Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \R^2 is 3. In contrast here we consider discontinuous differential systems in \R^2 defined in two half--planes separated by a straight line. In one half plane we have a general linear...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2017 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:182523 |
| Online Access: | https://ddd.uab.cat/record/182523 https://dx.doi.org/urn:doi:10.3934/dcdsb.2017203 |
| Access Level: | Open access |
| Keyword: | Cyclicity Discontinuous differential system Hopf bifurcation Limit cycles |
| Summary: | It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \R^2 is 3. In contrast here we consider discontinuous differential systems in \R^2 defined in two half--planes separated by a straight line. In one half plane we have a general linear center at the origin of \R^2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of \R^2. 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. |
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