Levandovskyy, Viktor; Pfister, Gerhard; Romanovski, Valery G.
September 2012
Communications on Pure & Applied Analysis;Sep2012, Vol. 11 Issue 5, p2029
Academic Journal
We describe an algorithmic approach to studying limit cycle bifurcations in a neighborhood of an elementary center or focus of a polynomial system. Using it we obtain an upper bound for cyclicity of a family of cubic systems. Then using a theorem by Christopher [3] we study bifurcation of limit cycles from each component of the center variety. We obtain also the sharp bound for the cyclicity of a generic time-reversible cubic system.


Related Articles

  • Bifurcation of Limit Cycles for 3D Lotka-Volterra Competitive Systems. Wang, Qinlong; Huang, Wentao; Wu, Haotao // Acta Applicandae Mathematica;Jun2011, Vol. 114 Issue 3, p207 

    Bifurcation of limit cycles is discussed for three-dimensional Lotka-Volterra competitive systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. Some new results are obtained for 6 classes 26-31 in Zeeman's...

  • Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations. Taborda, John Alexander; Arango, Ivan // Entropy;Jun2014, Vol. 16 Issue 6, p1949 

    In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and...

  • Limit Cycles Bifurcated from Some Z4-Equivariant Quintic Near-Hamiltonian Systems. Simin Qu; Cangxin Tang; Fengli Huang; Xianbo Sun // Abstract & Applied Analysis;2014, p1 

    We study the number and distribution of limit cycles of some planar Z4-equivariant quintic near-Hamiltonian systems. By the theories of Hopf and heteroclinic bifurcation, it is proved that the perturbed system can have 24 limit cycles with some new distributions. The configurations of limit...

  • Bifurcation in the stable manifold of the bioreactor with nth and mth order polynomial yields. Xuncheng Huang; Lemin Zhu // Journal of Mathematical Chemistry;Jul2009, Vol. 46 Issue 1, p199 

    The three dimensional chemostat with nth and mth order polynomial yields, instead of the particular one such as A + BS, A + BS2, A + BS3, A + BS4, A + BS2 + CS3 and A + BS n, is proposed. The existence of limit cycles in the two-dimensional stable manifold, the Hopf bifurcation and the stability...

  • Dulac-Cherkas function in a neighborhood of a structurally unstable focus of an autonomous polynomial system on the plane. Grin', A. // Differential Equations;Jan2014, Vol. 50 Issue 1, p1 

    We consider the problem of estimating the number of limit cycles and their localization for an autonomous polynomial system on the plane with fixed real coefficients and with a small parameter. At the origin, the system has a structurally unstable focus whose first Lyapunov focal quantity is...

  • The Number of Limit Cycles of a Polynomial System on the Plane. Chao Liu; Maoan Han // Abstract & Applied Analysis;2013, p1 

    We perturb the vector field x = -yC(x, y), y = xC(X, Y) with a polynomial perturbation of degree n, where C(x, y) = (1 - y2)m, and study the number of limit cycles bifurcating from the period annulus surrounding the origin.

  • A POINCARÉ BIFURCATION OF A CLASS OF HAMILTONIAN SYSTEM WITH CUBIC SOLUTIONS DISTURBED BY A CUBIC POLYNOMIAL. Chengbin Si; Yonghong Ren // Far East Journal of Applied Mathematics;Dec2013, Vol. 85 Issue 1/2, p1 

    The cubic function H(x, y) = xy2 + 2ey - x2 - (2 + e2 ) x - (1 + 2e2 ) = h is a Hamiltonian function of the following system: ẋ = 2xy + 2e, ẏ = (2 + e2 ) + 2x - y2, which can be bifurcated out at least four limit cycles after a cubic polynomial disturbance, i.e., B(2, 3) ≥ 4.

  • CENTRES AND LIMIT CYCLES FOR AN EXTENDED KUKLES SYSTEM. Hill, Joe M.; Lloyd, Noel G.; Pearson, Jane M. // Electronic Journal of Differential Equations;2007, Vol. 2007, p1 

    We present conditions for the origin to be a centre for a class of cubic systems. Some of the centre conditions are determined by finding complicated invariant functions. We also investigate the coexistence of fine foci and the simultaneous bifurcation of limit cycles from them.

  • Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems. Mönnigmann, M.; Marquardt, W. // Journal of Nonlinear Science;2002, Vol. 12 Issue 2, p85 

    Equilibrium solutions of systems of parameterized ordinary differential equations ... = f(x, α), x ∈ R[sup n], α ∈ R[sup m] can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical...


Read the Article


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

Try another library?
Sign out of this library

Other Topics