Limit cycles of a class of hybrid piecewise differential systems with a discontinuity line of L shape

Abstract

In this work we study a class of discontinuous hybrid piecewise differential systems formed by two Hamiltonian systems that we named piecewise hybrid Hamiltonian systems. More precisely, we consider the differential systems with Hamiltonian functions $H_1(x,y)=a_{1}x+a_{2}y+a_3{x}^2+a_{4}xy+a_5{y}^2+A,\text{if}(x,y)\in {\Sigma }^{+}$$H_2(x,y)=b_{1}x+b_{2}y+b_3{x}^2+b_{4}xy+b_5{y}^2+B,\text{if}(x,y)\in {\Sigma }^{-}$with reset maps $R_1\left(x\right)=sx$ on $x\ge 0$ and $R_2\left(y\right)=ry$ on $y\ge 0$ for $0<r,s<1$, and $A$, $B$ are either zero, or one of them is a nonzero homogeneous polynomial of degree 3, ${\Sigma }^{+}=\{(x,y)\in R^2:x\ge 0\text{and}y\ge 0\}$ and ${\Sigma }^{-}$ is the closure of $R^2\setminus {\Sigma }^{+}$.

We provide an upper bound for the maximum number of limit cycles that these hybrid piecewise differential systems can exhibit. In other words, we solve the extension of the 16th Hilbert problem to this class of hybrid differential systems.