doi: 10.3934/krm.2021036
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Related Models, 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.  Google Scholar

[2]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.   Google Scholar

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. 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.  Google Scholar

[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.   Google Scholar

[5]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. 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.  Google Scholar

[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.  Google Scholar

[7]

R. Ben-YishaiD. 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.  Google Scholar

[8]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.  Google Scholar

[9]

Y.-P. ChoiS.-Y. HaS. 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.  Google Scholar

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F. GiannuzziD. MarinazzoG. NardulliM. Pellicoro and S. Stramaglia, Phase diagram of a generalized winfree model, Physical Review E, 75 (2007), 051104.  doi: 10.1103/PhysRevE.75.051104.  Google Scholar

[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.  Google Scholar

[19]

S.-Y. HaM. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

S.-Y. HaH. 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.  Google Scholar

[23]

S.-Y. HaH. 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.  Google Scholar

[24]

S.-Y. HaH. 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.  Google Scholar

[25]

S.-Y. HaY. 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.  Google Scholar

[26]

S.-Y. HaD. KoJ. 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.  Google Scholar

[27]

S.-Y. HaD. KoJ. 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.  Google Scholar

[28]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[29]

S.-Y. HaD. 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.  Google Scholar

[30]

S.-Y. HaJ. 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.  Google Scholar

[31]

A. JadbabaieJ. 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.  Google Scholar

[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.  Google Scholar

[33]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.   Google Scholar

[34]

Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), 380-385.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

Z. LiY. 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.  Google Scholar

[39]

G. S. Medvedev, Stochastic stability of continuous time consensus protocols, SIAM J. Control Optim., 50 (2012), 1859-1885.  doi: 10.1137/100801457.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

O. E. Omel'chenkoM. WolfrumS. YanchukY. 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.  Google Scholar

[45]

W. OukilA. Kessi and P. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303.  Google Scholar

[46]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123.  Google Scholar

[47]

J. R. PhillipsH. S. J. van der ZantJ. 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.  Google Scholar

[48]

N. V. Swindale, The model for the formation of ocular dominance stripes, Neural Comput, 31 (2001), 1473-1494.   Google Scholar

[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.  Google Scholar

[50]

D. D. QuinnR. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

D. A. WileyS. H. Strogatz and M. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103.  doi: 10.1063/1.2165594.  Google Scholar

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.  Google Scholar

[2]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.   Google Scholar

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. 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.  Google Scholar

[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.   Google Scholar

[5]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. 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.  Google Scholar

[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.  Google Scholar

[7]

R. Ben-YishaiD. 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.  Google Scholar

[8]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.  Google Scholar

[9]

Y.-P. ChoiS.-Y. HaS. 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.  Google Scholar

[10]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F. GiannuzziD. MarinazzoG. NardulliM. Pellicoro and S. Stramaglia, Phase diagram of a generalized winfree model, Physical Review E, 75 (2007), 051104.  doi: 10.1103/PhysRevE.75.051104.  Google Scholar

[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.  Google Scholar

[19]

S.-Y. HaM. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

S.-Y. HaH. 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.  Google Scholar

[23]

S.-Y. HaH. 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.  Google Scholar

[24]

S.-Y. HaH. 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.  Google Scholar

[25]

S.-Y. HaY. 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.  Google Scholar

[26]

S.-Y. HaD. KoJ. 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.  Google Scholar

[27]

S.-Y. HaD. KoJ. 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.  Google Scholar

[28]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surv. Math. Sci., 3 (2016), 209-267.  doi: 10.4171/EMSS/17.  Google Scholar

[29]

S.-Y. HaD. 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.  Google Scholar

[30]

S.-Y. HaJ. 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.  Google Scholar

[31]

A. JadbabaieJ. 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.  Google Scholar

[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.  Google Scholar

[33]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.   Google Scholar

[34]

Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), 380-385.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

Z. LiY. 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.  Google Scholar

[39]

G. S. Medvedev, Stochastic stability of continuous time consensus protocols, SIAM J. Control Optim., 50 (2012), 1859-1885.  doi: 10.1137/100801457.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

O. E. Omel'chenkoM. WolfrumS. YanchukY. 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.  Google Scholar

[45]

W. OukilA. Kessi and P. Thieullen, Synchronization hypothesis in the Winfree model, Dyn. Syst., 32 (2017), 326-339.  doi: 10.1080/14689367.2016.1227303.  Google Scholar

[46]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123.  Google Scholar

[47]

J. R. PhillipsH. S. J. van der ZantJ. 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.  Google Scholar

[48]

N. V. Swindale, The model for the formation of ocular dominance stripes, Neural Comput, 31 (2001), 1473-1494.   Google Scholar

[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.  Google Scholar

[50]

D. D. QuinnR. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

D. A. WileyS. H. Strogatz and M. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103.  doi: 10.1063/1.2165594.  Google Scholar

Figure 1.  Special pair for $ S(\theta) $ and $ I(\theta) $
Figure 2.  Positions of $ \bar\theta $ and $ {\overline\theta}^\infty $ for case of $ (SI)(\theta) = -\sin\theta(1 +\cos\theta) $
Figure 3.  Uniform $ L^1 $-stability
Figure 4.  Point trajectory of $ \theta_1 $ and $ \theta_2 $ at $ (0.5, 0.5) $
Figure 5.  Comparison between stationary profile and natural frequency profile
Figure 6.  $ L^1 $-stability
Figure 7.  Point trajectory of $ \theta_1 $ and $ \theta_2 $ at $ (0.5, 0.5) $
Figure 8.  Comparison between steady state and natural frequency
Figure 9.  $ \frac{\theta(x, 20)}{20} $, two different view points
[1]

Daniela Calvetti, Jenni Heino, Erkki Somersalo, Knarik Tunyan. Bayesian stationary state flux balance analysis for a skeletal muscle metabolic model. Inverse Problems & Imaging, 2007, 1 (2) : 247-263. doi: 10.3934/ipi.2007.1.247

[2]

Pierre Degond, Sophie Hecht, Nicolas Vauchelet. Incompressible limit of a continuum model of tissue growth for two cell populations. Networks & Heterogeneous Media, 2020, 15 (1) : 57-85. doi: 10.3934/nhm.2020003

[3]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[4]

Davide Guidetti. Convergence to a stationary state of solutions to inverse problems of parabolic type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 711-722. doi: 10.3934/dcdss.2013.6.711

[5]

Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135

[6]

Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613

[7]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021

[8]

Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941

[9]

Seung-Yeal Ha, Hansol Park, Yinglong Zhang. Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration. Networks & Heterogeneous Media, 2020, 15 (3) : 427-461. doi: 10.3934/nhm.2020026

[10]

Alessandro Ciallella, Emilio N. M. Cirillo. Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time. Kinetic & Related Models, 2018, 11 (6) : 1475-1501. doi: 10.3934/krm.2018058

[11]

Qiying Hu, Chen Xu, Wuyi Yue. A unified model for state feedback of discrete event systems I: framework and maximal permissive state feedback. Journal of Industrial & Management Optimization, 2008, 4 (1) : 107-123. doi: 10.3934/jimo.2008.4.107

[12]

Seung-Yeal Ha, Doheon Kim, Bora Moon. Interplay of random inputs and adaptive couplings in the Winfree model. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3975-4006. doi: 10.3934/cpaa.2021140

[13]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[14]

Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239

[15]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039

[16]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931

[17]

Qiaojun Situ, Jinzhi Lei. A mathematical model of stem cell regeneration with epigenetic state transitions. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1379-1397. doi: 10.3934/mbe.2017071

[18]

Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071

[19]

Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805

[20]

Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055

2020 Impact Factor: 1.432

Article outline

Figures and Tables

[Back to Top]