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Bifurcations of Limit Cycles and Critical Periods

  • Valery Romanovski
  • Douglas Shafer
Chapter

In this chapter we consider systems of ordinary differential equations of the form
$$\dot u = \tilde U(u,v), \quad \dot v = \tilde V(u,v),$$
(6.1)
where u and v are real variables and \(\tilde U(u,v)\) and \(\tilde V(u,v)\) are polynomials for which max(deg,\(\tilde U\) deg\(\tilde V\)) ≤ n. The second part of the sixteenth of Hilbert’s well-known list of open problems posed in the year 1900 asks for a description of the possible number and relative locations of limit cycles (isolated periodic orbits) occurring in the phase portrait of such polynomial systems. The minimal uniform bound H (n) on the number of limit cycles for systems (6.1) (for some ӿxed n) is now known as the nth Hilbert number.

Keywords

Phase Portrait Critical Period Polynomial System Minimal Basis Quadratic System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Center for Applied Mathematics & Theorectical PhysicsUniversity of MariborSlovenia
  2. 2.Mathematics Dept.University of North CarolinaCharlotteUSA

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