Bifurcation of periodic orbits and its application for high-dimensional piecewise smooth near integrable systems with two switching manifolds

Li, Jing; Guo, Zhijun; Zhu, Shaotao; Gao, Ting

doi:10.1016/j.cnsns.2022.106840

Cited by 10 publications

(4 citation statements)

References 22 publications

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“…Toward this goal and to prove the theorem theoretically, Section 3 discusses that M(ξ), a Melnikov mapping whose order is 1, and the PF mappings are expressed and fulfilled by the originators of M(ξ). This work follows the discussions given recently in [33][34][35]. In terms of novelty, here by considering some piecewise smooth Liênard mappings with a small enough |ε|, a higher boundary for the LCNs bifurcated based on the annulus period is given for different values of n. The proof of the main result and the related discussions are furnished in Section 4.…”

Section: Background and Literaturementioning

confidence: 88%

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Zhang,

Yang,

Shateyi

2023

Mathematics

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Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS.

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Section: Background and Literaturementioning

confidence: 88%

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Zhang,

Yang,

Shateyi

2023

Mathematics

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“…using the above equality, where γ δ are constants. Differentiating (27) [ m−1 2 ] + 2 times using Lemma 3 furnishes us:…”

Section: By Proposition 1 We Know That Cmentioning

confidence: 99%

“…Their study delved into the continuity of periodic orbits, particularly when the unperturbed system itself possesses such orbits [24]. Other new results about periodic orbits of piecewise smooth systems can be traced in [25][26][27].…”

Section: Introductory Notesmentioning

confidence: 99%

Exploring Limit Cycle Bifurcations in the Presence of a Generalized Heteroclinic Loop

Zhang,

Shateyi

2023

Mathematics

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This work revisits the number of limit cycles (LCs) in a piecewise smooth system of Hamiltonian with a heteroclinic loop generalization, subjected to perturbed functions through polynomials of degree m. By analyzing the asymptotic expansion (AE) of the Melnikov function with first-order M(h) near the generalized heteroclinic loop (HL), we utilize the expansions of the corresponding generators. This approach allows us to establish both lower and upper bounds for the quantity of limit cycles in the perturbed system. Our analysis involves a combination of expansion techniques, derivations, and divisions to derive these findings.

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“…However, little attention has been paid to the periodic or multiple periodic vibration behaviors of cylindrical shells structure. The development of periodic solutions theory [9][10][11] is conducive to a deep understanding of the vibration characteristics of the system in different parameter regions, and provides guidance for the vibration reduction design of nonlinear dynamics theory.…”

Section: Introductionmentioning

confidence: 99%

Multiple periodic motions of a two degrees-of-freedom carbon fiber reinforced polymer laminated cylindrical shell

Gao,

Li,

Zhu

et al. 2023

Vib. proced.

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Carbon fiber reinforced polymer is a composite material, which is widely used in various engineering fields due to its excellent properties. We systematically discuss the influence of axial load amplitude parameters on the multiple periodic motions of carbon fiber reinforced polymer laminated cylindrical shell model. Based on the Melnikov vector function, the bifurcation regions of periodic orbits are obtained. It is found that the system has at most four periodic orbits under parameters conditions. Moreover, the phase portraits of periodic orbits are given by numerical simulation. The results offer an idea for parameter control of shell structure.

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Bifurcation of periodic orbits and its application for high-dimensional piecewise smooth near integrable systems with two switching manifolds

Cited by 10 publications

References 22 publications

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Exploring Limit Cycle Bifurcations in the Presence of a Generalized Heteroclinic Loop

Multiple periodic motions of a two degrees-of-freedom carbon fiber reinforced polymer laminated cylindrical shell

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