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On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems

Aziza Berbache (2023)

Mathematica Bohemica

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we...

On lopsided systems.

Javier Chavarriga, Nazim Salih (2002)

Extracta Mathematicae

On the approximation of the limit cycles function.

Cherkas, L., Grin, A., Schneider, K.R. (2007)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

On the number of limit cycles of a generalized Abel equation

Naeem Alkoumi, Pedro J. Torres (2011)

Czechoslovak Mathematical Journal

New results are proved on the maximum number of isolated $T$ -periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.

On the number of zeros of Melnikov functions

Sergey Benditkis, Dmitry Novikov (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order $k$ of the Melnikov function. The generic case $k = 1$ was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.