2 Phase and Amplitude Description of Complex Oscillatory Patterns …
25
approximately describe them by a set of phase and amplitude equations in the vicinity of the unperturbed limit cycle, which can be used for the analysis and control
of complex oscillatory dynamics in RD systems. As an application, we have considered entrainment of the oscillatory pattern with optimized periodic forcing and
feedback stabilization, which allows us to apply stronger forcing while keeping the
approximate phase-amplitude description valid and thereby realizing more stable
entrainment.
The method of phase reduction for infinite-dimensional dynamical systems has
also been developed for delay-differential equations exhibiting limit-cycle oscillations [13, 26], for nonlinear Fokker-Planck (integro-differential) equation describing
populations of coupled oscillators and excitable elements [11, 12], and also for fluid
systems exhibiting stable oscillatory convection [8–10, 33]. As we have formulated
for the RD systems, we can derive the amplitude equation also for such systems,
which can be used for analyzing their transient relaxation properties.
In this chapter, we have presented the theory only for a single oscillatory system in
the simplest lowest-order approximation with a single phase and a single amplitude.
Generalization to more complex cases is also possible along the lines presented
in this chapter. In general, even a single system may possess two or more phase
variables when it has additional continuous translational symmetries e.g. in spatial
directions [2, 10] and possesses a torus solution rather than a limit-cycle solution.
We may also need to consider two or more amplitude variables and consider the
case of complex Floquet eigenvalues. Moreover, although the phase is decoupled
from the amplitude at the lowest-order phase-amplitude description considered in
this chapter, this is not the case if we proceed to the next order; nonlinear phaseamplitude interactions can arise within a single system and, if coupled systems are
considered, phase and amplitude interactions of three- or more systems can generally
arise. Such higher-order interactions can be a source of intriguing complex dynamics
in nonlinear oscillatory systems; see, e.g., Refs. [1, 4, 14, 17, 29, 36, 37] for various
types of higher-order descriptions and their consequences.
Macroscopic oscillatory systems in the real world are often made up of spatially
distributed populations of interacting microsystems and modeled by PDEs including the RD equations. Development of the phase-amplitude framework for such
systems, together with the recent advance in the Koopman operator approach to nonlinear dynamical systems [23], will provide us with a unified viewpoint and practical
methods for the analysis, control, and design of such complex oscillatory systems.
Acknowledgements The author is grateful to A. Stefanovska and P. V. E. McClintock for invitation
to write this chapter and for their kind advice. This work is financially supported by JSPS KAKENHI
Grants JP17H03279, 18K03471, JP18H03287, and JST CREST JPMJCR1913.
25
approximately describe them by a set of phase and amplitude equations in the vicinity of the unperturbed limit cycle, which can be used for the analysis and control
of complex oscillatory dynamics in RD systems. As an application, we have considered entrainment of the oscillatory pattern with optimized periodic forcing and
feedback stabilization, which allows us to apply stronger forcing while keeping the
approximate phase-amplitude description valid and thereby realizing more stable
entrainment.
The method of phase reduction for infinite-dimensional dynamical systems has
also been developed for delay-differential equations exhibiting limit-cycle oscillations [13, 26], for nonlinear Fokker-Planck (integro-differential) equation describing
populations of coupled oscillators and excitable elements [11, 12], and also for fluid
systems exhibiting stable oscillatory convection [8–10, 33]. As we have formulated
for the RD systems, we can derive the amplitude equation also for such systems,
which can be used for analyzing their transient relaxation properties.
In this chapter, we have presented the theory only for a single oscillatory system in
the simplest lowest-order approximation with a single phase and a single amplitude.
Generalization to more complex cases is also possible along the lines presented
in this chapter. In general, even a single system may possess two or more phase
variables when it has additional continuous translational symmetries e.g. in spatial
directions [2, 10] and possesses a torus solution rather than a limit-cycle solution.
We may also need to consider two or more amplitude variables and consider the
case of complex Floquet eigenvalues. Moreover, although the phase is decoupled
from the amplitude at the lowest-order phase-amplitude description considered in
this chapter, this is not the case if we proceed to the next order; nonlinear phaseamplitude interactions can arise within a single system and, if coupled systems are
considered, phase and amplitude interactions of three- or more systems can generally
arise. Such higher-order interactions can be a source of intriguing complex dynamics
in nonlinear oscillatory systems; see, e.g., Refs. [1, 4, 14, 17, 29, 36, 37] for various
types of higher-order descriptions and their consequences.
Macroscopic oscillatory systems in the real world are often made up of spatially
distributed populations of interacting microsystems and modeled by PDEs including the RD equations. Development of the phase-amplitude framework for such
systems, together with the recent advance in the Koopman operator approach to nonlinear dynamical systems [23], will provide us with a unified viewpoint and practical
methods for the analysis, control, and design of such complex oscillatory systems.
Acknowledgements The author is grateful to A. Stefanovska and P. V. E. McClintock for invitation
to write this chapter and for their kind advice. This work is financially supported by JSPS KAKENHI
Grants JP17H03279, 18K03471, JP18H03287, and JST CREST JPMJCR1913.