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: \(dx/dt=a_{0}(t)+\sum _{i=1}^{n}a_{i}(t)|x|^{i}\) $dx/dt=a_0(t)+{\sum }_{i=1}^n{a}_i(t){|x|}^i$ , where \(n\in \mathbb {N}^{+}\) $n\in {\mathbb{N}}^{+}$ and each \(a_{i}(t)\) $a_i(t)$ is real, 1-periodic, and smooth function. We deal with two basic problems related to their limit cycles \(\big (\text {isolated solutions satisfying} x(0) = x(1)\big )\) $(\text{isolated solutions satisfying}x(0)=x(1))$ . First, we prove that, for any given \(n\in \mathbb {N}^{+}\) $n\in {\mathbb{N}}^{+}$ , there is no upper bound on the number of limit cycles of such equations. Second, we demonstrate that if \(a_{1}(t),\ldots , a_{n}(t)\) $a_1(t),\dots ,a_n(t)$ 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.