The geometry of bifurcation surfaces in parameter space. I. A walk through the pitchfork
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
1999
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| _version_ | 1866914100040171520 |
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| author | Ball, Rowena |
| author_facet | Ball, Rowena |
| contents | The classical pitchfork of singularity theory is a twice-degenerate bifurcation that typically occurs in dynamical system models exhibiting Z_2 symmetry. Non-classical pitchfork singularities also occur in many non-symmetric systems, where the total bifurcation environment is usually more complex. In this paper three-dimensional manifolds of critical points, or limit-point shells, are introduced by examining several bifurcation problems that contain a pitchfork as an organizing centre. Comparison of these surfaces shows that notionally equivalent problems can have significant positional differences in their bifurcation behaviour. As a consequence, the parameter range of jump, hysteresis, or phase transition phenomena in dynamical models (and the physical systems they purport to represent) is determined by other singularities that shape the limit-point shell. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_9910176 |
| institution | arXiv |
| publishDate | 1999 |
| record_format | arxiv |
| spellingShingle | The geometry of bifurcation surfaces in parameter space. I. A walk through the pitchfork Ball, Rowena Dynamical Systems Chaotic Dynamics Numerical Analysis Mathematical Physics 58Fxx The classical pitchfork of singularity theory is a twice-degenerate bifurcation that typically occurs in dynamical system models exhibiting Z_2 symmetry. Non-classical pitchfork singularities also occur in many non-symmetric systems, where the total bifurcation environment is usually more complex. In this paper three-dimensional manifolds of critical points, or limit-point shells, are introduced by examining several bifurcation problems that contain a pitchfork as an organizing centre. Comparison of these surfaces shows that notionally equivalent problems can have significant positional differences in their bifurcation behaviour. As a consequence, the parameter range of jump, hysteresis, or phase transition phenomena in dynamical models (and the physical systems they purport to represent) is determined by other singularities that shape the limit-point shell. |
| title | The geometry of bifurcation surfaces in parameter space. I. A walk through the pitchfork |
| topic | Dynamical Systems Chaotic Dynamics Numerical Analysis Mathematical Physics 58Fxx |
| url | https://arxiv.org/abs/math/9910176 |