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June  2019, 14(2): 317-340. doi: 10.3934/nhm.2019013

A local sensitivity analysis for the kinetic Kuramoto equation with random inputs

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Republic of Korea

3. 

School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

4. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Jinwook Jung

Received  May 2018 Revised  January 2019 Published  April 2019

We present a local sensivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime. In our proposed random kinetic Kuramoto equation (in short, RKKE), the random inputs are encoded in the coupling strength. For the deterministic case, it is well known that the kinetic Kuramoto equation exhibits asymptotic phase concentration for well-prepared initial data in the large coupling regime. To see a response of the system to the random inputs, we provide propagation of regularity, local-in-time stability estimates for the variations of the random kinetic density function in random parameter space. For identical oscillators with the same natural frequencies, we introduce a Lyapunov functional measuring the phase concentration, and provide a local sensitivity analysis for the functional.

Citation: Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks and Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013
References:
[1]

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. 

[2]

D. Aeyels and J. Rogge, Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941. 

[3]

G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp. doi: 10.1155/2015/850124.

[4]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.

[5]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.

[6]

J. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17pp. doi: 10.1063/1.4745197.

[7]

J. Buck and E. Buck, Biology of sychronous flashing of fireflies, Nature, 211 (1966), 562.

[8]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415.  doi: 10.1007/s10955-014-1005-z.

[9]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Comm. in Comp. Phys., 25 (2019), 508-531. 

[10]

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.

[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]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[13]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[14]

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.

[15]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol, 22 (1985), 1-9.  doi: 10.1007/BF00276542.

[16]

S.-Y. Ha and S. Jin, Local sensitivity analysis for the Cucker-Smale model with random inputs, Kinetic Relat. Models., 11 (2018), 859-889.  doi: 10.3934/krm.2018034.

[17]

S.-Y. HaS. Jin and J. Jung, A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs, J. Differential Equations, 265 (2018), 3618-3649.  doi: 10.1016/j.jde.2018.05.013.

[18]

S.-Y. Ha, S. Jin and J. Jung, Local sensitivity analysis for the Kuramoto mdoel with random inputs in a large coupling regime, Submitted.

[19]

S.-Y. HaJ. KimJ. Park and X. Zhang, Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media, 13 (2018), 297-322.  doi: 10.3934/nhm.2018013.

[20]

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., 4 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[21]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[22]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic and Related Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[23]

A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296-4301.

[24]

E. H. Kennard, Kinetic theory of gases. McGraw-Hill Book Company, New York and London, 1938.

[25]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[26]

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

[27]

C. Lancellotti, On the vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory and Statistical Physics, 34 (2005), 523-535.  doi: 10.1080/00411450508951152.

[28]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Science, 17 (2007), 309-347.  doi: 10.1007/s00332-006-0806-x.

[29]

R. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249-266.  doi: 10.1016/j.physd.2005.01.017.

[30]

R. Mirollo and S. H. Strogatz, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202.

[31]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation. In kinetic theories and the Boltzmann equation, Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), 60-110, Lecture Notes in Math., 1048, Springer, Berlin, 1984. doi: 10.1007/BFb0071878.

[32] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[33]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[34]

M. Verwoerd and O. Mason, A convergence result for the Kurmoto model with all-to-all couplings, SIAM J. Appl. Dyn. Syst., 10 (2011), 906-920.  doi: 10.1137/090771946.

[35]

M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), 417-453.  doi: 10.1137/080725726.

[36]

M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 134-160.  doi: 10.1137/070686858.

[37]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

show all references

References:
[1]

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. 

[2]

D. Aeyels and J. Rogge, Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941. 

[3]

G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp. doi: 10.1155/2015/850124.

[4]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.

[5]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.

[6]

J. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17pp. doi: 10.1063/1.4745197.

[7]

J. Buck and E. Buck, Biology of sychronous flashing of fireflies, Nature, 211 (1966), 562.

[8]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415.  doi: 10.1007/s10955-014-1005-z.

[9]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Comm. in Comp. Phys., 25 (2019), 508-531. 

[10]

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.

[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]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[13]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[14]

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.

[15]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol, 22 (1985), 1-9.  doi: 10.1007/BF00276542.

[16]

S.-Y. Ha and S. Jin, Local sensitivity analysis for the Cucker-Smale model with random inputs, Kinetic Relat. Models., 11 (2018), 859-889.  doi: 10.3934/krm.2018034.

[17]

S.-Y. HaS. Jin and J. Jung, A local sensitivity analysis for the kinetic Cucker-Smale equation with random inputs, J. Differential Equations, 265 (2018), 3618-3649.  doi: 10.1016/j.jde.2018.05.013.

[18]

S.-Y. Ha, S. Jin and J. Jung, Local sensitivity analysis for the Kuramoto mdoel with random inputs in a large coupling regime, Submitted.

[19]

S.-Y. HaJ. KimJ. Park and X. Zhang, Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media, 13 (2018), 297-322.  doi: 10.3934/nhm.2018013.

[20]

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., 4 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[21]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[22]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic and Related Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[23]

A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296-4301.

[24]

E. H. Kennard, Kinetic theory of gases. McGraw-Hill Book Company, New York and London, 1938.

[25]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin. 1984. doi: 10.1007/978-3-642-69689-3.

[26]

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

[27]

C. Lancellotti, On the vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory and Statistical Physics, 34 (2005), 523-535.  doi: 10.1080/00411450508951152.

[28]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Science, 17 (2007), 309-347.  doi: 10.1007/s00332-006-0806-x.

[29]

R. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249-266.  doi: 10.1016/j.physd.2005.01.017.

[30]

R. Mirollo and S. H. Strogatz, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202.

[31]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation. In kinetic theories and the Boltzmann equation, Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), 60-110, Lecture Notes in Math., 1048, Springer, Berlin, 1984. doi: 10.1007/BFb0071878.

[32] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[33]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[34]

M. Verwoerd and O. Mason, A convergence result for the Kurmoto model with all-to-all couplings, SIAM J. Appl. Dyn. Syst., 10 (2011), 906-920.  doi: 10.1137/090771946.

[35]

M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), 417-453.  doi: 10.1137/080725726.

[36]

M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 134-160.  doi: 10.1137/070686858.

[37]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

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