A note on linear differential equations with periodic coefficients
The authors consider the periodic linear differential equation \(\dot{x}=A(t)x\) and derive sufficient conditions for the asymptotic stability of the zero solution. They cover a class of systems, where \(A(t)\) is negative semidefinite, i.e. \(x^TA(t)x\leq 0\) for all \(x\) and \(t\), or nearly negative definite, i.e. \(x^TA(t)x\leq \varepsilon_0(t,x)+\varepsilon_1(t)x^2\) with conditions on \(\varepsilon_0\) and \(\varepsilon_1\). The proof uses the construction of a matrix-valued function \(P(t)\) that satisfies the matrix equation \[ Q=\dot{P}+PA+A^TP \] such that \(x^TP(t)x\) is a strict Lyapunov function. Moreover, the authors prove that any configuration of cycles can be realized as hyperbolic and asymptotically stable limit cycles of a polynomial vector field. This solves an open problem stated in [\textit{D. Peralta-Salas}, J. Differ. Equations 217, No. 1, 249--256 (2005; Zbl 1086.34029)].
scientific article; zbMATH DE number 5586851
scientific article