Geometric tools to determine the hyperbolicity of limit cycles

In this paper we present a new method to study limit cycles’ hyperbolicity. The main tool is the function ν = ([V,W] ∧ V )/(V ∧ W), where V is the vector field under investigation and W a transversal one. Our approach gives a high degree of freedom for choosing operators to study the stability. It is related to the divergence test, but provides more information on the system’s dynamics. We extend some previous results on hyperbolicity and apply our results to get limit cycles’ uniqueness. Liénard systems and conservative + dissipative systems are considered among the applications. © 2006 Elsevier Inc. All rights reserved.