26
H. Nakao
References
1. P. Ashwin, A. Rodrigues, Hopf normal form with SN symmetry and reduction to systems of
nonlinearly coupled phase oscillators. Phys. D 325, 14–24 (2016)
2. I.V. Biktasheva, D. Barkley, V.N. Biktashev, G.V. Bordyugov, A.J. Foulkes, Computation of
the response functions of spiral waves in active media Phys. Rev. E 79, 056702 (2009)
3. E. Brown, J. Moehlis, P. Holmes, On the phase reduction and response dynamics of neural
oscillator populations. Neural Comput. 16, 673 (2004)
4. B. Ermentrout, Y. Park, D. Wilson, Recent advances in coupled oscillator theory. Phil. Trans.
Roy. Soc. A 377, 20190092 (2019)
5. G.B. Ermentrout, D.H. Terman, Mathematical Foundations of Neuroscience (Springer, New
York, 2010)
6. L. Glass, M.C. Mackey, From Clocks to Chaos—The Rhythms of Life (Princeton University
Press, Princeton, 1988)
7. F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks (Springer, New York,
1997)
8. M. Iima, Jacobian-free algorithm to calculate the phase sensitivity function in the phase reduction theory and its applications to Kármán’s vortex street. Phys. Rev. E 99, 062203 (2019)
9. Y. Kawamura, H. Nakao, Collective phase description of oscillatory convection. Chaos 23,
043129 (2013)
10. Y. Kawamura, H. Nakao, Phase description of oscillatory convection with a spatially translational mode. Phys. D 295–296, 11–29 (2015)
11. Y. Kawamura, H. Nakao, K. Arai, H. Kori, Y. Kuramoto, Collective phase sensitivity. Phys.
Rev. Lett. 101, 024101 (2008)
12. Y. Kawamura, H. Nakao, Y. Kuramoto, Collective phase description of globally coupled
excitable elements. Phys. Rev. E 84, 046211 (2011)
13. K. Kotani, I. Yamaguchi, Y. Ogawa, Y. Jimbo, H. Nakao, G.B. Ermentrout, Adjoint method
provides phase response functions for delay-induced oscillations. Phys. Rev. Lett. 109, 044101
(2012)
14. K. Kotani, Y. Ogawa, S. Shirasaka, A. Akao, Y. Jimbo, H. Nakao, Nonlinear phase-amplitude
reduction of delay-induced oscillations. Phys. Rev. Res. 2, 033106 (2020)
15. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Dover, New York, 2003)
16. Y. Kuramoto, H. Nakao, On the concept of dynamical reduction—the case of coupled oscillators. Phil. Trans. Roy. Soc. A 377, 20190041 (2019)
17. I. León, D. Pazó, Phase reduction beyond the first order: The case of the mean-field complex
Ginzburg-Landau equation. Phys. Rev. E 100, 012211 (2019)
18. A. Mauroy, I. Mezi´ c, Y. Susuki (eds), The Koopman operator in systems and control. Lecture
Notes in Control and Information Sciences, vol. 484, (Springer, Cham, 2020)
19. A. Mauroy, I. Mezi´ c, Global stability analysis using the eigenfunctions of the Koopman operator. IEEE Trans. Autom. Control 61, 3356–3369 (2016)
20. A. Mauroy, I. Mezi´ c, Global computation of phase-amplitude reduction for limit-cycle dynamics. Chaos 28, 073108 (2018)
21. B. Monga, D. Wilson, T. Matchen, J. Moehlis, Phase reduction and phase-based optimal control
for biological systems: a tutorial. Biol. Cybern. 113, 11–46 (2019)
22. Nakao, H., Mezi´ c, I.: Koopman eigenfunctionals and phase-amplitude reduction of rhythmic
reaction—diffusion systems, in Proceedings of the SICE Annual Conference (2018), pp. 74–77
23. H. Nakao, I. Mezi´ c, Spectral Analysis of the Koopman Operator for Partial Differential Equations. Chaos 30, 113131 (2020)
24. H. Nakao, Phase reduction approach to synchronization of nonlinear oscillators. Contemp.
Phys. 57, 188–214 (2016)
25. H. Nakao, T. Yanagita, Y. Kawamura, Phase reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems. Phys. Rev. X 4, 021032 (2014)
26. V. Noviˇ cenko, K. Pyragas, Phase reduction of weakly perturbed limit cycle oscillations in
time-delay systems. Phys. D 241, 1090–1098 (2012)
H. Nakao
References
1. P. Ashwin, A. Rodrigues, Hopf normal form with SN symmetry and reduction to systems of
nonlinearly coupled phase oscillators. Phys. D 325, 14–24 (2016)
2. I.V. Biktasheva, D. Barkley, V.N. Biktashev, G.V. Bordyugov, A.J. Foulkes, Computation of
the response functions of spiral waves in active media Phys. Rev. E 79, 056702 (2009)
3. E. Brown, J. Moehlis, P. Holmes, On the phase reduction and response dynamics of neural
oscillator populations. Neural Comput. 16, 673 (2004)
4. B. Ermentrout, Y. Park, D. Wilson, Recent advances in coupled oscillator theory. Phil. Trans.
Roy. Soc. A 377, 20190092 (2019)
5. G.B. Ermentrout, D.H. Terman, Mathematical Foundations of Neuroscience (Springer, New
York, 2010)
6. L. Glass, M.C. Mackey, From Clocks to Chaos—The Rhythms of Life (Princeton University
Press, Princeton, 1988)
7. F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks (Springer, New York,
1997)
8. M. Iima, Jacobian-free algorithm to calculate the phase sensitivity function in the phase reduction theory and its applications to Kármán’s vortex street. Phys. Rev. E 99, 062203 (2019)
9. Y. Kawamura, H. Nakao, Collective phase description of oscillatory convection. Chaos 23,
043129 (2013)
10. Y. Kawamura, H. Nakao, Phase description of oscillatory convection with a spatially translational mode. Phys. D 295–296, 11–29 (2015)
11. Y. Kawamura, H. Nakao, K. Arai, H. Kori, Y. Kuramoto, Collective phase sensitivity. Phys.
Rev. Lett. 101, 024101 (2008)
12. Y. Kawamura, H. Nakao, Y. Kuramoto, Collective phase description of globally coupled
excitable elements. Phys. Rev. E 84, 046211 (2011)
13. K. Kotani, I. Yamaguchi, Y. Ogawa, Y. Jimbo, H. Nakao, G.B. Ermentrout, Adjoint method
provides phase response functions for delay-induced oscillations. Phys. Rev. Lett. 109, 044101
(2012)
14. K. Kotani, Y. Ogawa, S. Shirasaka, A. Akao, Y. Jimbo, H. Nakao, Nonlinear phase-amplitude
reduction of delay-induced oscillations. Phys. Rev. Res. 2, 033106 (2020)
15. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Dover, New York, 2003)
16. Y. Kuramoto, H. Nakao, On the concept of dynamical reduction—the case of coupled oscillators. Phil. Trans. Roy. Soc. A 377, 20190041 (2019)
17. I. León, D. Pazó, Phase reduction beyond the first order: The case of the mean-field complex
Ginzburg-Landau equation. Phys. Rev. E 100, 012211 (2019)
18. A. Mauroy, I. Mezi´ c, Y. Susuki (eds), The Koopman operator in systems and control. Lecture
Notes in Control and Information Sciences, vol. 484, (Springer, Cham, 2020)
19. A. Mauroy, I. Mezi´ c, Global stability analysis using the eigenfunctions of the Koopman operator. IEEE Trans. Autom. Control 61, 3356–3369 (2016)
20. A. Mauroy, I. Mezi´ c, Global computation of phase-amplitude reduction for limit-cycle dynamics. Chaos 28, 073108 (2018)
21. B. Monga, D. Wilson, T. Matchen, J. Moehlis, Phase reduction and phase-based optimal control
for biological systems: a tutorial. Biol. Cybern. 113, 11–46 (2019)
22. Nakao, H., Mezi´ c, I.: Koopman eigenfunctionals and phase-amplitude reduction of rhythmic
reaction—diffusion systems, in Proceedings of the SICE Annual Conference (2018), pp. 74–77
23. H. Nakao, I. Mezi´ c, Spectral Analysis of the Koopman Operator for Partial Differential Equations. Chaos 30, 113131 (2020)
24. H. Nakao, Phase reduction approach to synchronization of nonlinear oscillators. Contemp.
Phys. 57, 188–214 (2016)
25. H. Nakao, T. Yanagita, Y. Kawamura, Phase reduction approach to synchronization of spatiotemporal rhythms in reaction-diffusion systems. Phys. Rev. X 4, 021032 (2014)
26. V. Noviˇ cenko, K. Pyragas, Phase reduction of weakly perturbed limit cycle oscillations in
time-delay systems. Phys. D 241, 1090–1098 (2012)