On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

In this paper, we consider the bifurcation of limit cycles for generic L–V system ( ẋ=y+x2−y2±43xy,ẏ=−x+2xy) and B–T system (ẋ=y,ẏ=−x+x2) under perturbations of piecewise smooth polynomials with degree n. After linear transformation, we choose switching line y=0. By using Picard–Fuchs equations,...

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Veröffentlicht in:	Nonlinear analysis: real world applications 2019-10, Vol.49, p.137-158
Hauptverfasser:	Sui, Shiyou, Yang, Jihua, Zhao, Liqin
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Sprache:	eng
Schlagworte:	Bifurcations Competitive advantage Economic models First order Melnikov function Limit cycle Linear transformations Picard–Fuchs equation Piecewise smooth perturbation Polynomials Upper bounds
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creator	Sui, Shiyou Yang, Jihua Zhao, Liqin
description	In this paper, we consider the bifurcation of limit cycles for generic L–V system ( ẋ=y+x2−y2±43xy,ẏ=−x+2xy) and B–T system (ẋ=y,ẏ=−x+x2) under perturbations of piecewise smooth polynomials with degree n. After linear transformation, we choose switching line y=0. By using Picard–Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles (taking into account the multiplicity) for generic L–V system and B–T system are respectively 39n−72(n≥4),39,59,98(n=1,2,3) and 12n+3n2+5.
doi_str_mv	10.1016/j.nonrwa.2019.03.001
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After linear transformation, we choose switching line y=0. By using Picard–Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. 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After linear transformation, we choose switching line y=0. By using Picard–Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. 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subjects	Bifurcations Competitive advantage Economic models First order Melnikov function Limit cycle Linear transformations Picard–Fuchs equation Piecewise smooth perturbation Polynomials Upper bounds
title	On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials
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