Perturbations from an Elliptic Hamiltonian of Degree Four

By Freddy Dumortier, Chengzhi Li. 2001

127 citations recorded by the source corpus.

Abstract The paper deals with Lienard equations of the form x=y, y=P(x)+yQ(x) with P and Q polynomials of degree respectively 3 and 2. Attention goes to perturbations of the Hamiltonian vector field with an elliptic Hamiltonian of degree 4, exhibiting a cuspidal loop. It is proven that the least upper bound for the number of zeros of the related elliptic integral is four, and this upper bound is a sharp one.  This permits to prove the existence of Lienard equations of type (3, 2) with at least four limit cycles. The paper also contains a complete result on the respective number of “small” and “large” limit cycles.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/112538194dbe9acaa6bab1c856c8b8c798eacff0

Citations in the corpus

• Hilbert's 16th Problem and bifurcations of Planar Polynomial Vector Fields (2003, unknown): “In recent years, new progress have been achieved [Dumortier & Li, 2001; Li et al., 2000, 2001; Chow et al., 2001; etc.].”