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Twin Heteroclinic Connections of Reversible Systems

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Abstract

We examine smooth four-dimensional vector fields reversible under some smooth involution \(L\) that has a smooth two-dimensional submanifold of fixed points. Our main interest here is in the orbit structure of such a system near two types of heteroclinic connections involving saddle-foci and heteroclinic orbits connecting them. In both cases we found families of symmetric periodic orbits, multi-round heteroclinic connections and countable families of homoclinic orbits of saddle-foci. All this suggests that the orbit structure near such connections is very complicated. A non-variational version of the stationary Swift – Hohenberg equation is considered, as an example, where such structure has been found numerically.

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Fig. 1
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Notes

  1. In fact, points \(\Phi^{t_{1}}(b)\) and \(L\circ\Phi^{t_{1}}(b)\) will be different even if the orbit through \(b\) is symmetric, but its intersection point with \(\mathop{\rm Fix}(L)\) does not belong to \(N_{1}\).

  2. Similarly, one may consider the involution \(L_{\mu}\) smoothly depending on \(\mu\).

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ACKNOWLEDGMENTS

The authors thank D. Turaev for the valuable discussion that allowed us to understand more deeply some points of our work.

Funding

The authors acknowledge a financial support from the Russian Science Foundation (grant 22-11-00027). Numerical simulations of the paper were supported partially by Agreement 0729-2020-0036 of the Ministry of Science and Higher Education of the Russian Federation (L.M.L and K.N.T). The work of K.N.T. when examining the nonvariational Swift-Hohenberg equation was supported by the Russian Science Foundation (project 23-71-30008).

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Authors and Affiliations

  1. Frumkin Institute of Phys. Chemistry and Electrochemistry of RAS, pr. Leninskiy 31, 119071, Moscow, Russia

    Nikolay E. Kulagin

  2. HSE University, ul. Bolshaya Pecherskaya 25/12, 603155, Nizhny Novgorod, Russia

    Lev M. Lerman & Konstantin N. Trifonov

  3. Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, 603950, Nizhny Novgorod, Russia

    Konstantin N. Trifonov

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  1. Nikolay E. Kulagin
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MSC2010

34C23, 34C37, 37G40

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Kulagin, N.E., Lerman, L.M. & Trifonov, K.N. Twin Heteroclinic Connections of Reversible Systems. Regul. Chaot. Dyn. 29, 40–64 (2024). https://doi.org/10.1134/S1560354724010040

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