Bifurcation of limit cycles in a class of piecewise smooth generalized Abel equations with two asymmetric zones

Abstract This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfrac> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>θ</m:mi> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>B</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> $$\frac{\mathrm{d}x}{\mathrm{d}\theta }=A\left(\theta \right){x}^{p}+B\left(\theta \right){x}^{q},$$ where A and B are piecewise trigonometrical polynomials of degree m with two zones 0 ≤ θ &lt; θ 1 and θ 1 ≤ θ ≤ 2 π . By means of the first and second order analysis using the Melnikov theory and applying the new Chebyshev criterion that established by J. Huang, H. Liang, and X. Zhang, “A new Chebyshev criterion and its application to planar differential systems,” J. Differ. Equ., vol. 344, pp. 658–695, 2023, we estimate the maximum number of positive and negative limit cycles that such equations can have, and reveal how this maximum number, denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${H}_{{\theta }_{1}}\left(m\right)$ , is affected by the location of the separation line θ = θ 1 . For the equation of classical Abel type, our result not only includes the estimates provided in the recent paper (Huang et al., SIAM J. Appl. Dyn. Syst., 2020), i.e., H 2 π ( m ) ≥ 4 m − 2 for θ 1 = 2 π , but also shows that the equation in the discontinuous case can possess more than two times as many limit cycles as in the continuous case. More accurately, H π ( m ) ≥ 8 m + 2 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mn>14</m:mn> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>6</m:mn> </m:math> ${H}_{{\theta }_{1}}\left(m\right)\ge 14m-6$ for θ 1 ∈ (0, π ) ∪ ( π , 2 π ).