Bifurcations of Limit Cycles and Critical Periods

  • Valery Romanovski
  • Douglas Shafer

In this chapter we consider systems of ordinary differential equations of the form
$$\dot u = \tilde U(u,v), \quad \dot v = \tilde V(u,v),$$
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.


Phase Portrait Critical Period Polynomial System Minimal Basis Quadratic System 
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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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