Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center (Q281672)

Let \(\Omega\) be a simply connected convex region in the real plane. Let \(h(x,y)\) be a smooth function defined in \(\Omega\) such that \(\Sigma=\{(x,y) \in \Omega | h(x,y)=0\}\) is a regular curve. Consider a pair of vector fields NEWLINE\[NEWLINE X : \;\dot x = f_+(x,y), \;\;\dot y = g_+(x,y), \text{ if }(x,y)\in \Omega_+ = \{(x,y) \in \Omega | h(x,y) \geq 0\}, \atop Y : \;\dot x = f_-(x,y), \;\;\dot y = g_-(x,y), \text{ if } (x,y)\in \Omega_- = \{(x,y) \in \Omega | h(x,y) \leq 0\}, \eqno(1) NEWLINE\]NEWLINE where \(f_{\pm}, g_{\pm} \in C^r(\Omega)\) and \(r\) is a positive integer or \(\infty\) or \(\omega\) (\(C^{\omega}\) denotes the class of analytic functions). The authors study singular points of a certain type (called \(\Sigma\)-centers) of the piecewise-smooth vector field \(Z\) defined by formula (1) away of the discontinuous curve \(\Sigma\) and defined by the Filippov rule on \(\Sigma\).NEWLINENEWLINEA point \(p \in \Sigma\) is called \textit{\(\Sigma\)-center} of the field \(Z\), if there are no trajectories of \(Z\) entering the point \(p\) and all trajectories of \(Z\) are closed in a neighborhood of \(p\). A point \(p \in \Sigma\) is called a \(k\)-th contact point for the field \(X\) if \(X^i h(p) = 0\) for all \(i = 1, \dots, k-1\), and \(X^k h(p) \neq 0\). Similarly, \(p \in \Sigma\) is called a \(l\)-th contact point for the field \(Y\) if \(Y^i h(p) = 0\) for all \(i = 1, \dots, l-1\), and \(Y^l h(p) \neq 0\). A \(\Sigma\)-center \(p\) satisfying both these conditions is called \((k,l)\)-\(\Sigma\)-center. For example, the origin \(p=0\) is \((2,2)\)-\(\Sigma\)-center of the field NEWLINE\[NEWLINE X_0 : \;\dot x = -1, \;\;\dot y = 2x, \text{ if }y \geq 0, \atop Y_0 : \;\, \dot x = \phantom{-}1, \;\;\dot y = 2x, \text{ if } y \leq 0. \eqno(2) NEWLINE\]NEWLINENEWLINENEWLINEIt was previously known [\textit{C. A. Buzzi} et al., J. Math. Pures Appl. (9) 102, No. 1, 36--47 (2014; Zbl 1300.34032)] that the germ of any vector field (1) at \((2,2)\)-\(\Sigma\)-center \(p\) is \textit{topologically orbitally} equivalent to (2). That is, there exists an orientation preserving homeomorphism that transforms the orbits of (1) to the orbits of (2). In the present paper, the authors establish the same result for \((k,l)\)-\(\Sigma\)-center with any \(k,l\). They also show that there are vector fields (1) with \((k,l)\)-\(\Sigma\)-centers whose flows are topologically conjugate to that of (2), and also there exist vector fields (1) with \((2,2)\)-\(\Sigma\)-centers whose flows cannot be conjugate to that of (2). Applying these results, the authors study the number of limit cycles which can be bifurcated from \(\Sigma\)-center.