ON THE LYAPUNOV CONSTANTS OF PLANAR PIECEWISE SMOOTH SYSTEMS SEPARATED BY AN ANALYTICAL CURVE

In this paper, we consider the computations of Lyapunov constants of a class of planar piecewise analytical systems defined in two zones separated by an analytical curve $ y=\phi(x) $ with $ \phi(0)=0 $. Assume that the origin $ (0, 0) $ is a pseudo-focus of the system. We propose an extension of the classical polar coordinates for the subsystem with focus contact, and an extension of the $ (R, \theta, 1, 2) $-generalized polar coordinates for the subsystem with parabolic contact. Then we present the method on how to calculate the relevant Lyapunov constants. As applications, we present three planar piecewise quadratic systems. The first one is of parabolic-parabolic type separated by $ y=\sin^2 x $ which has four limit cycles bifurcated from $ (0, 0) $. The second one is of focus-parabolic type separated by $ y=e^x-1 $ which has five limit cycles bifurcated from $ (0, 0) $. The last one is of focus-focus type separated by $ y=\sin x $ which has five limit cycles bifurcated from $ (0, 0) $.