Estimates on the number of limit cycles of pendulum-like Abel equations up to second order analysis
Year: 2026
Type: article
Abstract: In this paper, we study the number of limit cycles of the perturbed pendulum-like generalized Abel equation π β’ π₯ π β’ π = ( π΄ β‘ ( π ) + π β’ π 1 β‘ ( π ) + π 2 β’ π 2 β‘ ( π ) ) β’ π₯ π + ( π β’ π 1 β‘ ( π ) + π 2 β’ π 2 β‘ ( π ) ) β’ π₯ π , where π , π β β€ , π΄ β‘ ( π ) = s i n β‘ π β
π΅ β‘ ( c o s β‘ π ) , B is a polynomial of degree n that does not vanish on [ β 1 , 1 ] , and π π β‘ ( π ) , π π β‘ ( π ) are trigonometric polynomials of degree m for π = 1 , 2 . By applying the theory of polynomials and the theory of Chebyshev systems, we give estimates for π β’ ( π π ) + and π β’ ( π π ) β , the maximum numbers of positive and negative zeros of the first non-vanishing Melnikov function π π of the equation, respectively. In particular, we prove that π β’ ( π 1 ) Β± = π + 1 , and π β’ ( π 2 ) Β± = 2 β’ π β 2 for π β₯ π + 1 . As a direct consequence, when π β₯ π + 1 , the equation admits at least 2 β’ π β 2 limit cycles bifurcating from one of the positive and negative periodic annuli. This estimate simultaneously improves several previous lower bounds for classical pendulum equations and whirling pendulum equations in the literature.
Authors: Jianfeng Huang, Haihua Liang
Institutions: Jinan University, Guangdong Polytechnic Normal University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7222976119