Global dynamics of stationary solutions of the extended Fisher-Kolmogorov equation
In this paper we study the fourth order differential equation d 4u /dt4 + q (d 2u /dt2 )+ u 3 - u = 0, which arises from the study of stationary solutions of the Extended Fisher-Kolmogorov equation. Denoting x = u, y = du/ dt , z = d 2u/ dt2 , v = d 3u/ dt3 this equation becomes equivalent to the po...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2011 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:150412 |
| Online Access: | https://ddd.uab.cat/record/150412 https://dx.doi.org/urn:doi:10.1063/1.3657425 |
| Access Level: | Open access |
| Keyword: | Integrability Extended Fisher-Kolmogorov equation Poincaré compactification Global dynamics |
| Summary: | In this paper we study the fourth order differential equation d 4u /dt4 + q (d 2u /dt2 )+ u 3 - u = 0, which arises from the study of stationary solutions of the Extended Fisher-Kolmogorov equation. Denoting x = u, y = du/ dt , z = d 2u/ dt2 , v = d 3u/ dt3 this equation becomes equivalent to the polynomial system ˙x = y, y˙ = z, z˙ = v, v˙ = x - qz - x 3 with (x, y, z, v) ∈ R 4 and q ∈ R. As usual, the dot denotes derivative with respect to the time t. Since the system has a first integral we can reduce our analysis to a family of systems on R 3 . We provide the global phase portrait of these systems in the Poincar'e ball (i.e. in the compactification of R 3 with the sphere S 2 of the infinity). |
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