Collective Behavior of a Ring of Van Der Pol Oscillators under Gaussian and Random Coupling Schemes
Affiliation(s)
Departamento de Materiales, Universidad Autonoma Metropolitana-Azcapotzalco, Mexico D.F., Mexico.
Departamento de Materiales, Universidad Autonoma Metropolitana-Azcapotzalco, Mexico D.F., Mexico.
ABSTRACT
The collective behavior of a ring of coupled identical van der Pol oscillators is numerically studied in this work. Constant, gaussian and random distributions of the coupling parameter along the ring are considered. Three values of the oscillators constant are assumed in order to cover from quasilinear to nonlinear dynamic performance. Single and multiple coupled frequencies are obtained using power spectra of the long term time series. Phase portraits are obtained from numerical simulations, and the coupled behavior is analyzed, compared and discussed.
The collective behavior of a ring of coupled identical van der Pol oscillators is numerically studied in this work. Constant, gaussian and random distributions of the coupling parameter along the ring are considered. Three values of the oscillators constant are assumed in order to cover from quasilinear to nonlinear dynamic performance. Single and multiple coupled frequencies are obtained using power spectra of the long term time series. Phase portraits are obtained from numerical simulations, and the coupled behavior is analyzed, compared and discussed.
KEYWORDS
Coupled Oscillators, Coupling Parameter, Gaussian Distribution, Oscillators Ring, Random Distribution, Van Der Pol Oscillator
Coupled Oscillators, Coupling Parameter, Gaussian Distribution, Oscillators Ring, Random Distribution, Van Der Pol Oscillator
Cite this paper
Barron, M. , Medina, D. and Hilerio, I. (2014) Collective Behavior of a Ring of Van Der Pol Oscillators under Gaussian and Random Coupling Schemes. Open Journal of Applied Sciences, 4, 383-391. doi: 10.4236/ojapps.2014.47036.
Barron, M. , Medina, D. and Hilerio, I. (2014) Collective Behavior of a Ring of Van Der Pol Oscillators under Gaussian and Random Coupling Schemes. Open Journal of Applied Sciences, 4, 383-391. doi: 10.4236/ojapps.2014.47036.
References
[1] Winfree, A.T. (1967) Biological Rhythms and the Behavior of Populations of Coupled Oscillators. Journal of Theoretical Biology, 16, 15-42.
http://dx.doi.org/10.1016/0022-5193(67)90051-3 [2] van der Pol, B. and van der Mark, J. (1928) The Heartbeat Considered as a Relaxation Oscillation, and an Electrical Model of the Heart. Philosophical Magazine, 6, 763-775. [3] dos Santos, A.M., Lopes, S.R. and Viana, R.L. (2004) Rhythm Synchronization and Chaotic Modulation of Coupled Van Der Pol Oscillators in a Model for the Heartbeat. Physica A, 338, 335-355.
http://dx.doi.org/10.1016/j.physa.2004.02.058 [4] Kuramoto, Y. (1984) Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-69689-3 [5] Camacho, E., Rand, R. and Howland H. (2004) Dynamics of Two Van Der Pol Oscillators Coupled via a Bath. International Journal of Solids and Structures, 41, 2133-2143.
http://dx.doi.org/10.1016/j.ijsolstr.2003.11.035 [6] Aggarwal, J.K. and Richie, C.G. (1966) On Coupled Van Der Pol Oscillators. IEEE Transactions on Circuit Theory, CT13, 465-466.
http://dx.doi.org/10.1109/TCT.1966.1082634 [7] Linkens, D.A. (1976) Stability of Entrainment Conditions for A Particular form of Mutually Coupled Van Der Pol Oscillators. IEEE Transactions on Circuits and Systems, 23, 113-121.
http://dx.doi.org/10.1109/TCS.1976.1084184 [8] Rand, R.H. and Holmes, P.J. (1980) Bifurcation of Periodic Motion in Two Weakly Coupled Van Der Pol Oscillators. International Journal of Non-Linear Mechanics, 15, 387-399.
http://dx.doi.org/10.1016/0020-7462(80)90024-4 [9] Storti, D.W. and Rand, R.H. (1982) Dynamics of Two Strongly Coupled Van Der Pol Oscillators. International Journal of Non-Linear Mechanics, 17, 143-152.
http://dx.doi.org/10.1016/0020-7462(82)90014-2 [10] Storti, D.W. and Rand, R.H. (1986) Dynamics of Two Strongly Coupled Relaxation Oscillators. SIAM Journal of Applied Mathematics, 46, 56-67.
http://dx.doi.org/10.1137/0146006 [11] Pastor, I., Perez, V.M., Encinas, F. and Guerra, J.M. (1993) Ordered and Chaotic Behavior of Two Coupled Van Der Pol Oscillators. Physical Review E, 48, 171-182.
http://dx.doi.org/10.1103/PhysRevE.48.171 [12] Nana, B. and Woafo, P. (2006) Synchronization in a Ring of Four Mutually Coupled Van De Pol Oscillators: Theory and experiment. Physical Review E, 74, 1-8, Article ID: 046213. [13] Woafo, P. and Kadji, H.G. (2004) Synchronized States in a Ring of Mutually Coupled Self-Sustained Electrical Oscillators. Physical Review E, 69, 1-9. [14] Barron, M.A., Sen, M. and Corona, E. (2008) Dynamics of Large Rings of Coupled Van der Pol Oscillators. In: Elleithy, K., Ed., Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering, Springer, Berlin, 346-349.
http://dx.doi.org/10.1007/978-1-4020-8735-6_65 [15] Barron, M.A. and Sen, M. (2013) Dynamic Behavior of a Large Ring of Coupled Self-Excited Oscillators. Journal of Computational and Nonlinear Dynamics, 8, 1-5. [16] Chapra, S.C. and Canale, R.P. (2010) Numerical Methods for Engineers. 6th Edition, McGraw-Hill, New York. [17] Pikovsky, A., Rosenblum, M. and Kurths, J. (2001) Synchronization, a Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511755743
[1] Winfree, A.T. (1967) Biological Rhythms and the Behavior of Populations of Coupled Oscillators. Journal of Theoretical Biology, 16, 15-42.
http://dx.doi.org/10.1016/0022-5193(67)90051-3 [2] van der Pol, B. and van der Mark, J. (1928) The Heartbeat Considered as a Relaxation Oscillation, and an Electrical Model of the Heart. Philosophical Magazine, 6, 763-775. [3] dos Santos, A.M., Lopes, S.R. and Viana, R.L. (2004) Rhythm Synchronization and Chaotic Modulation of Coupled Van Der Pol Oscillators in a Model for the Heartbeat. Physica A, 338, 335-355.
http://dx.doi.org/10.1016/j.physa.2004.02.058 [4] Kuramoto, Y. (1984) Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-69689-3 [5] Camacho, E., Rand, R. and Howland H. (2004) Dynamics of Two Van Der Pol Oscillators Coupled via a Bath. International Journal of Solids and Structures, 41, 2133-2143.
http://dx.doi.org/10.1016/j.ijsolstr.2003.11.035 [6] Aggarwal, J.K. and Richie, C.G. (1966) On Coupled Van Der Pol Oscillators. IEEE Transactions on Circuit Theory, CT13, 465-466.
http://dx.doi.org/10.1109/TCT.1966.1082634 [7] Linkens, D.A. (1976) Stability of Entrainment Conditions for A Particular form of Mutually Coupled Van Der Pol Oscillators. IEEE Transactions on Circuits and Systems, 23, 113-121.
http://dx.doi.org/10.1109/TCS.1976.1084184 [8] Rand, R.H. and Holmes, P.J. (1980) Bifurcation of Periodic Motion in Two Weakly Coupled Van Der Pol Oscillators. International Journal of Non-Linear Mechanics, 15, 387-399.
http://dx.doi.org/10.1016/0020-7462(80)90024-4 [9] Storti, D.W. and Rand, R.H. (1982) Dynamics of Two Strongly Coupled Van Der Pol Oscillators. International Journal of Non-Linear Mechanics, 17, 143-152.
http://dx.doi.org/10.1016/0020-7462(82)90014-2 [10] Storti, D.W. and Rand, R.H. (1986) Dynamics of Two Strongly Coupled Relaxation Oscillators. SIAM Journal of Applied Mathematics, 46, 56-67.
http://dx.doi.org/10.1137/0146006 [11] Pastor, I., Perez, V.M., Encinas, F. and Guerra, J.M. (1993) Ordered and Chaotic Behavior of Two Coupled Van Der Pol Oscillators. Physical Review E, 48, 171-182.
http://dx.doi.org/10.1103/PhysRevE.48.171 [12] Nana, B. and Woafo, P. (2006) Synchronization in a Ring of Four Mutually Coupled Van De Pol Oscillators: Theory and experiment. Physical Review E, 74, 1-8, Article ID: 046213. [13] Woafo, P. and Kadji, H.G. (2004) Synchronized States in a Ring of Mutually Coupled Self-Sustained Electrical Oscillators. Physical Review E, 69, 1-9. [14] Barron, M.A., Sen, M. and Corona, E. (2008) Dynamics of Large Rings of Coupled Van der Pol Oscillators. In: Elleithy, K., Ed., Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering, Springer, Berlin, 346-349.
http://dx.doi.org/10.1007/978-1-4020-8735-6_65 [15] Barron, M.A. and Sen, M. (2013) Dynamic Behavior of a Large Ring of Coupled Self-Excited Oscillators. Journal of Computational and Nonlinear Dynamics, 8, 1-5. [16] Chapra, S.C. and Canale, R.P. (2010) Numerical Methods for Engineers. 6th Edition, McGraw-Hill, New York. [17] Pikovsky, A., Rosenblum, M. and Kurths, J. (2001) Synchronization, a Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511755743