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Spatially structured networks of pulse-coupled phase oscillators on metric spaces
1. | Institute of Applied Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada |
2. | Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany |
References:
[1] |
J. Acebrón, L. Bonilla, C. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137.
doi: 10.1103/RevModPhys.77.137. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2008. |
[3] |
J. T. Ariaratnam, Collective Dynamics of the Winfree Model of Coupled Nonlinear Oscillators, PhD thesis, Cornell University, 2002. |
[4] |
J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett., 86 (2001), 4278-4281. |
[5] |
V. Arnold, Ordinary Differential Equations, Second printing of the 1992 edition. Universitext. Springer-Verlag, Berlin, 2006. |
[6] |
K. Athreya and S. Lahiri, Measure Theory and Probability Theory, Springer, 2006. |
[7] |
L. Basnarkov and V. Urumov, Critical exponents of the transition from incoherence to partial oscillation death in the Winfree model, J. Stat. Mech., 2009 (2009), P10014.
doi: 10.1088/1742-5468/2009/10/P10014. |
[8] |
R. Bellman, The stability of solutions of linear differential equations, Duke. Math. J., 10 (1943), 643-647.
doi: 10.1215/S0012-7094-43-01059-2. |
[9] |
R. Beurle, Properties of a mass of cells capable of regenerating pulses, Philos. T. R. Soc. B, 240 (1956), 55-94.
doi: 10.1098/rstb.1956.0012. |
[10] |
V. Bogachev, Measure Theory, Vol. I, II. Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[11] |
A. P. Boris Makarov, Real Analysis: Measures, Integrals and Applications, Springer, 2013.
doi: 10.1007/978-1-4471-5122-7. |
[12] |
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[13] |
J. D. Crawford and K. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings, Physica D, 125 (1999), 1-46.
doi: 10.1016/S0167-2789(98)00235-8. |
[14] |
D. Drachman, Do we have brain to spare? Neurology, 64 (2005), 2004-2005.
doi: 10.1212/01.WNL.0000166914.38327.BB. |
[15] |
B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams and A. McRobie, Modeling walker synchronization on the millennium bridge, Phys. Rev. E, 75 (2007), 021110.
doi: 10.1103/PhysRevE.75.021110. |
[16] |
T. Eisner, Stability of Operators and Operator Semigroups, Birkhäuser, 2010. |
[17] |
F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Phys. Rev. E, 75 (2007), 051104.
doi: 10.1103/PhysRevE.75.051104. |
[18] |
P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002), 191-216.
doi: 10.1016/S0167-2789(01)00374-8. |
[19] |
J. Griffith, A field theory of neural nets: I: Derivation of field equations, B. Math. Biol., 25 (1963), 111-120.
doi: 10.1007/BF02477774. |
[20] |
J. J. Grobler, A note on the theorems of Jentzsch-Perron and Frobenius, in Indagationes Mathematicae (Proceedings), 90 (1987), 381-391. |
[21] |
P. Hertel, Continuum Physics, Springer, 2012.
doi: 10.1007/978-3-642-29500-3. |
[22] |
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990. |
[23] |
G. Katriel, Stability of synchronized oscillations in networks of phase-oscillators, Discret. Contin. Dyn. B, 5 (2005), 353-364.
doi: 10.3934/dcdsb.2005.5.353. |
[24] |
K. Kowalski, Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems, World Scientific, 1994.
doi: 10.1142/9789814354127. |
[25] |
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Springer, 39 (1975), 420-422. |
[26] |
Y. Kuramoto and H. Arakai, Chemical Oscillations, Waves and Turbulence, Springer, 1984.
doi: 10.1007/978-3-642-69689-3. |
[27] |
C. Laing, The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D, 238 (2009), 1569-1588.
doi: 10.1016/j.physd.2009.04.012. |
[28] |
S. Lang, Real and Functional Analysis, Third edition. Graduate Texts in Mathematics, 142. Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0897-6. |
[29] |
W. S. Lee, J. G. Restrepo, E. Ott and T. M. Antonsen, Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times, Chaos, 21 (2011), 023122.
doi: 10.1063/1.3596697. |
[30] | |
[31] |
L. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific, 2007.
doi: 10.1142/9789814261012. |
[32] |
D. D. Quinn, R. H. Rand and S. H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E, 75 (2007), 036218.
doi: 10.1103/PhysRevE.75.036218. |
[33] |
H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. |
[34] |
Y. Shiogai and Y. Kuramoto, Wave propagation in nonlocally coupled oscillators with noise, Prog. Theor. Phys. Supp., 150 (2003), 435-438.
doi: 10.1143/PTPS.150.435. |
[35] |
R. M. Smeal, G. B. Ermentrout and J. A. White, Phase-response curves and synchronized neural networks, Philos. T. Roy. Soc. B, 365 (2010), 2407-2422.
doi: 10.1098/rstb.2009.0292. |
[36] |
S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, 2003. |
[37] |
S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt and E. Ott, Crowd synchrony on the Millennium Bridge, Nature, 438 (2005), 43-44.
doi: 10.1038/43843a. |
[38] |
B. R. Trees, V. Saranathan and D. Stroud, Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model, Phys. Rev. E, 71 (2005), 016215.
doi: 10.1103/PhysRevE.71.016215. |
[39] |
K. Wiesenfeld, P. Colet and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), 1563-1569.
doi: 10.1103/PhysRevE.57.1563. |
[40] |
A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
doi: 10.1016/0022-5193(67)90051-3. |
[41] |
A. Winfree, The Geometry of Biological Time, Second edition. Interdisciplinary Applied Mathematics, 12. Springer-Verlag, New York, 2001. |
[42] |
A. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997.
doi: 10.1007/978-3-642-60637-3. |
show all references
References:
[1] |
J. Acebrón, L. Bonilla, C. Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137.
doi: 10.1103/RevModPhys.77.137. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2008. |
[3] |
J. T. Ariaratnam, Collective Dynamics of the Winfree Model of Coupled Nonlinear Oscillators, PhD thesis, Cornell University, 2002. |
[4] |
J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett., 86 (2001), 4278-4281. |
[5] |
V. Arnold, Ordinary Differential Equations, Second printing of the 1992 edition. Universitext. Springer-Verlag, Berlin, 2006. |
[6] |
K. Athreya and S. Lahiri, Measure Theory and Probability Theory, Springer, 2006. |
[7] |
L. Basnarkov and V. Urumov, Critical exponents of the transition from incoherence to partial oscillation death in the Winfree model, J. Stat. Mech., 2009 (2009), P10014.
doi: 10.1088/1742-5468/2009/10/P10014. |
[8] |
R. Bellman, The stability of solutions of linear differential equations, Duke. Math. J., 10 (1943), 643-647.
doi: 10.1215/S0012-7094-43-01059-2. |
[9] |
R. Beurle, Properties of a mass of cells capable of regenerating pulses, Philos. T. R. Soc. B, 240 (1956), 55-94.
doi: 10.1098/rstb.1956.0012. |
[10] |
V. Bogachev, Measure Theory, Vol. I, II. Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[11] |
A. P. Boris Makarov, Real Analysis: Measures, Integrals and Applications, Springer, 2013.
doi: 10.1007/978-1-4471-5122-7. |
[12] |
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[13] |
J. D. Crawford and K. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings, Physica D, 125 (1999), 1-46.
doi: 10.1016/S0167-2789(98)00235-8. |
[14] |
D. Drachman, Do we have brain to spare? Neurology, 64 (2005), 2004-2005.
doi: 10.1212/01.WNL.0000166914.38327.BB. |
[15] |
B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams and A. McRobie, Modeling walker synchronization on the millennium bridge, Phys. Rev. E, 75 (2007), 021110.
doi: 10.1103/PhysRevE.75.021110. |
[16] |
T. Eisner, Stability of Operators and Operator Semigroups, Birkhäuser, 2010. |
[17] |
F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized Winfree model, Phys. Rev. E, 75 (2007), 051104.
doi: 10.1103/PhysRevE.75.051104. |
[18] |
P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002), 191-216.
doi: 10.1016/S0167-2789(01)00374-8. |
[19] |
J. Griffith, A field theory of neural nets: I: Derivation of field equations, B. Math. Biol., 25 (1963), 111-120.
doi: 10.1007/BF02477774. |
[20] |
J. J. Grobler, A note on the theorems of Jentzsch-Perron and Frobenius, in Indagationes Mathematicae (Proceedings), 90 (1987), 381-391. |
[21] |
P. Hertel, Continuum Physics, Springer, 2012.
doi: 10.1007/978-3-642-29500-3. |
[22] |
R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1990. |
[23] |
G. Katriel, Stability of synchronized oscillations in networks of phase-oscillators, Discret. Contin. Dyn. B, 5 (2005), 353-364.
doi: 10.3934/dcdsb.2005.5.353. |
[24] |
K. Kowalski, Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems, World Scientific, 1994.
doi: 10.1142/9789814354127. |
[25] |
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Springer, 39 (1975), 420-422. |
[26] |
Y. Kuramoto and H. Arakai, Chemical Oscillations, Waves and Turbulence, Springer, 1984.
doi: 10.1007/978-3-642-69689-3. |
[27] |
C. Laing, The dynamics of chimera states in heterogeneous Kuramoto networks, Physica D, 238 (2009), 1569-1588.
doi: 10.1016/j.physd.2009.04.012. |
[28] |
S. Lang, Real and Functional Analysis, Third edition. Graduate Texts in Mathematics, 142. Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0897-6. |
[29] |
W. S. Lee, J. G. Restrepo, E. Ott and T. M. Antonsen, Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times, Chaos, 21 (2011), 023122.
doi: 10.1063/1.3596697. |
[30] | |
[31] |
L. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific, 2007.
doi: 10.1142/9789814261012. |
[32] |
D. D. Quinn, R. H. Rand and S. H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E, 75 (2007), 036218.
doi: 10.1103/PhysRevE.75.036218. |
[33] |
H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. |
[34] |
Y. Shiogai and Y. Kuramoto, Wave propagation in nonlocally coupled oscillators with noise, Prog. Theor. Phys. Supp., 150 (2003), 435-438.
doi: 10.1143/PTPS.150.435. |
[35] |
R. M. Smeal, G. B. Ermentrout and J. A. White, Phase-response curves and synchronized neural networks, Philos. T. Roy. Soc. B, 365 (2010), 2407-2422.
doi: 10.1098/rstb.2009.0292. |
[36] |
S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, 2003. |
[37] |
S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt and E. Ott, Crowd synchrony on the Millennium Bridge, Nature, 438 (2005), 43-44.
doi: 10.1038/43843a. |
[38] |
B. R. Trees, V. Saranathan and D. Stroud, Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model, Phys. Rev. E, 71 (2005), 016215.
doi: 10.1103/PhysRevE.71.016215. |
[39] |
K. Wiesenfeld, P. Colet and S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E, 57 (1998), 1563-1569.
doi: 10.1103/PhysRevE.57.1563. |
[40] |
A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
doi: 10.1016/0022-5193(67)90051-3. |
[41] |
A. Winfree, The Geometry of Biological Time, Second edition. Interdisciplinary Applied Mathematics, 12. Springer-Verlag, New York, 2001. |
[42] |
A. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997.
doi: 10.1007/978-3-642-60637-3. |
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