ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Global Bifurcations of Limit Cycles


V. A. Gaiko

Belarussian State University of Informatics and Radioelectronics
Department of Mathematics
Belarus, 220090 Minsk, Koltsov Str.49–305


Two-dimensional polynomial dynamical systems are considered. The main problem of the qualitative theory of such systems is Hilbert's Sixteenth Problem: Problem. Find the maximum number and relative position of limit cycles. There are three bifurcations of limit cycles:

  • Andronov-Hopf bifurcation (from a singular point of the center or focus type);
  • separatrix cycle bifurcation (from a homoclinic or heteroclinic orbit);
  • multiple limit cycle bifurcation.
  • All these bifurcations are local ones. We consider only a neighborhood of either the point or the separatrix cycle, or the multiple limit cycle studying only the corresponding sufficiently small neighborhood in the parameter space. The analysis of each of these bifurcations individually does not yield a complete solution of the Problem even in the simplest (quadratic) case of nonlinear polynomial systems. We connect all local bifurcations of limit cycles, develop a global bifurcation theory of polynomial dynamical systems and on the example of quadratic systems outline a global approach to the solution of Hilbert's 16th Problem. We discuss also possibilities of application of the obtained results to higher-dimensional dynamical systems and for the global qualitative investigation of mathematical models from microelectronics and ecology. First, we recall some previous results. Using Erugin's ideas on the qualitative investigation on the whole (both on the whole phase plane and on the whole parameter space) and his two-isocline method, we construct canonical systems with field-rotation parameters and give a geometric interpretation of all four cases of center for the quadratic systems:   1) axial symmetry;   2) local symmetry (zero divergence) on the whole phase plane (Hamiltonian case);   3) orthogonality of asymptotes of hyperbolas forming the family of isoclines (Lotka-Volterra case);   4) orthogonality of asymptotes of saddles at infinity. Basing on the center cases and applying the field-rotation parameters, we carry out the classification of separatrix cycles with the corresponding division of the parameter space and control the multiple limit cycle bifurcations. Then, using the Wintner-Perko termination principle, we give a sketch of proof of the following conjecture: Conjecture 1. There exists no quadratic system having a swallow-tail bifurcation surface of multiplicity-four limit cycles in its parameter space. In other words, a quadratic system cannot have neither a multiplicity-four limit cycle nor four limit cycles around a singular point (focus) and the maximum multiplicity or the maximum number of limit cycles surrounding a focus is equal to three. The proof is carried out by contradiction. We suppose that our canonical system containing three field-rotation parameters has four limit cycles around the origin; then we get into some three-dimensional domain of the field-rotation parameters being restricted by some conditions on the rest two parameters corresponding to the definite case of singular points in the phase plane. This three-parameter domain of four limit cycles is bounded by three fold bifurcation surfaces forming a swallow-tail bifurcation surface of multiplicity-four limit cycles. It can be shown that the corresponding maximal one-parameter family of multiplicity-four limit cycles cannot be cyclic and terminates either at the origin or on some separatrix cycle surrounding the origin. Since we know absolutely precisely at least the cyclicity of the singular point (the result by N.N.Bautin) which is equal to three, we have got a contradiction with the termination principle stating that the multiplicity of limit cycles cannot be higher than the multiplicity (cyclicity) of the singular point in which they terminate. This contradiction concludes the proof. Since we know the concrete properties of all three field-rotation parameters in the canonical system and, besides, we are able to control simultaneously bifurcations of limit cycles around different singular points, we can formulate also Conjecture 2. The maximum number of limit cycles in a quadratic system is equal to four and the only possible their distribution is (3:1).

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