A new Chebyshev family with applications to Abel equations

DOI10.1016/j.jde.2011.06.010zbMath1233.41003OpenAlexW2025644638MaRDI QIDQ652496

Armengol Gasull, Joan Torregrosa, Chengzhi Li

Publication date: 14 December 2011

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jde.2011.06.010

zbMATH Keywords

periodic solutionWronskianChebyshev systemAbel equationintegral Gram determinantnumber of zeroes of real analytic functions

Mathematics Subject Classification ID

Bifurcation theory for ordinary differential equations (34C23) Best approximation, Chebyshev systems (41A50) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07)

Related Items (17)

On the number of limit cycles for perturbed pendulum equations ⋮ A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities ⋮ Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities ⋮ Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves ⋮ A Chebyshev criterion with applications ⋮ The limit cycle bifurcations of a whirling pendulum with piecewise smooth perturbations ⋮ Bifurcation of a Kind of 1D Piecewise Differential Equation and Its Application to Piecewise Planar Polynomial Systems ⋮ Analysis of Abel-type nonlinear integral equations with weakly singular kernels ⋮ Bifurcation of a Kind of Piecewise Smooth Generalized Abel Equation via First and Second Order Analyses ⋮ Limit cycles of piecewise differential equations on the cylinder ⋮ Abelian integrals and limit cycles ⋮ A geometric criterion for equation \(\dot{x} = \sum\nolimits_{i = 0}^m a_i(t) x^i\) having at most \(m\) isolated periodic solutions ⋮ New family of abelian integrals satisfying Chebyshev property ⋮ On the Number of Limit Cycles in Generalized Abel Equations ⋮ Periodic orbits from second order perturbation via rational trigonometric integrals ⋮ Further study on Horozov-Iliev's method of estimating the number of limit cycles ⋮ A new Chebyshev criterion and its application to planar differential systems

Cites Work

• On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
• Lower bounds for the number of limit cycles of trigonometric Abel equations
• Limit Cycles for a Class of Abel Equations
• Centennial History of Hilbert's 16th Problem
• LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS
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