| Autor: | Marcin Bobieński, Henryk Żoładek |
|---|---|
| Rok vydání: | 2003 |
| Předmět: |
Numerical Analysis
Control and Optimization Algebra and Number Theory Differential equation Mathematical analysis Perturbation (astronomy) Rational function symbols.namesake Control and Systems Engineering Limit cycle Bounded function symbols Elliptic integral Vector field Hamiltonian (quantum mechanics) Mathematics Mathematical physics |
| Zdroj: | Journal of Dynamical and Control Systems. 9:265-310 |
| ISSN: | 1079-2724 |
| Popis: | We consider polynomial vector fields of the form \.{x} = 2_y + zR(x,y), \.{y} = 3x^2 - 3 + zS(x,y), \.{z} = A_x,y)z, z \epsilon \Ropf^v, and their polynomial perturbations of degree ln. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y2 − x3 + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R1(h)I1 + ⋅⋅⋅ + Re(h)Ie, where Rj(h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x,y) we estimate the number of zeros of the integral by const ⋅n. |
| Databáze: | OpenAIRE |
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