More limit cycles for complex differential equations with three monomials
Year: 2024
Type: article
Abstract: In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, $ \dot z=Az^k\bar{z}^l+Bz^m\bar{z}^n + C z^p\bar{z}^q,$ being $k,l,m,n, p, q$ non-negative integers and $A,B,C \in\mathbb{C}.$ More concretely, if $N=\max(k+l,m+n, p+q)$ and $H_3(N)\in\mathbb{N}\cup\{\infty\}$ denotes the maximum number of limit cycles of the above equations, we show that for $N\ge4,$ $H_3(N)\ge N-3$ and that for some values of $N$ this new lower bound is $N+1.$ We also present examples with many limit cycles and different configurations. Finally, we show that $H_3(2)\ge 2$ and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles.
Authors: M.J. Álvarez, B. Coll, A. Gasull, R. Prohens
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,050
OpenAlex ID: W4403536394