Abstract
The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.
Highlights
A classical problem in the study of qualitative theory of planar differential equations in the plane is the 16th Hilbert Problem, related to the bifurcation of limit cycles in a polynomial class of fixed degree
This section is devoted to the study of Liénard systems and it is divided into three parts
We use the Lie bracket method to deduce the structure of Lyapunov and period constants of a Liénard system starting with an odd and an even degree monomials on its first differential equation
Summary
A classical problem in the study of qualitative theory of planar differential equations in the plane is the 16th Hilbert Problem, related to the bifurcation of limit cycles Let us consider a real polynomial system of differential equations in the plane whose origin is a nondegenerate monodromic equilibrium point, so the matrix associated to the differential system evaluated at the origin has zero trace and positive determinant It is a well-known fact that, by a suitable change of coordinates and time rescaling, it can be written in the form x = αx − y + X (x, y) =: P(x, y), (1). For all the results developed in this work, we will consider a fixed transversal section x defined as the positive x-axis, this is the semi-axis {(x, 0), x > 0} This restriction is necessary for the study of the period function because when the origin of (1) is not a center we can construct an isochronous section, as can be seen in [3].
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