On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z 3 Invariant Quintic Perturbations

Wu, Yu; Han, Mao
May 2007
Acta Mathematica Sinica;May2007, Vol. 23 Issue 5, p869
Academic Journal
A cubic system having three homoclinic loops perturbed by Z 3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.


Related Articles

  • DISTRIBUTION OF RATIONAL MAPS WITH A PREPERIODIC CRITICAL POINT. Dujardin, Romain; Favre, Charles // American Journal of Mathematics;Aug2008, Vol. 130 Issue 4, p979 

    Let {fλ} be a family of rational maps of a fixed degree, with a marked critical point c(λ). Under a natural assumption, we first prove that the hypersurfaces of parameters for which c(λ) is periodic converge as a sequence of positive closed (1, 1) currents to the bifurcation current...

  • Bifurcation of the equilibrium states of a weightless liquid bridge. Slobozhanin, Lev A.; Iwan, J. // Physics of Fluids;Jul97, Vol. 9 Issue 7, p1893 

    Examines the bifurcation of the solutions of the nonlinear equilibrium problem of a weightless liquid bridge with a free surface pinned to the edges of two coaxial equidimensional circular disks. Study of the bifurcation in the neighborhood of the stability boundary for axisymmetric critical...

  • The Logistic Mapping. Piña, E. // AIP Conference Proceedings;2005, Vol. 757 Issue 1, p237 

    The Logistic Mapping is studied from a particular perspective. Firstly an identification is made of this mapping with the Newton iterating mapping to find the roots of a corresponding function. With this approach one identifies the fixed points and the periodic points of a stable cycle as zeros...

  • The Hindmarsh–Rose neuron model: Bifurcation analysis and piecewise-linear approximations. Storace, Marco; Linaro, Daniele; de Lange, Enno // Chaos;Sep2008, Vol. 18 Issue 3, p033128 

    This paper provides a global picture of the bifurcation scenario of the Hindmarsh–Rose model. A combination between simulations and numerical continuations is used to unfold the complex bifurcation structure. The bifurcation analysis is carried out by varying two bifurcation parameters...

  • Anticipatory Systems near Bifurcation Points. Béda, Péter B. // AIP Conference Proceedings;2006, Vol. 839 Issue 1, p610 

    Bifurcation in the sense of applied mathematics happens when the system is on the boundary of one set of equivalent states. Generally a system undergoing bifurcation is in a critical state. Any small change in the parameters may result essentially different behaviors. This phenomenon is called...

  • RANK ONE CHAOS IN SWITCH-CONTROLLED PIECEWISE LINEAR CHUA'S CIRCUIT. WANG, QIUDONG; OKSASOGLU, ALI // Journal of Circuits, Systems & Computers;Oct2007, Vol. 16 Issue 5, p769 

    In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in...

  • Bifurcation in coupled Hopf oscillators. Sterpu, Mihaela; Rocşoreanu, Carmen // AIP Conference Proceedings;2006, Vol. 835 Issue 1, p133 

    Two identical dynamical systems, representing the normal form corresponding to the Hopf bifurcation, were coupled using two parameters. The 4D dynamical system obtained possesses additional equilibria. Our study concerns the bifurcations of this system around the origin. We found that Hopf...

  • Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle. Zhao, Li; Li, De // Acta Mathematica Sinica;Mar2014, Vol. 30 Issue 3, p411 

    In this paper, we consider Liénard systems of the form where b ∈ ℝ, 0 < |∈| ≪ 1, ( α, β, γ) ∈ D ∈ ℝ and D is bounded. We prove that for | b| ≫ 1 ( b < 0) the least upper bound of the number of isolated zeros of the related Abelian integrals...

  • Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. Anishchenko, V.; Nikolaev, S.; Kurths, J. // Chaos;Sep2008, Vol. 18 Issue 3, p037123 

    We study synchronization of a resonant limit cycle on a two-dimensional torus with an external harmonic signal. The regime of the resonant limit cycle is realized in a system of two coupled Van der Pol oscillators; we consider the resonances 1:1 and 1:3. We analyze the influence of coupling...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics