Skip to main content
SpringerLink
Log in

Chaos in Coupled Heteroclinic Cycles Between Weak Chimeras

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

Heteroclinic cycles are widely used in neuroscience in order to mathematically describe different mechanisms of functioning of the brain and nervous system. Heteroclinic cycles and interactions between them can be a source of different types of nontrivial dynamics. For instance, as it was shown earlier, chaotic dynamics can appear as a result of interaction via diffusive couplings between two stable heteroclinic cycles between saddle equilibria. We go beyond these findings by considering two coupled stable heteroclinic cycles rotating in opposite directions between weak chimeras. Such an ensemble can be mathematically described by a system of six phase equations. Using two-parameter bifurcation analysis, we investigate the scenarios of emergence and destruction of chaotic dynamics in the system under study.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

REFERENCES

  1. Bick, Ch., Goodfellow, M., Laing, C. R., and Martens, E. A., Understanding the Dynamics of Biological and Neural Oscillator Networks through Exact Mean-Field Reductions: A Review, J. Math. Neurosc., 2020, vol. 10, no. 1, Art. 9, 43 pp.

    Article  MathSciNet  Google Scholar 

  2. Pikovsky, A., Rosenblum, M., and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Sci. Ser., vol. 12, New York: Cambridge Univ. Press, 2002.

    Google Scholar 

  3. Strogatz, S., Exploring Complex Networks, Nature, 2001, vol. 410, no. 6825, pp. 268–276.

    Article  ADS  CAS  PubMed  Google Scholar 

  4. Breakspear, M., Dynamic Models of Large-Scale Brain Activity, Nature Neurosci., 2017, vol. 20, no. 3, pp. 340–352.

    Article  CAS  PubMed  Google Scholar 

  5. Winfree, A. T., The Geometry of Biological Time, Biomath., vol. 8, Berlin: Springer, 1980.

    Book  Google Scholar 

  6. Belykh, V. N., Petrov, V. S., and Osipov, G. V., Dynamics of the Finite-Dimensional Kuramoto Model: Global and Cluster Synchronization, Regul. Chaotic Dyn., 2015, vol. 20, no. 1, pp. 37–48.

    Article  ADS  MathSciNet  Google Scholar 

  7. Barabash, N. V., Belykh, V. N., Osipov, G. V., and Belykh, I. V., Partial Synchronization in the Second-Order Kuramoto Model: An Auxiliary System Method, Chaos, 2021, vol. 31, no. 11, Paper No. 113113, 12 pp.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  8. Ashwin, P. and Burylko, O., Weak Chimeras in Minimal Networks of Coupled Phase Oscillators, Chaos, 2015, vol. 25, no. 1, 013106, 9 pp.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  9. Bick, Ch. and Ashwin, P., Chaotic Weak Chimeras and Their Persistence in Coupled Populations of Phase Oscillators, Nonlinearity, 2016, vol. 29, no. 5, pp. 1468–1476.

    Article  ADS  MathSciNet  Google Scholar 

  10. Omel’chenko, O. E., The Mathematics behind Chimera States, Nonlinearity, 2018, vol. 31, no. 5, R121–R164.

    Article  ADS  MathSciNet  Google Scholar 

  11. Afraimovich, V., Ashwin, P., and Kirk, V., Robust Heteroclinic and Switching Dynamics, Dyn. Syst., 2010, vol. 25, no. 3, pp. 285–286.

    Article  Google Scholar 

  12. Ashwin, P., Karabacak, Ö., and Nowotny, Th., Criteria for Robustness of Heteroclinic Cycles in Neural Microcircuits, J. Math. Neurosci., 2011, vol. 1, Art. 13, 18 pp.

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

  13. Komarov, M. A., Osipov, G. V., Suykens, J. A. K., Sequentially Activated Clusters in Neural Networks, Europhys. Lett., 2009, vol. 86, no. 6, 60006.

    Article  ADS  Google Scholar 

  14. Nekorkin, V. I., Dmitrichev, A. S., Kasatkin, D. V., and Afraimovich, V. S., Relating the Sequential Dynamics of Excitatory Neural Networks to Synaptic Cellular Automata, Chaos, 2011, vol. 21, no. 4, 043124, 13 pp.

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Nekorkin, V. I., Dmitrichev, A. S., Kasatkin, D. V., and Afraimovich, V. S., Reducing the Sequential Dynamics of Excitatory Neural Networks to Cellular Automata, JETP Lett., 2012, vol. 95, no. 9, pp. 492–496; see also: Pis’ma v Zh. Èksper. Teoret. Fiz., 2012, vol. 95, no. 9, pp. 557-561.

    Article  ADS  CAS  Google Scholar 

  16. Afraimovich, V. S., Hsu, S.-B., and Lin, H.-E., Chaotic Behavior of Three Competing Species of May – Leonard Model under Small Periodic Perturbations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2001, vol. 11, no. 2, pp. 435–447.

    Article  MathSciNet  Google Scholar 

  17. Afraimovich, V. S., Zhigulin, V. P., and Rabinovich, M. I., On the Origin of Reproducible Sequential Activity in Neural Circuits, Chaos, 2004, vol. 14, no. 4, pp. 1123–1129.

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  18. Afraimovich, V. S., Rabinovich, M. I., and Varona, P., Heteroclinic Contours in Neural Ensembles and the Winnerless Competition Principle, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2004, vol. 14, no. 4, pp. 1195–1208.

    Article  MathSciNet  Google Scholar 

  19. Komarov, M. A., Osipov, G. V., and Zhou, C. S., Heteroclinic Contours in Oscillatory Ensembles, Phys. Rev. E, 2013, vol. 87, no. 2, 022909, 11 pp.

    Article  ADS  CAS  Google Scholar 

  20. Levanova, T. A., Komarov, M. A., and Osipov, G. V., Sequential Activity and Multistability in an Ensemble of Coupled Van der Pol Oscillators, Eur. Phys. J. Special Topics, 2013, vol. 222, no. 10, pp. 2417–2428.

    Article  ADS  Google Scholar 

  21. Mikhaylov, A., Komarov, M., Levanova, T., and Osipov, G., Sequential Switching Activity in Ensembles of Inhibitory Coupled Oscillators, Europhys. Lett., 2013, vol. 101, no. 2, 20009, 5 pp.

    Article  ADS  CAS  Google Scholar 

  22. Dellnitz, M., Field, M., Golubitsky, M., Hohmann, A., and Ma, J., Cycling Chaos, IEEE Trans. on Circuits and Systems 1: Fundamental Theory and Applications, 1995, vol. 42, no. 10, pp. 821–823.

    Article  Google Scholar 

  23. Levanova, T. A., Osipov, G. V., and Pikovsky, A., Coherence Properties of Cycling Chaos, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 8, pp. 2734–2739.

    Article  ADS  MathSciNet  Google Scholar 

  24. Afraimovich, V., Young, T. R, and Rabinovich, M. I., Hierarchical Heteroclinics in Dynamical Model of Cognitive Processes: Chunking, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2014, vol. 24, no. 10, 1450132, 15 pp.

    Article  MathSciNet  Google Scholar 

  25. Afraimovich, V. S., Zaks, M. A., and Rabinovich, M. I., Mind-to-Mind Heteroclinic Coordination: Model of Sequential Episodic Memory Initiation, Chaos, 2018, vol. 28, no. 5, 053107, 15 pp.

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  26. Bick, Ch., Heteroclinic Switching between Chimeras, Phys. Rev. E, 2018, vol. 97, no. 5, 050201, 5 pp.

    Article  ADS  PubMed  Google Scholar 

  27. Bick, Ch., Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations, J. Nonlinear Sci., 2019, vol. 29, no. 6, pp. 2547–2570.

    Article  ADS  MathSciNet  Google Scholar 

  28. Bick, Ch. and Lohse, A., Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks, J. Nonlinear Sci., 2019, vol. 29, no. 6, pp. 2571–2600.

    Article  ADS  MathSciNet  CAS  Google Scholar 

  29. Li, D., Cross, M. C., Zhou, Ch., and Zheng, Zh., Quasiperiodic, Periodic, and Slowing-Down States of Coupled Heteroclinic Cycles, Phys. Rev. E, 2012, vol. 85, no. 1, 016215, 8 pp.

    Article  ADS  Google Scholar 

  30. Voit, M., Veneziale, S., and Meyer-Ortmanns, H., Coupled Heteroclinic Networks in Disguise, Chaos, 2020, vol. 30, no. 8, 083113, 11 pp.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  31. Pikovsky, A. and Nepomnyashchy, A., Chaos in Coupled Heteroclinic Cycles and Its Piecewise-Constant Representation, Phys. D, 2023, vol. 452, Paper No. 133772, 19 pp.

    Article  MathSciNet  Google Scholar 

  32. Ashwin, P. and Rodrigues, A., Hopf Normal Form with \(S_{N}\) Symmetry and Reduction to Systems of Nonlinearly Coupled Phase Oscillators, Phys. D, 2016, vol. 325, pp. 14–24.

    Article  MathSciNet  Google Scholar 

  33. León, I. and Pazó, D., Phase Reduction beyond the First Order: The Case of the Mean-Field Complex Ginzburg – Landau Equation, Phys. Rev. E, 2019, vol. 100, no. 1, 012211, 13 pp.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  34. Datseris, G., DynamicalSystems.jl: A Julia Software Library for Chaos and Nonlinear Dynamics, J. Open Source Software, 2018, vol. 3, no. 23, 598, 5 pp.

    Article  ADS  Google Scholar 

  35. Dhooge, A., Govaerts, W., Kuznetsov, Yu. A., Meijer, H. G. E., and Sautois, B., New Features of the Software \(\mathtt{MatCont}\) for Bifurcation Analysis of Dynamical Systems, Math. Comput. Model. Dyn. Syst., 2008, vol. 14, no. 2, pp. 147–175.

    Article  MathSciNet  Google Scholar 

  36. Borisov, A. V., Jalnine, A. Yu., Kuznetsov, S. P., Sataev, I. R., and Sedova, J. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532.

    Article  ADS  MathSciNet  Google Scholar 

  37. Borisov, A. V., Kazakov, A. O., and Sataev, I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718–733.

    Article  ADS  MathSciNet  Google Scholar 

  38. Belykh, V. N., Belykh, I. V., and Hasler, M., Connection Graph Stability Method for Synchronized Coupled Chaotic Systems, Phys. D, 2004, vol. 195, no. 1–2, pp. 159–187.

    Article  MathSciNet  Google Scholar 

  39. Belykh, V. N., Belykh, I. V., and Hasler, M., Blinking Model and Synchronization in Small-World Networks with a Time-Varying Coupling, Phys. D, 2004, vol. 195, no. 1–2, pp. 188–206.

    Article  MathSciNet  Google Scholar 

  40. Barabash, N. V. and Belykh, V. N., Synchronization Thresholds in an Ensemble of Kuramoto Phase Oscillators with Randomly Blinking Couplings, Radiophys. Quantum El., 2018, vol. 60, no. 9, pp. 761–768; see also: Izv. Vyssh. Uchebn. Zaved. Radiofizika, 2017, vol. 60, no. 9, pp. 851-858.

    Article  ADS  Google Scholar 

  41. Belykh, I. V., Brister, B. N., and Belykh, V. N., Bistability of Patterns of Synchrony in Kuramoto Oscillators with Inertia, Chaos, 2016, vol. 26, no. 9, 094822, 11 pp.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  42. Brister, B. N., Belykh, V. N., and Belykh, I. V., When Three Is a Crowd: Chaos from Clusters of Kuramoto Oscillators with Inertia, Phys. Rev. E, 2020, vol. 101, no. 6, 062206, 17 pp.

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  43. Belykh, V. N., Homoclinic and Heteroclinic Trajectories of a Family of Multidimensional Dynamical Systems, Proc. Steklov Inst. Math., 1997, vol. 216, pp. 14–25; see also: Tr. Mat. Inst. Steklova, 1997, vol. 216, pp. 20-31.

    MathSciNet  Google Scholar 

  44. Belykh, V. N. and Pankratova, E. V., Chaotic Dynamics of Two Van der Pol – Duffing Oscillators with Huygens Coupling, Regul. Chaotic Dyn., 2010, vol. 15, no. 2–3, pp. 274–284.

    Article  ADS  MathSciNet  Google Scholar 

  45. Belykh, V. N., Bifurcation of Separatrices of a Saddle of the Lorenz System, Differ. Uravn., 1984, vol. 20, no. 10, pp. 1666–1674 (Russian).

    MathSciNet  Google Scholar 

  46. Belykh, V., Belykh, I., Colding-Jørgensen, M., and Mosekilde, E., Homoclinic Bifurcations Leading to the Emergence of Bursting Oscillations in Cell Models, Eur. Phys. J. E, 2000, vol. 3, no. 3, pp. 205–219.

    Article  CAS  Google Scholar 

  47. Belykh, V., Belykh, I., and Mosekilde, E., Hyperbolic Plykin Attractor Can Exist in Neuron Models, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2005, vol. 15, no. 11, pp. 3567–3578.

    Article  MathSciNet  Google Scholar 

  48. Belykh, V. N., Pankratova, E. V., and Mosekilde, E., Dynamics and Synchronization of Noise Perturbed Ensembles of Periodically Activated Neuron Cells, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2008, vol. 18, no. 9, pp. 2807–2815.

    Article  MathSciNet  Google Scholar 

  49. Barabash, N. V., Levanova, T. A., and Belykh, V. N., Ghost Attractors in Blinking Lorenz and Hindmarsh – Rose Systems, Chaos, 2020, vol. 30, no. 8, 081105, 7 pp.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors thank Prof. A. S. Pikovsky and Prof. G. V. Osipov for useful discussions.

Funding

This work was supported by the Ministry of Science and Education of the Russian Federation, Contract no. FSRW-2020-0036 (A.E.E. and E.A.G.) and RSF grant 22-12-00348 (T.A.L.).

Author information

Authors and Affiliations

  1. Lobachevsky University, pr. Gagarin 23, 603022, Nizhny Novgorod, Russia

    Artyom E. Emelin, Evgeny A. Grines & Tatiana A. Levanova

Authors
  1. Artyom E. Emelin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evgeny A. Grines
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tatiana A. Levanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Artyom E. Emelin, Evgeny A. Grines or Tatiana A. Levanova.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

PUBLISHER’S NOTE

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

MSC2010

65P20

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Emelin, A.E., Grines, E.A. & Levanova, T.A. Chaos in Coupled Heteroclinic Cycles Between Weak Chimeras. Regul. Chaot. Dyn. 29, 205–217 (2024). https://doi.org/10.1134/S1560354724010131

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354724010131

Keywords

Search

Navigation