Estimates on the number of limit cycles of pendulum-like Abel equations up to second order analysis

In this paper, we study the number of limit cycles of the perturbed pendulum-like generalized Abel equation 𝑑 ⁢ 𝑥 𝑑 ⁢ 𝜃 = ( 𝐴 ⁡ ( 𝜃 ) + 𝜀 ⁢ 𝑃 1 ⁡ ( 𝜃 ) + 𝜀 2 ⁢ 𝑃 2 ⁡ ( 𝜃 ) ) ⁢ 𝑥 𝑝 + ( 𝜀 ⁢ 𝑄 1 ⁡ ( 𝜃 ) + 𝜀 2 ⁢ 𝑄 2 ⁡ ( 𝜃 ) ) ⁢ 𝑥 𝑞 , where 𝑝 , 𝑞 ∈ ℤ , 𝐴 ⁡ ( 𝜃 ) = s i n ⁡ 𝜃 ⋅ 𝐵 ⁡ ( c o s ⁡ 𝜃 ) , B is a polynomial of degree n that does not vanish on [ − 1 , 1 ] , and 𝑃 𝑖 ⁡ ( 𝜃 ) , 𝑄 𝑖 ⁡ ( 𝜃 ) are trigonometric polynomials of degree m for 𝑖 = 1 , 2 . By applying the theory of polynomials and the theory of Chebyshev systems, we give estimates for 𝑍 ⁢ ( 𝑀 𝑖 ) + and 𝑍 ⁢ ( 𝑀 𝑖 ) − , the maximum numbers of positive and negative zeros of the first non-vanishing Melnikov function 𝑀 𝑖 of the equation, respectively. In particular, we prove that 𝑍 ⁢ ( 𝑀 1 ) ± = 𝑚 + 1 , and 𝑍 ⁢ ( 𝑀 2 ) ± = 2 ⁢ 𝑚 − 2 for 𝑚 ≥ 𝑛 + 1 . As a direct consequence, when 𝑚 ≥ 𝑛 + 1 , the equation admits at least 2 ⁢ 𝑚 − 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.