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Heteroclinic Cycles in Nature

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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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  1.  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.

REFERENCES

  1. Afraimovich, V.S., Zhigulin, V.P., and Rabinovich, M.I., On the origin of reproducible sequential activity in neural circuits, Chaos, 2004, vol. 14, pp. 1123–1129.

    Article  Google Scholar 

  2. Agliari, A. and Vachadze, G., Homoclinic and heteroclinic bifurcations in an overlapping generations model with credit market imperfection, Comput. Econ., 2011, vol. 38, pp. 241–260.

    Article  Google Scholar 

  3. Aguiar, M.A.D. and Castro, S.B.S.D., Chaotic switching in a two-person game, Physica D., 2010, vol. 239, pp. 1598–1609.

    Article  Google Scholar 

  4. Arnold, V.I., Teoriya katastrof (Catastrophe Ttheory), Moscow: Nauka, 1990.

  5. Ashwin, P., Cova, E., and Tavakol, R., Transverse instability for non-normal parameters, Nonlinearity, 1999, vol. 12, pp. 563–577.

    Article  Google Scholar 

  6. Aurnou, J.M. and Olson, P.L., Experiments on Rayleigh-Bénard convection, magnetoconvection and rotating magnetoconvection in liquid gallium, J. Fluid Mech., 2001, vol. 430, pp. 283–307.

    Article  Google Scholar 

  7. Berhanu, M., et al., Magnetic field reversals in an experimental turbulent dynamo, Europhys. Lett., 2007, vol. 77, p. 59001.

    Article  Google Scholar 

  8. Billyard, A.P., Coley, A.A., and Lidsey, J.E., Cyclical behavior in early universe cosmologies, J. Math. Phys., 2000, vol. 41, pp. 6277–6283.

    Article  Google Scholar 

  9. Bossolini, E., Brons, M., and Uldall, K., Singular limit analysis of a model for earthquake faulting, Nonlinearity, 2017, vol. 30, pp. 2805–2834.

    Article  Google Scholar 

  10. Bratus, A.S., Novozhilov, A.S., and Platonov, A.P., Dinamicheskiye sistemy i modeli v biologii (Dynamic Systems and Models in Biology), Moscow: FIZMATLIT, 2009.

  11. Busse, F.R. and Clever, R.M., Heteroclinic cycles and phase turbulence, in Pattern Formation in Continuous and Coupled Systems, Golubitsky, M., Eds., New York: Springer, 1999.

    Google Scholar 

  12. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford: Claredon Press, 1961.

  13. Chertovskih, R., Gama, S.M.A., Podvigina, O., and Zheligovsky, V., Dependence of magnetic field generation by thermal convection on the rotation rate: a case study, Physica D., 2010, vol. 239, pp. 1188–1209.

  14. Chertovskih, R., Chimanski, E.V., and Rempel, E.L., Route to hyperchaos in Rayleigh- Bénard convection, Europhys. Lett., 2015, vol. 112, 14001.

    Article  Google Scholar 

  15. Chertovskih, R., Rempel, E.L., and Chimanski, E.V., Magnetic field generation by intermittent convection, Phys. Lett. A, 2017, vol. 381, pp. 3300–3306.

    Article  Google Scholar 

  16. Cimatoribus, A.A., Drijfhout, S.S., Livina, V., and van der Schrier, G., Dansgaard-Oeschger events: tipping points in the climate system, Climate of the Past, 2012, vol. 8, pp. 4269–4294.

    Article  Google Scholar 

  17. Coley, A.A., Dynamical Systems and Cosmology, Dordrecht: Springer, 2003.

  18. Crommelin, D.T., Homoclinic dynamics: a scenario for atmospheric ultralow-frequency variability, J. Atmos. Sci., 2002, vol. 59, pp. 1533–1549.

    Article  Google Scholar 

  19. Crommelin, D.T., Regime transitions and heteroclinic connections in a barotropic atmosphere, J. Atmos. Sci., 2003, vol. 60, pp. 229–246.

    Article  Google Scholar 

  20. Crommelin, D.T., Opsteegh, J.D., and Verhulst, F., A mechanism for atmospheric regime behaviour, J. Atmos. Sci., 2004, vol. 61, pp. 1406–1419.

    Article  Google Scholar 

  21. Crucifix, M., Oscillators and relaxation phenomena in Pleistocene climate theory, Phil. Trans. R. Soc. A., 2012, vol. 370, pp. 1140–1165.

    Article  Google Scholar 

  22. Gershuni, G.Z., and Zhukhovitsky, E.M., Konvektivnaya ustoychivost' neszhimayemoy zhidkosti (Convective Stability of an Incompressible Fluid), Moscow: Nauka, 1972.

  23. Glatzmaier, G.A. and Roberts, P.H., A three-dimensional self-consistent computer simulation of a geomagnetic field reversal, Nature, 1995, vol. 377, pp. 203–209.

    Article  Google Scholar 

  24. Glatzmaier, G.A. and Roberts, P.H., Simulating the geodynamo, Contemp. Phys., 1997, vol. 38, pp. 269–288.

    Article  Google Scholar 

  25. Grasman, J., Asymptotic methods for relaxation oscillations and applications, Applied Mathematical Sciences, vol. 63, New York: Springer, 1987.

    Book  Google Scholar 

  26. Hauert, C., De Monte, S., Hofbauer, J., and Sigmundn, K., Replicator dynamics for optional public good games, J. Theor. Biol., 2002, vol. 218, pp. 187–194.

    Article  Google Scholar 

  27. Hofbauer, J. and Sigmund, K., The Theory of Evolution and Dynamical Systems, Cambridge: Cambridge Univ., 1988.

    Google Scholar 

  28. Hogan, S.J., Heteroclinic bifurcations in damped rigid block motion, Proc. R. Soc. Lond. A., 1992, vol. 439, pp. 155–162.

    Article  Google Scholar 

  29. Holmes, P., Symmetries, heteroclinic cycles and intermittency in fluid flow, in Turbulence in Fluid Flows, Sell, G.R., Eds., New York: Springer, 1993, pp. 49–58.

    Google Scholar 

  30. James, I.N. and James, P.M., Ultra-low-frequency variability in a simple atmospheric circulation model, Nature, 1989, vol. 342, pp. 53–55.

    Article  Google Scholar 

  31. Keeling, M.J., Rohani, P., and Grenfell, B.T., Seasonally forced disease dynamics explored as switching between attractors, Physica D., 2001, vol. 148, pp. 317–335.

    Article  Google Scholar 

  32. Koon, W.S., Lo, M.W., Marsden, J.E., and Ross, S.D., Heteroclinic connections between periodic orbits and resonance transitions in Celestial mechanics, Chaos, 2000, vol. 10, pp. 427–459.

    Article  Google Scholar 

  33. Krasil’nikov, P., Fast non-resonance rotations of spacecraft in restricted three body problem with magnetic torques, Int. J. Non-Linear Mech., 2015, vol. 73, pp. 43–50.

    Article  Google Scholar 

  34. Krasil’nikov, P.S. and Podvigina, O.M., On evolution of obliquity in a non-resonant planetary system, Vestn. Udmurt. Univ.,Matem. Mekhan. Komp’yut. Nauki, 2018, vol. 28, no. 4, pp. 549–564.

    Google Scholar 

  35. Krasilnikov, P.S. and Zakharova, E.E., Non-resonant rotations of a satellite relative to the center of mass in a restricted problem of bodies, Kosm. Issled., 1993, vol. 31, no. 6, pp. 11–21.

    Google Scholar 

  36. Krupa, M., and Melbourne, I., Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynam. Sys., 1995, vol. 15, pp. 121–147.

    Article  Google Scholar 

  37. Krupa, M. and Melbourne, I., Asymptotic stability of heteroclinic cycles in systems with symmetry II, Proc. Roy. Soc. Edinburgh, 2004, vol. 134A, pp. 1177–1197.

    Article  Google Scholar 

  38. Küppers, G. and Lortz, D., Transition from laminar convection to thermal turbulence in a rotating fluid layer, J. Fluid Mech., 1969, vol. 35, pp. 609–620.

    Article  Google Scholar 

  39. Laj, C. and Kissel, C., An impending geomagnetic transition? Hints from the past, Frontiers Earth Science, 2015, vol. 3, p. 6.

    Article  Google Scholar 

  40. Lotka, A.J., Contribution to the Theory of Periodic Reaction, J. Phys. Chem., 1910, vol. 14, pp. 271–274.

    Article  Google Scholar 

  41. May, R.M. and Leonard, W., Nonlinear aspects of competition between three species, SlAMJ. Appl. Math., 1975, vol. 29, pp. 243–252.

    Google Scholar 

  42. Melbourne, I., Proctor, M.R.E., and Rucklidge, A.M., A heteroclinic model of geodynamo reversals and excursions, in Dynamo and Dynamics, a Mathematical Challenge, Chossat, P., Armbruster, D., and Oprea, I., Eds., Dordrecht: Kluwer, 2001, pp. 363–370.

    Google Scholar 

  43. Merril, R.T., McEllhiny, M.W., and McFadden, Ph.L., The magnetic field of the Earth, in Paleomagnetism, the Core and the Deep Mantle, San Diego: Academic Press, 1996.

    Google Scholar 

  44. Mikhailov, A.O., Komarov, M.A., and Osipov, G.V., Sequential switching activity in an ensemble of non-identical Poincare systems, Izv.Univ. PND, 2013, vol. 21, pp. 79–91.

    Google Scholar 

  45. Nore, C., Moisy, F., and Quartier, L., Experimental observation of near heteroclinic cycles in the von Karman swirling flow, Phys. Fluids, 2005, vol. 17, 064103.

    Article  Google Scholar 

  46. Nowotny, T. and Rabinovich, M.I., Dynamical origin of independent spiking and bursting activity in neural microcircuits, Phys. Rev. Lett., 2007, vol. 98, 128106.

    Article  Google Scholar 

  47. Petrelis, F. and Fauve, S., Mechanics for magnetic field reversals, Phil. Trans. R. Soc. A., 2010, vol. 368, pp. 1595–1605.

    Article  Google Scholar 

  48. Platt, N., Spiegel, E.A., and Tresser, C. On-off intermittency: A mechanism for bursting, Phys. Rev. Lett., 1993, vol. 70, pp. 279–282.

    Article  Google Scholar 

  49. Podvigina, O., A route to magnetic field reversals: an example of an ABC-forced non-linear dynamo, Geophys. Astrophys. Fluid Dynam., 2003, vol. 97, pp. 149–174.

    Article  Google Scholar 

  50. Podvigina, O.M., Convective stability of a conductive fluid layer rotating a in external magnetic field, Izv.,Mekh. Zhidk. Gaza, 2009, vol. 4, pp. 29–39.

    Google Scholar 

  51. Podvigina, O.M., Stability of rolls in rotating magnetoconvection in a layer with no-slip electrically insulating horizontal boundaries, Phys. Rev. E., 2010, vol. 81, 056322.

    Article  Google Scholar 

  52. Podvigina, O., Stability and bifurcations of heteroclinic cycles of type Z, Nonlinearity, 2012, vol. 25, pp. 1887–1917.

    Article  Google Scholar 

  53. Podvigina, O., Classification and stability of simple homoclinic cycles in \({{\mathbb{R}}^{5}}\), Nonlinearity, 2013, vol. 26, pp. 1501–1528.

    Article  Google Scholar 

  54. Podvigina, O.M. and Ashwin, P.B., On local attraction properties and a stability index for heteroclinic connections, Nonlinearity, 2011, vol. 24, pp. 887–929.

    Article  Google Scholar 

  55. Podvigina, O., Ashwin, P., and Hawker, D., Modelling instability of ABC flow using a mode interaction between steady and Hopf bifurcations with rotational symmetries of the cube, Physica D., 2006, vol. 215, pp. 62–79.

    Article  Google Scholar 

  56. Putelat, T., Dawes, J.H.P., and Champneys, A.R, A phase-plane analysis of localized frictional waves, Proc. R. Soc. A., 2017, vol. 473l 20160606.

    Article  Google Scholar 

  57. Pykh, Yu.A., Obobshchennyye sistemy Lotki-Vol’terra: teoriya i prilozheniya (Generalized Lotka-Volterra Systems: Theory and Applications), St. Petersburg: SPb GIPSR, 2017.

  58. Sagnotti, L., Scardia, G., Giaccio, B., Liddicoat, J.C., Nomade, S., Renne, P.R., and Sprain, C.J., Extremely rapid directional change during Matuyama–Brunhes geomagnetic polarity reversal, Geophys. J. Int., 2014, vol. 199, pp. 1110–1124.

    Article  Google Scholar 

  59. Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V., and Chua, L., Metody kachestvennoy teorii v nelineynoy dinamike (Methods of Qualitative Theory in Nonlinear Dynamics), Part 2, Moscow-Izhevsk: NITS “Regulyarnaya i khaoticheskaya dinamika,” Inst. Comp. Res., 2009.

  60. Stone, E., Gorman, M., el-Hamdi, M., and Robbins, K.A., Identification of intermittent ordered patterns as heteroclinic connections, Phys. Rev. Lett., 1996, vol. 76, pp. 2061–2064.

    Article  Google Scholar 

  61. Szmolyan, P. and Wechselberger, M., Relaxation oscillations in \({{\mathbb{R}}^{3}}\), J. Differential Equations, 2004, vol. 200, pp. 69–104.

    Article  Google Scholar 

  62. Szolnoki, A., Mobilia, M., Jiang, L.-L., Szczesny, B., Rucklidge, A.M., and Perc, M., Cyclic dominance in evolutionary games: a review, J. R. Soc. Interface, 2014, vol. 11, 20140735.

    Article  Google Scholar 

  63. Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. Acad. Lincei Roma, 1926, vol. 2, pp. 31–113.

    Google Scholar 

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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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