# American Institute of Mathematical Sciences

• Previous Article
Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing
• NHM Home
• This Issue
• Next Article
Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions
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. 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. [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. Benedetto, E. 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. Benedetto, E. 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. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-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. Carrillo, L. 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. 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. [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. Ha, S. 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. Ha, J. Kim, J. 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. 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., 4 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10. [21] S.-Y. Ha, D. Ko, J. 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. Pikovsky, M. 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. 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. [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. Benedetto, E. 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. Benedetto, E. 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. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-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. Carrillo, L. 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. 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. [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. Ha, S. 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. Ha, J. Kim, J. 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. 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., 4 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10. [21] S.-Y. Ha, D. Ko, J. 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. Pikovsky, M. 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.
 [1] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic and Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [2] Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553 [3] Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks and Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001 [4] Woojoo Shim. On the generic complete synchronization of the discrete Kuramoto model. Kinetic and Related Models, 2020, 13 (5) : 979-1005. doi: 10.3934/krm.2020034 [5] Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027 [6] Seung-Yeal Ha, Bora Moon. Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement. Kinetic and Related Models, 2020, 13 (5) : 889-931. doi: 10.3934/krm.2020031 [7] Xiaoxue Zhao, Zhuchun Li. Synchronization of a Kuramoto-like model for power grids with frustration. Networks and Heterogeneous Media, 2020, 15 (3) : 543-553. doi: 10.3934/nhm.2020030 [8] Tingting Zhu. Emergence of synchronization in Kuramoto model with frustration under general network topology. Networks and Heterogeneous Media, 2022, 17 (2) : 255-291. doi: 10.3934/nhm.2022005 [9] Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic and Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037 [10] Georgios I. Papayiannis. Robust policy selection and harvest risk quantification for natural resources management under model uncertainty. Journal of Dynamics and Games, 2022, 9 (2) : 203-217. doi: 10.3934/jdg.2022004 [11] Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527 [12] Andrew J. Majda, Michal Branicki. Lessons in uncertainty quantification for turbulent dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3133-3221. doi: 10.3934/dcds.2012.32.3133 [13] Jing Li, Panos Stinis. Mori-Zwanzig reduced models for uncertainty quantification. Journal of Computational Dynamics, 2019, 6 (1) : 39-68. doi: 10.3934/jcd.2019002 [14] H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, Emma Thorpe. Uncertainty quantification in modeling HIV viral mechanics. Mathematical Biosciences & Engineering, 2015, 12 (5) : 937-964. doi: 10.3934/mbe.2015.12.937 [15] Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1 (2) : 157-176. doi: 10.3934/fods.2019007 [16] Richard Archibald, Feng Bao, Yanzhao Cao, He Zhang. A backward SDE method for uncertainty quantification in deep learning. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022062 [17] Michael Herty, Elisa Iacomini. Uncertainty quantification in hierarchical vehicular flow models. Kinetic and Related Models, 2022, 15 (2) : 239-256. doi: 10.3934/krm.2022006 [18] Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks and Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787 [19] Chun-Hsiung Hsia, Chang-Yeol Jung, Bongsuk Kwon. On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3319-3334. doi: 10.3934/dcdsb.2018322 [20] Joseph D. Skufca, Erik M. Bollt. Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks. Mathematical Biosciences & Engineering, 2004, 1 (2) : 347-359. doi: 10.3934/mbe.2004.1.347

2021 Impact Factor: 1.41