Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

Abstract

We provide lower bounds for the maximum number of limit cycles for the m-piecewise discontinuous polynomial differential equations x = y + sgn(gm (x, y))F (x), y = -x, where the zero set of the function sgn(gm (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane in sectors of angle 2π/m, and sgn(z) denotes the sign function.