Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3
Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either y2 = x3, or y2 = x2(x + 1), or y2 = x2(x - 1). We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic y2 = x2(x + 1). We prove that the...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:274781 |
| Acceso en línea: | https://ddd.uab.cat/record/274781 https://dx.doi.org/urn:doi:10.1142/S0218127423500037 |
| Access Level: | acceso abierto |
| Palabra clave: | Poincaré disk Global phase portrait Singular curve Cubic curve Polycycles Canonical regions Separatrices |
| Sumario: | Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either y2 = x3, or y2 = x2(x + 1), or y2 = x2(x - 1). We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic y2 = x2(x + 1). We prove that there are 63 different topological phase portraits for such quadratic polynomial differential systems. We control all the bifurcations among these distinct topological phase portraits. These systems have no limit cycles. Only three phase portraits have a center, 19 of these phase portraits have one polycycle, three of these phase portraits have two polycycles. The maximum number of separartices that have these phase portraits is 26 and the minimum number is nine, the maximum number of canonical regions of these phase portraits is seven and the minimum is three. |
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