invariant real straight lines. https://arxiv.org/abs/1808.01553

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