Limit cycle bifurcations for piecewise smooth integrable differential systems

Yang, Jihua; Zhao, Liang

doi:10.3934/dcdsb.2017123

Cited by 14 publications

(10 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…Liu Fei et al [21] studied the cyclic response characteristics of a class of piecewise nonlinear elastic damped double constraint systems. Yang et al [22] studied a class of piecewise smooth integrable non-Hamiltonian systems with centers. Yu [23] used the Melnikov function method to study the maximum number of limit cycles bifurcated from the periodic cycle domain of the nonlinear center of a class of discontinuous generalized Lienard differential systems.…”

Section: Introductionmentioning

confidence: 99%

Study on a Class of Piecewise Nonlinear Systems with Fractional Delay

Wang

Chen

et al. 2021

Shock and Vibration

View full text Add to dashboard Cite

In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given.

show abstract

“…Proof. Applying Theorem 1.1, it is easy to see that system (19) has at most 4 limit cycles by the averaging method of second order. Now we prove 4 limit cycles can appear.…”

mentioning

confidence: 99%

“…For example, the authors in [11] developed the Melnikov function method for piecewise smooth planar systems, and established a formula for the first order Melnikov function. From [7,18,19], we know that one can consider the number of limit cycles for piecewise polynomial systems by using the method of first order Melnikov function in Hopf and generalized homoclinic bifurcations. Recently the authors of [17] applied the Melnikov function theory to high-dimensional piecewise smooth near-integrable systems and gave a formula for the first order Melnikov vector function.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

Liu¹,

Han²

2020

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

show abstract

Limit cycle bifurcations for piecewise smooth integrable differential systems

Abstract: In this paper, we study a class of piecewise smooth integrable non-Hamiltonian systems, which has a center. By using the first order Melnikov function, we give an exact number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n.

Cited by 14 publications

References 16 publications

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

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

Study on a Class of Piecewise Nonlinear Systems with Fractional Delay

Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

Contact Info

Product

Resources

About