Abstract
It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in ℝ2 is 3. In contrast here we consider discontinuous differential systems in ℝ2 defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of ℝ2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of ℝ2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.
Original language | English |
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Pages (from-to) | 3953-3965 |
Journal | Discrete and Continuous Dynamical Systems - Series B |
Volume | 22 |
Issue number | 10 |
DOIs | |
Publication status | Published - 1 Dec 2017 |
Keywords
- Cyclicity
- Discontinuous differential system
- Hopf bifurcation
- Limit cycle