×

Heteroclinic ratchets in networks of coupled oscillators. (English) Zbl 1208.34057

Summary: We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning \(\Delta \) between certain pairs of oscillators by a breaking of the phase locking for arbitrary \(\Delta >0\) but not for \(\Delta \leq 0\). Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on the symmetry or the number of oscillators but depends on the specific connection structure of the coupled system.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations

Software:

XPPAUT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aguiar, M.A.D., Dias, A.P.S., Golubitsky, M., Leite, M.C.A.: Homogenous coupled cell networks with S 3-symmetric quotient. DCDS Supplement, pp. 1–9 (2007) · Zbl 1163.34305
[2] Aguiar, M.A.D., Ashwin, P., Dias, A.P.S., Field, M.: Robust heteroclinic cycles in coupled cell systems: identical cells with asymmetric inputs. Preprint (2009)
[3] Ashwin, P., Borresen, J.: Encoding via conjugate symmetries of slow oscillations for globally coupled oscillators. Phys. Rev. E 70(2), 026203 (2004) · doi:10.1103/PhysRevE.70.026203
[4] Ashwin, P., Borresen, J.: Discrete computation using a perturbed heteroclinic network. Phys. Lett. A 347(4–6), 208–214 (2005) · Zbl 1195.34053 · doi:10.1016/j.physleta.2005.08.013
[5] Ashwin, P., Swift, J.W.: The dynamics of n weakly coupled identical oscillators. J. Nonlinear Sci. 2(1), 69–108 (1992) · Zbl 0872.58049 · doi:10.1007/BF02429852
[6] Ashwin, P., Burylko, O., Maistrenko, Y., Popovych, O.: Extreme sensitivity to detuning for globally coupled phase oscillators. Phys. Rev. Lett. 96(5), 054102 (2006) · doi:10.1103/PhysRevLett.96.054102
[7] Ashwin, P., Orosz, G., Wordsworth, J., Townley, S.: Dynamics on networks of clustered states for globally coupled phase oscillators. SIAM J. Appl. Dyn. Syst. 6(4), 728–758 (2007) · Zbl 1167.34326 · doi:10.1137/070683969
[8] Ashwin, P., Burylko, O., Maistrenko, Y.: Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Physica D 237, 454–466 (2008) · Zbl 1178.34041 · doi:10.1016/j.physd.2007.09.015
[9] Busse, F.H., Clever, R.M.: Nonstationary convection in a rotating system. In: Müller, U., Roesner, K.G., Schmidt, B. (eds.) Recent Developments in Theoretical and Experimental Fluid Dynamics, pp. 376–385. Springer, Berlin (1979) · Zbl 0411.76060
[10] Ermentrout, G.B.: A Guide to XPPAUT for Researchers and Students. SIAM, Pittsburgh (2002) · Zbl 1003.68738
[11] Feng, B.Y., Hu, R.: A survey on homoclinic and heteroclinic orbits. Appl. Math. E-Notes 3, 16–37 (2003) (electronic) · Zbl 1028.34045
[12] Golubitsky, M., Stewart, I.: The Symmetry Perspective. Birkhäuser, Basel (2002) · Zbl 1031.37001
[13] Golubitsky, M., Stewart, I.: Nonlinear dynamics of networks: The groupoid formalism. Bull. Am. Math. Soc. (N.S.) 43(3), 305–364 (2006) (electronic) · Zbl 1119.37036 · doi:10.1090/S0273-0979-06-01108-6
[14] Golubitsky, M., Pivato, M., Stewart, I.: Interior symmetry and local bifurcation in coupled cell networks. Dyn. Syst. 19(4), 389–407 (2004) · Zbl 1067.37066 · doi:10.1080/14689360512331318006
[15] Guckenheimer, J., Holmes, P.: Structurally stable heteroclinic cycles. Math. Proc. Camb. Philos. Soc. 103, 189–192 (1988) · Zbl 0645.58022 · doi:10.1017/S0305004100064732
[16] Hansel, D., Mato, G., Meunier, C.: Clustering and slow switching in globally coupled phase oscillators. Phys. Rev. E 48(5), 3470–3477 (1993) · doi:10.1103/PhysRevE.48.3470
[17] Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998) · Zbl 0914.90287
[18] Kiss, I.Z., Rusin, C.G., Kori, H., Hudson, J.L.: Engineering complex dynamical structures: Sequential patterns and desynchronization. Science 316, 1886–1889 (2007) · Zbl 1226.93068 · doi:10.1126/science.1140858
[19] Kori, H., Kuramoto, Y.: Slow switching in globally coupled oscillators: Robustness and occurence through delayed coupling. Phys. Rev. E 63, 046214 (2001) · doi:10.1103/PhysRevE.63.046214
[20] Krupa, M.: Robust heteroclinic cycles. J. Nonlinear Sci. 7(2), 129–176 (1997) · Zbl 0879.58054 · doi:10.1007/BF02677976
[21] Krupa, M., Melbourne, I.: Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergod. Theory. Dyn. Syst. 15, 121–147 (1995) · Zbl 0818.58025 · doi:10.1017/S0143385700008270
[22] Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984) · Zbl 0558.76051
[23] Melbourne, I.: Intermittency as a codimension-three phenomenon. J. Dyn. Differ. Equ. 1(4), 347–367 (1989) · Zbl 0689.58029 · doi:10.1007/BF01048454
[24] Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985) · Zbl 0595.58028 · doi:10.1007/BF01212280
[25] Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.: Network motifs: Simple building blocks of complex networks. Science 298, 824–827 (2002) · doi:10.1126/science.298.5594.824
[26] Rabinovich, M.I., Huerta, R., Varona, P., Afraimovich, V.S.: Generation and reshaping of sequences in neural systems. Biol. Cybern. 95, 519–536 (2006a) · Zbl 1114.92016 · doi:10.1007/s00422-006-0121-5
[27] Rabinovich, M.I., Varona, P., Selverston, A.I., Abarbanel, H.D.I.: Dynamical principles in neuroscience. Rev. Mod. Phys. 95, 519–536 (2006b)
[28] Sakaguchi, H., Kuramoto, Y.: A soluble active rotator model showing phase transitions via mutual entrainment. Prog. Theor. Phys. 76(3), 576–581 (1986) · doi:10.1143/PTP.76.576
[29] Sporns, O., Kötter, R.: Motifs in brain networks. PLoS Biol. 2(11), 1910–1918 (2004) · doi:10.1371/journal.pbio.0020369
[30] Stone, E., Holmes, P.: Random perturbations of heteroclinic attractors. SIAM J. Appl. Math. 50(3), 726–743 (1990) · Zbl 0702.58038 · doi:10.1137/0150043
[31] Strogatz, S.H.: From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20 (2000) · Zbl 0983.34022 · doi:10.1016/S0167-2789(00)00094-4
[32] Zhai, Y.M., Kiss, I.Z., Daido, H., Hudson, J.L.: Extracting order parameters from global measurements with application to coupled electrochemical oscillators. Physica D 205, 57–69 (2005) · Zbl 1087.34005 · doi:10.1016/j.physd.2004.09.017
[33] Zhigulin, P.Z.: Dynamical motifs: Building blocks of complex dynamics in sparsely connected random networks. Phys. Rev. Lett. 92(23), 238701 (2004) · doi:10.1103/PhysRevLett.92.238701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.