| Abstract: |
| This talk establishes a rigorous connection between the hyperbolicity of algebraic limit cycles and Abelian integrals for planar polynomial differential systems admitting invariant algebraic curves. Our results are non-perturbative, i.e., independent of any small parameter, and thus extend the classical Abelian-integral criterion beyond the perturbative framework. We prove that the vanishing of the associated Abelian integral $I(h)$ is a necessary condition for a periodic orbit to lie on an irreducible invariant algebraic curve. Moreover, for a class of systems under suitable constraints, we show that an algebraic limit cycle is hyperbolic if and only if $I`(h)\not=0$. As an application, we study Kukles systems of arbitrary degree and obtain existence results for algebraic (reversible) limit cycles. By analyzing the zeros of the relevant Abelian integrals, we further demonstrate the coexistence of the algebraic cycle with additional limit cycles, revealing rich global dynamics for this family. |
|