Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

Wei, Minzhi; Cai, Junning; Zhu, Hongxiang

doi:10.1155/2019/6943563

Discrete Dynamics in Nature and Society

2019

DOI: 10.1155/2019/6943563

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Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

Minzhi Wei

Junning Cai

Hongxiang Zhu

Abstract: In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.

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2023

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“…Using the Melnikov function method the results for some Lienard systems with n = 4 and m = 3 were obtained in [13], [14], [15], with n = 7, m = 4 -in [17], with n = 4 and m = 3, 100 -in [16].…”

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confidence: 99%

Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part

Basov¹,

Zhukov²

2023

Preprint

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Two classes of two-dimensional time-periodic systems of ordinary differential equations with a small parameter ε > 0 in the perturbed part, which is continuous and, for ε = 0, analytic in zero, are studied. Depending on the presence or absence of the common factor ε, these classes contain the system with "fast" or "slow" time. The unperturbed part of these systems is generated by the hamiltonianThe universal method, meaning it can be applied to any hamiltonian, called the method of the generating tori splitting (GTS method), is developed and applied to the research of such systems. For arbitrary system of any class, this method allows to find the sets of the initial values for the solutions of the corresponding unperturbed system, and for each such set, to provide the explicit conditions on the system perturbations independent of the parameter. Any such set, that satisfies the aforementioned conditions, determines the solution of the unperturbed system, which parametrizes the generating cycle. The obtained cycle is a generatrix of an invariant cylindrical surface. It is proven that the system has twoperiodic invariant surface, homeomorphic to torus, if time is factored with the respect to the period, in the small with the respect to ε neighbourhood of this surface. The formula and the asymptotic extension are provided for this surface, the number of properties is discovered.An example of the set of systems with eleven invariant tori and a perturbation, which average value is a three-term polynomial of the third degree, is constructed as a demonstration of the practical use of the GTS method.The GTS method is an universal alternative to the so-called method of detection functions and the Melnikov function method, which are used in studies concerning the weakened XVI Hilbert's problem on the evaluation of a number of limit cycles of autonomous systems with the hamiltonian unperturbed part. The GTS method allows not only to evaluate the lower bound of the analogue of the Hilbert's cyclicity value, which determines the amount of the invariant tori in the periodic systems with "slow" time, but also to solve the same problem for the periodic systems of any even degree with the common factor ε in its right-hand side. The results can be put into practice, while researching the systems of the ordinary differential equations of the second degree, which describe the oscillations of the weaklycoupled oscillators.

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mentioning

confidence: 99%

Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part

Basov¹,

Zhukov²

2023

Preprint

View full text Add to dashboard Cite

show abstract

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Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

Cited by 1 publication

References 30 publications

Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part

Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part

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