Necessary conditions for a limit cycle and its basin of attraction

✍ Scribed by Peter Giesl

• Book ID: 103847048
• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 437 KB
• Volume: 56
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2003.07.020

✦ Synopsis

In this paper we consider a general di erential equation of the form ẋ=f(x) with f ∈ C 1 (R n ; R n ) and n ¿ 2. Borg, Hartman, Leonov and others have studied su cient conditions for the existence, uniqueness and exponential stability of a periodic orbit and for a set to belong to its basin of attraction. They used a certain contraction property of the ow with respect to the Euclidian or a Riemannian metric. In this paper we also prove su cient conditions including upper bounds for the Floquet exponents of the periodic orbit. Moreover, we show the necessity of these conditions using Floquet theory and a Lyapunov function.