For citation:
Glyzin D. S., Glyzin S. D., Kolesov A. Y. Hunt for chimeras in fully coupled networks of nonlinear oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 2, pp. 152-175. DOI: 10.18500/0869-6632-2022-30-2-152-175
Hunt for chimeras in fully coupled networks of nonlinear oscillators
The purpose of this work is to study the dynamic properties of solutions to special systems of ordinary differential equations, called fully connected networks of nonlinear oscillators. Methods. A new approach to obtain periodic regimes of the chimeric type in these systems is proposed, the essence of which is as follows. First, in the case of a symmetric network, a simpler problem is solved of the existence and stability of quasi-chimeric solutions — periodic regimes of two-cluster synchronization. For each of these modes, the set of oscillators falls into two disjoint classes. Within these classes, full synchronization of oscillations is observed, and every two oscillators from different classes oscillate asynchronously. Results. On the basis of the proposed methods, it is separately established that in the transition from a symmetric system to a general network, the periodic regimes of two-cluster synchronization can be transformed into chimeras. Conclusion. The main statements of the work concerning the emergence of chimeras were obtained analytically on the basis of an asymptotic study of a model example. For this example, the notion of a canonical chimera is introduced and the statement about the existence and stability of solutions of chimeric type in the case of asymmetry of the network is proved. All the results presented are extended to a continuous analogue of the corresponding system. The obtained results are illustrated numerically.
- Kuramoto Y, Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems. 2002;5(4):380–385.
- Abrams DM, Strogatz SH. Chimera states for coupled oscillators. Phys. Rev. Lett. 2004;93(17): 174102. DOI: 10.1103/PhysRevLett.93.174102.
- Panaggio MJ, Abrams DM. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity. 2015;28(3):R67. DOI: 10.1088/0951-7715/28/3/R67.
- Sethia GC, Sen A. Chimera states: The existence criteria revisited. Phys. Rev. Lett. 2014;112(14): 144101. DOI: 10.1103/PhysRevLett.112.144101.
- Schmidt L, Krischer K. Clustering as a prerequisite for chimera states in globally coupled systems. Phys. Rev. Lett. 2015;114(3):034101. DOI: 10.1103/PhysRevLett.114.034101.
- Laing CR. Chimeras in networks with purely local coupling. Phys. Rev. E. 2015;92(5):050904. DOI: 10.1103/PhysRevE.92.050904.
- Laing CR. Chimeras in networks of planar oscillators. Phys. Rev. E. 2010;81(6):066221. DOI: 10.1103/PhysRevE.81.066221.
- Zakharova A, Kapeller M, Scholl E. Chimera death: Symmetry breaking in dynamical networks. Phys. Rev. Lett. 2014;112(15):154101. DOI: 10.1103/PhysRevLett.112.154101.
- Omelchenko I, Zakharova A, Hovel P, Siebert J, Scholl E. Nonlinearity of local dynamics promotes multi-chimeras. Chaos. 2015;25(8):083104. DOI: 10.1063/1.4927829.
- Omelchenko I, Omelchenko OE, Hovel P, Scholl E. When nonlocal coupling between oscillators becomes stronger: Patched synchrony or multichimera states. Phys. Rev. Lett. 2013;110(22):224101. DOI: 10.1103/PhysRevLett.110.224101.
- Sakaguchi H. Instability of synchronized motion in nonlocally coupled neural oscillators. Phys. Rev. E. 2006;73(3):031907. DOI: 10.1103/PhysRevE.73.031907.
- Hizanidis J, Kanas V, Bezerianos A, Bountis T. Chimera states in networks of nonlocally coupled Hindmarsh–Rose neuron models. International Journal of Bifurcation and Chaos. 2014;24(3): 1450030. DOI: 10.1142/S0218127414500308.
- Zakharova A. Chimera Patterns in Networks: Interplay between Dynamics, Structure, Noise, and Delay. Berlin: Springer; 2020. 233 p. DOI: 10.1007/978-3-030-21714-3.
- Glyzin SD, Kolesov AY, Rozov NK. Self-excited relaxation oscillations in networks of impulse neurons. Russian Mathematical Surveys. 2015;70(3):383–452. DOI: 10.1070/RM2015v070n03ABEH004951.
- Glyzin SD, Kolesov AY, Rozov NK. Periodic two-cluster synchronization modes in completely connected genetic networks. Differential Equations. 2016;52(2):157–176. DOI: 10.1134/S0012266116020038.
- Kolesov AY, Rozov NK. Invariant Tori of Nonlinear Wave Equations. Moscow: Fizmatlit; 2004. 408 p. (in Russian).
- Mishchenko EF, Sadovnichii VA, Kolesov AY, Rozov NK. Autowave Processes in Nonlinear Media with Diffusion. Moscow: Fizmatlit; 2010. 400 p. (in Russian).
- Daleckii JL, Krein MG. Stability of Solutions of Differential Equations in Banach Space. Providence: American Mathematical Society; 2002. 386 p.
- 2652 reads