Abstract
Heteroclinic cycle is an invariant of a dynamical system comprised of steady states (or more general invariant subsets) and heteroclinic trajectories. The behavior of a dynamical system with a heteroclinic cycle is intermittent: a typical trajectory stays for a long time close to a steady state while the transitions between the states occur much faster. Intermittency is present in various physical phenomena, e.g., Earth’s atmospheric circulation, climate variations considered on long time intervals, evolution of species, distribution of diseases, behavior of the Earth’s magnetic field and many others. In this paper, we consider the examples of this natural system and the respective mathematical models possessing heterocliic cycles.
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Notes
Let a heteroclinic trajectory \([{{{\xi }}_{j}} \to {{{\xi }}_{{j + 1}}}]\) belong to invariant subspace \({{P}_{j}}\) and let \({{{\xi }}_{{j + 1}}}\) be stable in the interior of \({{P}_{j}}\) and attract the part of unstable manifold \({{{\xi }}_{j}}\) that lies in \({{P}_{j}}.\) Under small perturbations in the right-hand side of (1) which preserve invariance of \({{P}_{j}},\)\({{{\xi }}_{{j + 1}}}\) remains a stable steady state the trajectories coming from \({{{\xi }}_{j}}\) are drawn to.
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ACKNOWLEDGMENTS
I am grateful to CMUP for their hospitality during my visit to Porto in January to April 2019.
Funding
The work was partly carried out at the University of Porto with the support from CMUP (Centro de Matemática da Universidade do Porto, UID/MAT/00144/2019) funded by the FCT (Fundação para a Ciência ea Tecnologia, Portugal) together with the National (MCTES) and European Foundations under the FEDER program (Fundo Europeu de Desenvolvimento Regional/European Regional Development Fund) COMPETE 2020 (the PT2020 partnership agreement) as well as under the STRIDE projects (NORTE-01-0145-FEDER-000033) funded by FEDER (NORTE 2020) and MAGIC (POCI-01-0145-FEDER-032485) funded by FEDER through COMPETE 2020–POCI.
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Translated by M. Nazarenko
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Podvigina, O.M. Heteroclinic Cycles in Nature. Izv., Phys. Solid Earth 56, 117–124 (2020). https://doi.org/10.1134/S1069351320010115
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DOI: https://doi.org/10.1134/S1069351320010115