Differential systems, the mapping over period for which is represented by a product of three exponential matrices (Q868813)

Let \(\varphi(t;t_0,x_0)\) denote the solution to the system \[ x^{\prime} =X(t,x),\quad t\in\mathbb R,\quad x \in\mathbb R^{n} x(t_0)=x_{0} \] where \(X\) is a continuously differentiable function. The function \(F(t,x)=\varphi(-t;t,x)\) is called the reflecting function for this system. For a linear system \[ x^{\prime}=P(t)x,\quad t\in\mathbb R,\quad x\in\mathbb R^{n} \] its reflecting function is linear and given by \(F(t,x)=F(t)x\), where \(F(t):=\Phi(-t)\Phi^{-1}(t)\), and \(\Phi (t)\) is the fundamental matrix of the linear system. The matrix \(F(t)\) is called the reflecting matrix of this linear system. The author studies the case when the reflecting matrix \(F(t)\) can be written as the product of three exponential matrices in the form \[ F(t)=e^{\alpha(t)A}e^{\beta(t)B}e^{-\alpha(-t)A}, \] where \(\alpha(t)\) and \(\beta(t)\) are arbitrary scalar functions, and, \(A\) and \(B\) are constant \(n\times n\) matrices. He deduces sufficient conditions for existence and stability of periodic solutions to a periodic ordinary differential system.