# American Institute of Mathematical Sciences

December  2021, 14(6): 1003-1033. doi: 10.3934/krm.2021036

## Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 2 Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea 3 Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea 4 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

*Corresponding author: Myeongju Kang

Received  November 2020 Revised  August 2021 Published  December 2021 Early access  November 2021

We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a $L^1$-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.

Citation: Seung-Yeal Ha, Myeongju Kang, Bora Moon. Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics. Kinetic and Related Models, 2021, 14 (6) : 1003-1033. doi: 10.3934/krm.2021036
##### References:
 [1] D. M. Abrams and S. H. Strogatz, Chimera states in a ring of nonlocally coupled oscillators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 21-37.  doi: 10.1142/S0218127406014551. [2] J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. [3] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374. [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] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105. [6] N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), 1230004, 29 pp. doi: 10.1142/S0218202512300049. [7] R. Ben-Yishai, D. Hansel and H. Sompolinsky, Traveling waves and the processing of weakly tuned inputs in a cortical network module, J. Comput. Neurosci., 4 (1997), 985-999.  doi: 10.1023/A:1008816611284. [8] J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0. [9] Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011. [10] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [11] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884. [12] P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989–1021. doi: 10.1007/s10955-008-9529-8. [13] F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators, SIAM J. Control Optim., 50 (2012), 1616-1642.  doi: 10.1137/110851584. [14] F. Dörfler and F. Bullo, Synchronization in complex network of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012. [15] F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X. [16] J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Automatic Control, 49 (2004), 1465-1476.  doi: 10.1109/TAC.2004.834433. [17] F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized winfree model, Physical Review E, 75 (2007), 051104.  doi: 10.1103/PhysRevE.75.051104. [18] T. Girnyk, M. Hasler and Y. Maistrenko, Multistability of twisted states in non-locally coupled Kuramoto-type models, Chaos, 22 (2012), 013114, 10 pp. doi: 10.1063/1.3677365. [19] S.-Y. Ha, M. Kang and B. Moon, On the emerging asymptotic patterns of the Winfree model with frustrations, Nonlinearity, 34 (2021), 2454-2482.  doi: 10.1088/1361-6544/abb9f8. [20] S.-Y. Ha and D. Kim, Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay, J. Math. Phys., 59 (2018), 112702, 20 pp. doi: 10.1063/1.5017063. [21] S.-Y. Ha, D. Kim and B. Moon, Interplay of Random Inputs and Adaptive Couplings in the Winfree Model, Communications on Pure and Applied Analysis, 2021. [22] S.-Y. Ha, H. Kim and J. Park, Remarks on the complete synchronization for the Kuramoto model with frustrations, Anal. Appl. (Singap.), 16 (2018), 525-563.  doi: 10.1142/S0219530517500130. [23] S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10. [24] S.-Y. Ha, H. K. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators, Nonlinarity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441. [25] S.-Y. Ha, Y. Kim and Z. Li, Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration, SIAM J. Appl. Dyn. Syst., 13 (2014), 466-492.  doi: 10.1137/130926559. [26] S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergence of partial locking states from the ensemble of Winfree oscillators, Quart. Appl. Math., 75 (2017), 39-68.  doi: 10.1090/qam/1448. [27] S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations., 260 (2016), 4203-4236.  doi: 10.1016/j.jde.2015.11.008. [28] S.-Y. Ha, D. Ko, J. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17. [29] S.-Y. Ha, D. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst., 17 (2018), 581-625.  doi: 10.1137/17M1112959. [30] S.-Y. Ha, J. Park and S. W. Ryoo, Emergence of phase-locked states for the Winfree model in a large coupling regime, Discrete Contin. Dyn. Syst., 35 (2015), 3417-3436.  doi: 10.3934/dcds.2015.35.3417. [31] A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781. [32] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [33] Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420. [34] Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), 380-385. [35] C. R. Laing, Derivation of a neural field model from a network of theta neurons, Physical Review. E, 90 (2014), 010901.  doi: 10.1103/PhysRevE.90.010901. [36] C. R. Laing and C. C. Chow, Stationary bumps in networks of spiking neurons, Neural Comput., 31 (2001), 1473-1494.  doi: 10.1162/089976601750264974. [37] R.-D. Li and T. Erneux, Preferential instability in arrays of coupled lasers, Phys. Rev. A, 46 (1992), 4252-4260.  doi: 10.1103/PhysRevA.46.4252. [38] Z. Li, Y. Liu and X. Xue, Convergence and stability of generalized gradient systems by Lojasiewicz inequality with application in continuum Kuramoto model, Discrete Contin. Dyn. Syst., 39 (2019), 345-367.  doi: 10.3934/dcds.2019014. [39] G. S. Medvedev, Stochastic stability of continuous time consensus protocols, SIAM J. Control Optim., 50 (2012), 1859-1885.  doi: 10.1137/100801457. [40] G. S. Medvedev, The continuum limit of the Kuramoto model on sparse random graphs, Commun. Math. Sci., 17 (2019), 883-898.  doi: 10.4310/CMS.2019.v17.n4.a1. [41] G. S. Medvedev, The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Anal., 46 (2014), 2743-2766.  doi: 10.1137/130943741. [42] G. S. Medvedev, The nonlinear heat equation on W-random graphs, Arch. Ration. Mech. Anal., 212 (2014), 781-803.  doi: 10.1007/s00205-013-0706-9. [43] G. S. Medvedev and S. Zhuravytska, The geometry of spontaneous spiking in neuronal networks, J. Nonlinear Sci., 22 (2012), 689-725.  doi: 10.1007/s00332-012-9125-6. [44] O. E. Omel'chenko, M. Wolfrum, S. Yanchuk, Y. L. Maistrenko and O. Sudakov, Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators, Phys. Rev. E, 85 (2012), 036210.  doi: 10.1103/PhysRevE.85.036210. [45] W. Oukil, A. Kessi and P. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303. [46] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123. [47] J. R. Phillips, H. S. J. van der Zant, J. White and T. P. Orlando, Influence of induced magnetic fields on the static properties of Josephson-junction arrays, Phys. Rev. B, 47 (1993), 5219-5229.  doi: 10.1103/PhysRevB.47.5219. [48] N. V. Swindale, The model for the formation of ocular dominance stripes, Neural Comput, 31 (2001), 1473-1494. [49] J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828. [50] D. D. Quinn, R. H. Rand and S. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E, 75 (2007), 036218.  doi: 10.1103/PhysRevE.75.036218. [51] D. D. Quinn, R. H. Rand and S. Strogatz, Synchronization in the Winfree model of coupled nonlinear interactions, A. ENOC 2005 Conference, Eindhoven, Netherlands, (2005), 7–12. [52] W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Automat. Control, 50 (2005), 655-661.  doi: 10.1109/TAC.2005.846556. [53] S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Phys. D, 74 (1994), 197-253.  doi: 10.1016/0167-2789(94)90196-1. [54] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3. [55] D. A. Wiley, S. H. Strogatz and M. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103.  doi: 10.1063/1.2165594.

show all references

##### References:
 [1] D. M. Abrams and S. H. Strogatz, Chimera states in a ring of nonlocally coupled oscillators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 21-37.  doi: 10.1142/S0218127406014551. [2] J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. [3] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374. [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] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105. [6] N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), 1230004, 29 pp. doi: 10.1142/S0218202512300049. [7] R. Ben-Yishai, D. Hansel and H. Sompolinsky, Traveling waves and the processing of weakly tuned inputs in a cortical network module, J. Comput. Neurosci., 4 (1997), 985-999.  doi: 10.1023/A:1008816611284. [8] J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0. [9] Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011. [10] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [11] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884. [12] P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989–1021. doi: 10.1007/s10955-008-9529-8. [13] F. Dörfler and F. Bullo, Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators, SIAM J. Control Optim., 50 (2012), 1616-1642.  doi: 10.1137/110851584. [14] F. Dörfler and F. Bullo, Synchronization in complex network of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012. [15] F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X. [16] J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Automatic Control, 49 (2004), 1465-1476.  doi: 10.1109/TAC.2004.834433. [17] F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro and S. Stramaglia, Phase diagram of a generalized winfree model, Physical Review E, 75 (2007), 051104.  doi: 10.1103/PhysRevE.75.051104. [18] T. Girnyk, M. Hasler and Y. Maistrenko, Multistability of twisted states in non-locally coupled Kuramoto-type models, Chaos, 22 (2012), 013114, 10 pp. doi: 10.1063/1.3677365. [19] S.-Y. Ha, M. Kang and B. Moon, On the emerging asymptotic patterns of the Winfree model with frustrations, Nonlinearity, 34 (2021), 2454-2482.  doi: 10.1088/1361-6544/abb9f8. [20] S.-Y. Ha and D. Kim, Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay, J. Math. Phys., 59 (2018), 112702, 20 pp. doi: 10.1063/1.5017063. [21] S.-Y. Ha, D. Kim and B. Moon, Interplay of Random Inputs and Adaptive Couplings in the Winfree Model, Communications on Pure and Applied Analysis, 2021. [22] S.-Y. Ha, H. Kim and J. Park, Remarks on the complete synchronization for the Kuramoto model with frustrations, Anal. Appl. (Singap.), 16 (2018), 525-563.  doi: 10.1142/S0219530517500130. [23] S.-Y. Ha, H. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10. [24] S.-Y. Ha, H. K. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators, Nonlinarity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441. [25] S.-Y. Ha, Y. Kim and Z. Li, Large-time dynamics of Kuramoto oscillators under the effects of inertia and frustration, SIAM J. Appl. Dyn. Syst., 13 (2014), 466-492.  doi: 10.1137/130926559. [26] S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergence of partial locking states from the ensemble of Winfree oscillators, Quart. Appl. Math., 75 (2017), 39-68.  doi: 10.1090/qam/1448. [27] S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations., 260 (2016), 4203-4236.  doi: 10.1016/j.jde.2015.11.008. [28] S.-Y. Ha, D. Ko, J. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17. [29] S.-Y. Ha, D. Ko and Y. Zhang, Emergence of phase-locking in the Kuramoto model for identical oscillators with frustration, SIAM J. Appl. Dyn. Syst., 17 (2018), 581-625.  doi: 10.1137/17M1112959. [30] S.-Y. Ha, J. Park and S. W. Ryoo, Emergence of phase-locked states for the Winfree model in a large coupling regime, Discrete Contin. Dyn. Syst., 35 (2015), 3417-3436.  doi: 10.3934/dcds.2015.35.3417. [31] A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781. [32] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [33] Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420. [34] Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), 380-385. [35] C. R. Laing, Derivation of a neural field model from a network of theta neurons, Physical Review. E, 90 (2014), 010901.  doi: 10.1103/PhysRevE.90.010901. [36] C. R. Laing and C. C. Chow, Stationary bumps in networks of spiking neurons, Neural Comput., 31 (2001), 1473-1494.  doi: 10.1162/089976601750264974. [37] R.-D. Li and T. Erneux, Preferential instability in arrays of coupled lasers, Phys. Rev. A, 46 (1992), 4252-4260.  doi: 10.1103/PhysRevA.46.4252. [38] Z. Li, Y. Liu and X. Xue, Convergence and stability of generalized gradient systems by Lojasiewicz inequality with application in continuum Kuramoto model, Discrete Contin. Dyn. Syst., 39 (2019), 345-367.  doi: 10.3934/dcds.2019014. [39] G. S. Medvedev, Stochastic stability of continuous time consensus protocols, SIAM J. Control Optim., 50 (2012), 1859-1885.  doi: 10.1137/100801457. [40] G. S. Medvedev, The continuum limit of the Kuramoto model on sparse random graphs, Commun. Math. Sci., 17 (2019), 883-898.  doi: 10.4310/CMS.2019.v17.n4.a1. [41] G. S. Medvedev, The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Anal., 46 (2014), 2743-2766.  doi: 10.1137/130943741. [42] G. S. Medvedev, The nonlinear heat equation on W-random graphs, Arch. Ration. Mech. Anal., 212 (2014), 781-803.  doi: 10.1007/s00205-013-0706-9. [43] G. S. Medvedev and S. Zhuravytska, The geometry of spontaneous spiking in neuronal networks, J. Nonlinear Sci., 22 (2012), 689-725.  doi: 10.1007/s00332-012-9125-6. [44] O. E. Omel'chenko, M. Wolfrum, S. Yanchuk, Y. L. Maistrenko and O. Sudakov, Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators, Phys. Rev. E, 85 (2012), 036210.  doi: 10.1103/PhysRevE.85.036210. [45] W. Oukil, A. Kessi and P. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303. [46] D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123. [47] J. R. Phillips, H. S. J. van der Zant, J. White and T. P. Orlando, Influence of induced magnetic fields on the static properties of Josephson-junction arrays, Phys. Rev. B, 47 (1993), 5219-5229.  doi: 10.1103/PhysRevB.47.5219. [48] N. V. Swindale, The model for the formation of ocular dominance stripes, Neural Comput, 31 (2001), 1473-1494. [49] J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828. [50] D. D. Quinn, R. H. Rand and S. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E, 75 (2007), 036218.  doi: 10.1103/PhysRevE.75.036218. [51] D. D. Quinn, R. H. Rand and S. Strogatz, Synchronization in the Winfree model of coupled nonlinear interactions, A. ENOC 2005 Conference, Eindhoven, Netherlands, (2005), 7–12. [52] W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Automat. Control, 50 (2005), 655-661.  doi: 10.1109/TAC.2005.846556. [53] S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Phys. D, 74 (1994), 197-253.  doi: 10.1016/0167-2789(94)90196-1. [54] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3. [55] D. A. Wiley, S. H. Strogatz and M. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103.  doi: 10.1063/1.2165594.
Special pair for $S(\theta)$ and $I(\theta)$
Positions of $\bar\theta$ and ${\overline\theta}^\infty$ for case of $(SI)(\theta) = -\sin\theta(1 +\cos\theta)$
Uniform $L^1$-stability
Point trajectory of $\theta_1$ and $\theta_2$ at $(0.5, 0.5)$
Comparison between stationary profile and natural frequency profile
$L^1$-stability
Point trajectory of $\theta_1$ and $\theta_2$ at $(0.5, 0.5)$
Comparison between steady state and natural frequency
$\frac{\theta(x, 20)}{20}$, two different view points
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