Bounded Solutions of Dynamical Systems in the Plane under the Conditions of Instability

✍ Scribed by J. Kalas; J. Osička

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 404 KB
• Volume: 170
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19941700111

✦ Synopsis

Abstract

In this paper the asymptotic behaviour of the solutions of x' = A(t)x + h(t,x) under the assumptions of instability is studied, A(t) and h(t,x) being a square matrix and a vector function, respectively. The conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are obtained. The method: the system is recasted to an equation with complex conjugate coordinates and this equation is studied by means of a suitable Lyapunov function and by virtue of the Wazevski topological method. Applications to a nonlinear differential equation of the second order are given.