The Limit Cycles for a Class of Non-autonomous Piecewise Differential Equations

In this paper, we study a class of non-autonomous piecewise differential equations defined as follows: dx/dt=a_0(t)+∑ _i=1^na_i(t)|x|^i , where n∈ℕ^+ and each a_i(t) is real, 1-periodic, and smooth function. We deal with two basic problems related to their limit cycles (isolated solutions satisfying x(0) = x(1) ) . First, we prove that, for any given n∈ℕ^+ , there is no upper bound on the number of limit cycles of such equations. Second, we demonstrate that if a_1(t),… , a_n(t) do not change sign and have the same sign in the interval [0, 1], then the equation has at most two limit cycles. We provide a comprehensive analysis of all possible configurations of these limit cycles. In addition, we extend the result of at most two limit cycles to a broader class of general non-autonomous piecewise polynomial differential equations and offer a criterion for determining the uniqueness of the limit cycle within this class of equations.