Limit cycles of discontinuous piecewise polynomial vector fields

Carvalho, Tiago; Llibre, Jaume; Tonon, Durval J.

doi:10.1016/j.jmaa.2016.11.048

Cited by 9 publications

(8 citation statements)

References 26 publications

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“…and all the results of Theorem 1.1 for N = 1 is just the ones in [9,Theorem 1]. However, in this paper we consider the higher order perturbations of the center (1.1) and, by using the higher order averaging theory we obtain new results for N ≥ 2 as stated in Theorem 1.1.…”

Section: And This Upper Bound Is Reached Formentioning

confidence: 75%

“…On the other hand in [9] Carvalho, Llibre and Tonon considered the first order discontinuous piecewise polynomial perturbations of the center (1.1) with m ≥ 1 in k zones separated by k rays originating at the origin. According to [9, Theorem 1], for given m, k and degree n the first order averaging theory at most provides n crossing limit cycles bifurcating from the unperturbed periodic orbits and this number is reached.…”

Section: And This Upper Bound Is Reached Formentioning

confidence: 99%

“…According to [9, Theorem 1], for given m, k and degree n the first order averaging theory at most provides n crossing limit cycles bifurcating from the unperturbed periodic orbits and this number is reached. If we restrict the perturbation considered in [9] to the case that k = 4 and the four rays are exactly the coordinate axes, then it corresponds to the first order case of system (1.2), i.e.…”

Section: And This Upper Bound Is Reached Formentioning

confidence: 99%

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Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

Li¹,

Llibre²

2021

CPAA

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In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center <inline-formula><tex-math id="M1">\begin{document}$ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m\ge0 $\end{document}</tex-math></inline-formula> under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> with the discontinuity set <inline-formula><tex-math id="M4">\begin{document}$ \{(x, y)\in\mathbb{R}^2: xy = 0\} $\end{document}</tex-math></inline-formula>. Using the averaging theory up to any order <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, we give upper bounds for the maximum number of limit cycles in the function of <inline-formula><tex-math id="M6">\begin{document}$ m, n, N $\end{document}</tex-math></inline-formula>. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.

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Section: And This Upper Bound Is Reached Formentioning

confidence: 75%

Section: And This Upper Bound Is Reached Formentioning

confidence: 99%

Section: And This Upper Bound Is Reached Formentioning

confidence: 99%

See 1 more Smart Citation

Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

Li¹,

Llibre²

2021

CPAA

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“…Piecewise smooth differential system is a kind of important non-smooth system which is based on non-smooth model. In the past few decades, many authors have been devoted to study the number of limit cycles of piecewise smooth differential systems with two zones separated by a switching line, see [3,4,7,9,13,15,17,21,22,25,27] and the references quoted therein. The ways used in the aforementioned works are Melnikov function established in [8,17] and averaging method developed in [18,19].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Yang

2020

Chaos, Solitons & Fractals

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This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves y = ±x 2 . By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles which bifurcate from the period annulus around the origin under n-th degree polynomial perturbations. In the case n = 1, we obtain that at least 4 (resp. 3) limit cycles can bifurcate from the period annulus if the switching curves are y = ±x 2 (resp. y = x 2 or y = −x 2 ). The results also show that the number of switching curves affects the number of limit cycles.

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“…These oscillations are named limit cycles, e.g., RLC electrical circuit with a nonlinear resistor and Van der Pol equation. Limit cycles are special phenomenon of nonlinear networks and have been widely investigated; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Limit Cycles Investigation for a Class of Nonlinear Systems via Differential and Integral Inequalities

Sun¹

2017

IJTSRD

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In this paper, the existence of limit cycles of nonlinear systems is explored. Based on the time domain approach with differential inequalities, the phenomenon of the stable limit cycle can be accurately verified for such nonlinear Furthermore, the exponentially stable limit cycles frequency of oscillation, and guaranteed convergence rate can be correctly calculated. Finally, numerical simulations are provided to demonstrate feasibility and effectiveness of the main result

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Limit cycles of discontinuous piecewise polynomial vector fields

Cited by 9 publications

References 26 publications

Limit cycles of piecewise polynomial differential systems with the discontinuity line <i>xy</i> = 0

Limit cycles of piecewise polynomial differential systems with the discontinuity line <i>xy</i> = 0

Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Limit Cycles Investigation for a Class of Nonlinear Systems via Differential and Integral Inequalities

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