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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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. 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]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. 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]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

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