Orbital normal forms for a class of three-dimensional systems with an application to Hopf-zero bifurcation analysis of Fitzhugh-Nagumo system
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arXiv
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| Format: | Preprint |
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2024
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| author | Algaba, A Fuentes, N Gamero, E García, C |
| author_facet | Algaba, A Fuentes, N Gamero, E García, C |
| contents | We consider a class of three-dimensional systems having an equilibrium point at the origin, whose principal part is of the form (-Dy h(x, y), Dx h(x,y), f(x,y))^T . This principal part, which has zero divergence and does not depend on the third variable z, is the coupling of a planar Hamiltonian vector field Xh(x,y)=(-Dy h(x, y), Dx h(x,y))^T with a one-dimensional system. We analyze the quasi-homogeneous orbital normal forms for this kind of systems, by introducing a new splitting for quasi-homogeneous three-dimensional vector fields. The obtained results are applied to the nondegenerate Hopf-zero singularity that falls into this kind of systems. Beyond the Hopf-zero normal form, a parametric normal form is obtained, and the analytic expressions for the normal form coefficients are provided. Finally, the results are applied to a case of the three-dimensional Fitzhugh-Nagumo system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbital normal forms for a class of three-dimensional systems with an application to Hopf-zero bifurcation analysis of Fitzhugh-Nagumo system Algaba, A Fuentes, N Gamero, E García, C Dynamical Systems 14J60 (Primary) 14F05, 14J26 (Secondary) G.1.7 We consider a class of three-dimensional systems having an equilibrium point at the origin, whose principal part is of the form (-Dy h(x, y), Dx h(x,y), f(x,y))^T . This principal part, which has zero divergence and does not depend on the third variable z, is the coupling of a planar Hamiltonian vector field Xh(x,y)=(-Dy h(x, y), Dx h(x,y))^T with a one-dimensional system. We analyze the quasi-homogeneous orbital normal forms for this kind of systems, by introducing a new splitting for quasi-homogeneous three-dimensional vector fields. The obtained results are applied to the nondegenerate Hopf-zero singularity that falls into this kind of systems. Beyond the Hopf-zero normal form, a parametric normal form is obtained, and the analytic expressions for the normal form coefficients are provided. Finally, the results are applied to a case of the three-dimensional Fitzhugh-Nagumo system. |
| title | Orbital normal forms for a class of three-dimensional systems with an application to Hopf-zero bifurcation analysis of Fitzhugh-Nagumo system |
| topic | Dynamical Systems 14J60 (Primary) 14F05, 14J26 (Secondary) G.1.7 |
| url | https://arxiv.org/abs/2409.18079 |