Abstract
This paper is devoted to the investigation of Abel equation \(\dot{x}=S(t,x)=\sum ^3_{i=0}a_i(t)x^i\), where \(a_i\in \mathrm C^{\infty }([0,1])\). A solution x(t) with \(x(0)=x(1)\) is called a periodic solution. And an orbit \(x=x(t)\) is called a limit cycle if x(t) is a isolated periodic solution. By means of Lagrange interpolation formula, we give a criterion to estimate the number of limit cycles of the equation. This criterion is only concerned with S(t, x) on three non-intersecting curves. Applying our main result, we prove that the maximum number of limit cycles of the equation is 4 if \(a_2(t)a_0(t)<0\). To the best of our knowledge, this is a nontrivial supplement for a classical result which says that the equation has at most 3 limit cycles when \(a_2(t)\ne 0\) and \(a_0(t)\equiv 0\). We also study a planar polynomial system with homogeneous nonlinearities:
where \(a\in \mathbb R\) and \(P_n, Q_n\) are homogeneous polynomials of degree \(n\ge 2\). Denote by \(\psi (\theta )=\cos (\theta )\cdot Q_n\big (\cos (\theta ),\sin (\theta )\big )-\sin (\theta )\cdot P_n\big (\cos (\theta ),\sin (\theta )\big )\). We prove that if \((n-1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0\), then the polynomial system has at most 1 limit cycle surrounding the origin, and the multiplicity is no more than 2.
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The first author is supported by the NSF of China (No. 11401255) and the China Scholarship Council (No. 201606785007), the Fundamental Research Funds for the Central Universities (No. 21614325) and the China Scholarship Council (No. 201606785007). The second author is supported by the NSF of Guangdong Province (No. 2015A030313669).
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Huang, J., Liang, H. Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves. Nonlinear Differ. Equ. Appl. 24, 47 (2017). https://doi.org/10.1007/s00030-017-0469-3
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DOI: https://doi.org/10.1007/s00030-017-0469-3