Nonsmooth folds as tipping points
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912200256389120 |
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| author | Simpson, D. J. W. |
| author_facet | Simpson, D. J. W. |
| contents | A nonsmooth fold is where an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation the leading-order truncation to the system in general has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ODEs, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus if the equilibrium or limit cycle is attracting the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of the bifurcation structures of the truncated systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04587 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonsmooth folds as tipping points Simpson, D. J. W. Dynamical Systems 37G35, 37G15 A nonsmooth fold is where an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation the leading-order truncation to the system in general has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ODEs, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus if the equilibrium or limit cycle is attracting the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of the bifurcation structures of the truncated systems. |
| title | Nonsmooth folds as tipping points |
| topic | Dynamical Systems 37G35, 37G15 |
| url | https://arxiv.org/abs/2406.04587 |