Generalized synchronization in a system of several non-autonomous oscillators coupled by a medium. (English) Zbl 1340.34180

Summary: An abstract theory on general synchronization of a system of several oscillators coupled by a medium is given. By generalized synchronization we mean the existence of an invariant manifold that allows a reduction of the dimension. The case of a concrete system modeling the dynamics of a chemical solution on two containers connected to a third container is studied. Conditions under which synchronization occurs are given. Our theoretical results are complemented with a numerical study.


34D06 Synchronization of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
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[1] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, Springer, 1977. · Zbl 1306.00013 · doi:10.1137/140973773
[2] Hatcher, A.: Algebraic Topology. Cambridge University Press, 2002. · Zbl 1044.55001 · doi:10.1017/s0013091503214620
[3] Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, 1990. · Zbl 0801.15001 · doi:10.1017/cbo9780511840371
[4] Katriel, G.: Synchronization of oscillators coupled through an environment. Physica D 237 (2008), 2933-2944. · Zbl 1184.34060 · doi:10.1016/j.physd.2008.04.015
[5] Margheri, A., Martins, R.: Generalized synchronization in linearly coupled time periodic systems. J. Differential Equations 249 (2010), 3215-3232. · Zbl 1364.34080 · doi:10.1016/j.jde.2010.09.005
[6] Morais, G.: Dinâmica de Osciladores Acoplados. PhD Thesis, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2013.
[7] Pantaleone, J.: Synchronization of metronomes. Am. J. Phys. 70 (2002), 992-1000. · doi:10.1119/1.1501118
[8] Smith, R. A.: Poincaré index theorem concerning periodic orbits of differential equations. Proc. London Math. Soc. s3-48 (1984), 2, 341-362. · Zbl 0509.34046 · doi:10.1112/plms/s3-48.2.341
[9] Smith, R. A.: Massera’s convergence theorem for periodic nonlinear differential equations. J. Math. Anal. Appl 120 (1986), 679-708. · Zbl 0603.34033 · doi:10.1016/0022-247X(86)90189-7
[10] Wazewski, T.: Sur un principle topologique de l’examen de l’allure asymptotique des integrales des Equations differentielles ordinaires. Ann. Soc. Polon Math. 20 (1947), 279-313. · Zbl 0032.35001
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