A cubic system with thirteen limit cycles

Li, Chengzhi; Liu, Changjian; Yang, Jiazhong

doi:10.1016/j.jde.2009.01.038

Cited by 120 publications

(86 citation statements)

References 6 publications

(9 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For cubic polynomial systems, many results have been obtained on the low bound of the Hilbert number. So far, the best result for cubic systems is Hð3Þ P 13 [7,8]. Note that the 13 limit cycles obtained in [7,8] are distributed around several singular points.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

Tian

2014

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

“…So far, the best result for cubic systems is Hð3Þ P 13 [7,8]. Note that the 13 limit cycles obtained in [7,8] are distributed around several singular points. This number is believed to be below the maximal number which can be obtained for generic cubic systems.…”

Section: Introductionmentioning

confidence: 99%

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

Tian

2014

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

“…So, up to now, the best result of possible estimation for number H(2) of limit cycles in quadratic system is H(2) ≥ 4 and it is finite (for cubic systems H(3) ≥ 13 [Li et al, 2009], and in the work [Han & Li, 2012] a lower estimate is given for the Hilbert number H(n): it grows at least as rapidly as (2ln2) −1 (n + 2) 2 ln(n + 2) for all large n).…”

Section: Th Hilbertmentioning

confidence: 99%

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Леонов

Kuznetsov

2013

Int. J. Bifurcation Chaos

770

414

View full text Add to dashboard Cite

From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors.At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes.Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit.This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods. †

show abstract

“…Recently, it has been proved that such Z 2 -equivariant system can have a 13th limit cycle at infinity [9]. Also, another cubic system with different structure has been reported to exhibit 13 limit cycles [8]. In [10], 11 cases have been specified to be integrable systems, among which one is a Hamiltonian system, and all other ten cases are integrable, non-Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of Limit Cycles in Cubic Integrable Z2-Equivariant Planar Vector Fields

Han

2010

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

In this paper, we study bifurcation of limit cycles in cubic planar integrable, non-Hamiltonian systems. The systems are assumed to be Z 2 -equivariant with two symmetric centers. Particular attention is given to bifurcation of limit cycles in the neighborhood of the two centers under cubic perturbations. Such integrable systems can be classified as 11 cases. It is shown that different cases have different number of limit cycles and the maximal number is 10. The method used in the paper relies on focus value computation.

show abstract

A cubic system with thirteen limit cycles

Abstract: We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.

Cited by 120 publications

References 6 publications

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Bifurcation of Limit Cycles in Cubic Integrable Z2-Equivariant Planar Vector Fields

Contact Info

Product

Resources

About