On the structural stability relative to the space of linear differential equations with periodic coefficients

V. Sh. Roitenberg

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 5, pp. 27-31

Résumé

Let $\textrm{LE}^n_\omega$ be the Banach space of linear non-homogeneous differential equations of order $n$ with $\omega$-periodic coefficients. We prove the following statements. The equation $l\in \textrm{LE}^n_\omega$ is structurally stable in the phase space $\Phi^2:=\mathbf{R}^n\times\mathbf{R}/\omega \mathbf{Z}(n\geq2)$ if and only if its multiplicators do not belong to the unit circle. The set of all structurally stable equations is everywhere dense in $\textrm{LE}^n_\omega$. The equation $l\in \textrm{LE}^n_\omega$ is structurally stable in the phase space $\bar{\Phi}^2:=\mathbf{RP}^2\times\mathbf{R}/\omega \mathbf{Z}$ if and only if its multiplicators are real, different and distinct from $\pm 1$. We describe also the topological equivalence classis of structurally stable in $\bar{\Phi}^2$ equations.

Détail

Keywords: linear differential equations, periodic coefficients, projective plane, structurally stable equations, multiplicators.