On the limit cycles of a class of piecewise linear differential systems in \(\mathbb R^4\) with two zones (Q433608)

The authors study the maximum number of limit cycles of the \(4\)-dimensional continuous piecewise linear vector field NEWLINE\[NEWLINE x'=A_0x+\varepsilon F(x), NEWLINE\]NEWLINE where \(\varepsilon\) is a small parameter, \(A_0\) is an elliptic matrix with eigenvalues \(+- i\) (in the Jordan normal form) and \(F:\mathbb R^4\rightarrow\mathbb R^4\) is given by \(F(x)=Ax+\varphi(k^T x)b\) for a matrix \(A\), \(k,b\in\mathbb R^4\setminus \{0\}\) and \(\varphi:\mathbb R\rightarrow \mathbb R\) is the piecewise linear function such that \(\varphi(s)=0\) for \(s\in (\infty,1)\) and \(\varphi(s)=ms\) for \(s\in[1,\infty)\).NEWLINENEWLINEThe main result is that the upper bound for the number of limit cycles of the above defined system is three, and there are systems having exactly three limit cycles. The proof is based on the non-smooth averaging theory.