Abstract
This article deals with the general ideas of almost global synchronization of Kuramoto coupled oscillators and synchronizing graphs. It reviews the main existing results and gives some new results about the complexity of the problem. It is proved that any connected graph can be transformed into a synchronized one by making suitable groups of twin vertices. As a corollary it is deduced that any connected graph is the induced subgraph of a synchronizing graph. This implies a big structural complexity of synchronizability. Finally the former is applied to find a two integer parameter family G(a,b) of connected graphs such that if b is the k-th power of 10, the synchronizability of G(a,b) is equivalent to find the k-th digit in the expansion in base 10 of the square root of 2. Thus, the complexity of classify G(a,b) is of the same order than the computation of square root of 2. This is the first result so far about the computational complexity of the synchronizability problem.
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References
Kuramoto, Y.: International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39, p. 420 (1975)
Winfree, A.: The geometry of biological time. Springer, Heidelberg (1980)
Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled nonlinear oscillators. Physica D (143), 1–20 (2000)
van der Pol, B.: The nonlinear theory of electrical oscillations. Proc. of the Institute of Radio Engineers 22(9), 1051–1086 (1934)
Jadbabaie, A., Barahona, M., Motee, N.: On the stability of the Kuramoto model of coupled nonlinear oscillators. In: Proc. of the American Control Conference (2004)
Monzón, P., Paganini, F.: Global considerations on the kuramoto model of sinusoidally coupled oscillators. In: Proc. of the 44th IEEE CDC and ECC, Sevilla, Spain, pp. 3923–3928 (2005)
Lin, Lin: The mathematical research for the Kuramoto model of the describing neuronal synchrony in the brain. Commun. Nonlinear Sci. Numer. Simulat. 14, 3258–3260 (2009)
Chopra, Spong: On exponential synchronization of Kuramoto coupled oscillators. IEEE Transactions on Automatic Control 54(2), 353–357 (2009)
Monzón, P.: Almost global stabilityt of dynamical systems. Ph.D. dissertation, Udelar, Uruguay (2006)
Canale, E., Monzón, P.: Gluing Kuramoto coupled oscillators networks. In: Proc. of the 46th IEEE Conference on Decision and Control, New Orleans, USA, pp. 4596–4601 (2007)
Canale, E., Monzón, P.: Synchronizing graphs, In: Workshop on Spectral Graph Theory with applications on computer science, combinatorial optimization and chemistry (2008)
Canale, E., Monzón, P., Robledo, F.: Global Synchronization Properties for Different Classes of Underlying Interconnection Graphs for Kuramoto Coupled Oscillators. In: Lee, Y.-h., Kim, T.-h., Fang, W.-c., Ślęzak, D. (eds.) FGIT 2009. LNCS, vol. 5899, pp. 104–111. Springer, Heidelberg (2009)
Mallada, Tang: Synchronization of Phase-coupled oscillators with arbitrary topology. In: American Control Conference (2010)
Biggs, N.: Algebraic Graph theory. Cambridge University Press, Cambridge (1993)
Khalil, H.: Nonlinear Systems. Prentice-Hall, Ed. Prentice-Hall, Englewood Cliffs (1996)
Canale, E., Monzón, P.: On the characterization of Families of Synchronizing Graphs of Kuramoto Coupled Oscillators. In: IFAC Workshop on Estimation and Control of Networked Systems (NecSys 2009), Venice, Italy (2009)
Canale, E., Monzón, P.: Global properties of Kuramoto bidirectionally coupled oscillators in a ring structure. In: IEEE International Conference on Control Applications (CCA) & International Symposium on Intelligent Control (ISIC), Saint Petersburg, Russia, pp. 183–188 (2009)
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Canale, E., Monzón, P., Robledo, F. (2010). On the Complexity of the Classification of Synchronizing Graphs. In: Kim, Th., Yau, S.S., Gervasi, O., Kang, BH., Stoica, A., Ślęzak, D. (eds) Grid and Distributed Computing, Control and Automation. GDC CA 2010 2010. Communications in Computer and Information Science, vol 121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17625-8_19
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DOI: https://doi.org/10.1007/978-3-642-17625-8_19
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