Bifurcation of Limit Cycles and Isochronous Centers for a Quartic System

Huang, Wentao; Chen, Aiyong; Xu, Qiujin

doi:10.1142/s0218127413501721

Cited by 9 publications

(6 citation statements)

References 15 publications

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“…For the center-focus determination and Hopf bifurcation on the center manifold, there have been some valid research approaches, such as the averaging theory considered in Llibre et al and Barreira et al, 18,19 the technique of inverse Jacobi multiplier studied in Buica et al, 20,21 the simplest normal form method given in Tian and Yu,22 and the formal first integral method given in Edneral et al 23 Here, we apply the formal series method introduced in Wang et al, 8 as an upgraded version of the method given in Liu and Li and Liu,24,25 which are also very valid to study the Hopf bifurcation, for two-dimensional system, some results are obtained, see eg, previous studies, [26][27][28][29] for three-dimensional system, see eg, previous studies. [30][31][32] The method in Wang et al 8 mainly consider the three-dimensional system (4) as follows:…”

Section: Methods To Study the 3d Hopf Bifurcationmentioning

confidence: 99%

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

Huang

Wang

Chen

2019

Math Methods in App Sciences

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In this paper, Hopf bifurcation and center problem for a generic three-dimensional Chua's circuit system are studied. Applying the formal series method of computing singular point quantities to investigate the two cases of the generic circuit system, we find necessary conditions for the existence of centers on a local center manifold for the systems, then Darboux method is applied to show the sufficiency. Further, we determine the maximum number of limit cycles that can bifurcate from the corresponding equilibrium via Hopf bifurcation. KEYWORDS3D circuit system, Hopf bifurcation, symbolic computing, singular point quantities and center MSC CLASSIFICATION 34C07; 34C151988

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Section: Methods To Study the 3d Hopf Bifurcationmentioning

confidence: 99%

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

Huang

Wang

Chen

2019

Math Methods in App Sciences

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“…In literature, 31 a method was given to compute complex period constants of planar systems with an algorithm that is linear recursive and avoids complex integral computation. The application of this method can been seen in previous works, 32,33 for instance. Recently, this method has been extended and extended to three-dimensional systems in reference.…”

Section: Introductionmentioning

confidence: 96%

Bifurcation of limit cycles and isochronous centers on center manifolds for four‐dimensional systems

Huang

Wang

et al. 2021

Math Methods in App Sciences

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This paper is concerned with the bifurcation of limit cycles and isochronous centers on center manifolds for four-dimensional nonlinear dynamic systems, which have a pair of purely imaginary eigenvalues and a pair of eigenvalues with negative real part for the Jacobian matrix corresponding to the singularity situated at the origin. In order to investigate the isochronous centers for four-dimensional systems, a new algorithm to calculate so-called isochronous constants is given. Moreover, with the calculation of the isochronous constants, the conditions for the existence of an isochronous center are determined in a way which does not need first computing the center manifolds of the four-dimensional system. Finally, as an application, we investigate a class of quadratic polynomial systems for a model of airfoil and prove that the system can have five small limit cycles around the origin on the center manifold. Moreover, we apply our new methodology to find the sufficient and necessary conditions for the origin to be an isochronous center restricted to the center manifold, illustrating the reliability and availability of the new algorithm.

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“…However, in general cubic systems, the maximal number M (3) is still open, many results have been obtained on its low bound [15,16], so far, the best result is M (3) ≥ 12 [17]. For other relevant results, one can see [18,19] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Multiple limit cycles bifurcation and Jacobi stability for a class of segmented disc dynamo system

Liu,

Wang,

Huang

2024

Preprint

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In this paper, the multiple bifurcation of limit cycles for a segmented disc dynamo system is studied. The formal series method for calculating the singular point quantities is applied to determine the highest order focus value at Hopf bifurcation point. For two cases of the segmented disc dynamo system, namely the system with or without friction coefficient (abbr. SDDF- or SDD-model), the maximum number of limit cycles is obtained at the symmetrical equilibrium points under the condition of synchronous perturbation respectively. At the same time, the parameters condition is classified for exact number of limit cycles near each weak focus. Finally, we find that all equilibrium points of the heart model are Jacobi unstable under certain parameter values.

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Bifurcation of Limit Cycles and Isochronous Centers for a Quartic System

Cited by 9 publications

References 15 publications

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

Bifurcation of limit cycles and isochronous centers on center manifolds for four‐dimensional systems

Multiple limit cycles bifurcation and Jacobi stability for a class of segmented disc dynamo system

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