On simple linear differential systems with an even matrix

E. V. Musafirov

Trudy Instituta matematiki, Tome 18 (2010) no. 2, pp. 93-98

Résumé

Conditions of simplicity of linear differential systems with an even coefficient matrix are obtained. Fundamental matrixes of solutions of linear differential systems $\dot{x}=2P(t)x$ and $\dot{x}=-2P(-t)x$ are expressed by means of reflective matrix $F(t)$ of simple system $\dot{x}=P(t)x$, $t\in\mathbb{R}$, $x\in\mathbb{R}^n$. Fundamental matrixes of solutions of systems $\dot{x}=-2kP(t)x$, $k\in\mathbb{Z}$ and $\dot{x}=-2P(t)x+\dot{P}(t)P^{-1}(t)x$ are also expressed by means of $F(t)$ under condition of evenness of matrix $P(t)$. Equivalence (in terms of coincidence of reflective functions) of last system and a simple system $\dot{x}=-2P(t)x$ with an even coefficient matrix is proved.