Canard cycles of non-linearly regularized piecewise smooth vector fields
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| Format: | Preprint |
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2025
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| _version_ | 1866915354886799360 |
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| author | De Maesschalck, Peter Huzak, Renato Perez, Otavio Henrique |
| author_facet | De Maesschalck, Peter Huzak, Renato Perez, Otavio Henrique |
| contents | The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for upper bounds and the existence of limit cycles of canard type, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of piecewise linear center such that for any integer $k>0$ it has at least $k+1$ limit cycles, for a suitably chosen monotonic transition function $φ_k:\mathbb{R}\rightarrow\mathbb{R}$. We prove a similar result for regularized invisible-invisible fold-fold singularities of type II$_2$. Canard cycles of dodging layer are also considered, and we prove that such limit cycles undergo a saddle-node bifurcation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18099 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canard cycles of non-linearly regularized piecewise smooth vector fields De Maesschalck, Peter Huzak, Renato Perez, Otavio Henrique Dynamical Systems 34D15 The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for upper bounds and the existence of limit cycles of canard type, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of piecewise linear center such that for any integer $k>0$ it has at least $k+1$ limit cycles, for a suitably chosen monotonic transition function $φ_k:\mathbb{R}\rightarrow\mathbb{R}$. We prove a similar result for regularized invisible-invisible fold-fold singularities of type II$_2$. Canard cycles of dodging layer are also considered, and we prove that such limit cycles undergo a saddle-node bifurcation. |
| title | Canard cycles of non-linearly regularized piecewise smooth vector fields |
| topic | Dynamical Systems 34D15 |
| url | https://arxiv.org/abs/2506.18099 |