Phase-Amplitude Reduction of Limit Cycling Systems

Abstract

Optimization and control of collective dynamics in rhythmic systems have attracted increasing interest, and various methods have been developed on the basis of the phase reduction framework for limit cycle oscillators. The phase reduction relies on the notion of the isochron, which represents the set of system states that share the same asymptotic phase. However, the phase reduction does not take into account the amplitude degrees of freedom representing deviations of the system state from the limit cycle attractor, to which some rich and nontrivial transient behaviors of the oscillators are attributed. In this chapter, after a brief introduction of the phase reduction framework, a phase-amplitude reduction framework that is applicable to transient dynamics far from the limit cycle attractor is formulated. A rigorous theoretical background for defining the amplitudes is provided by the notion of the isostable, which naturally complements the isochron in the sense that both of them can be understood from a unified viewpoint of the spectral properties of the Koopman operator. The utility of the proposed phase-amplitude reduction framework is illustrated by evaluating the optimal injection timing of a weak control input that efficiently suppresses deviations of the system state from the limit cycle attractor.

The text of Sect. 15.3 is adapted from [78], with permission of AIP Publishing.

Appendix: Laplace Average

 

In this appendix, we propose a simpler version of the Laplace average [49, 54]. The following results also hold for complex Koopman eigenvalues $\lambda$. When $\mathbf {F}$ is a nonresonant analytic vector field, a generic scalar-valued observable function f can be expressed as [55]

$$\begin{aligned} f(\mathbf {x}) = \sum _{k\in \mathbb {Z},\ \mathbf {m}\in \mathbb {Z}_{\ge 0}^{N-1}}{a_{k,\mathbf {m}}(f)\phi _{\lambda _{k,\mathbf {m}}}(\mathbf {x})}, \end{aligned}$$

(15.65)

where $\mathbf {m}=(m_2,m_3,\ldots ,m_N)$. The complex coefficients $a_{k,\mathbf {m}}(f) \in \mathbb {C}$ are called the Koopman modes, $\lambda _{k,\mathbf {m}} = k\lambda _1 + \sum _{i=2}^N{m_i \lambda _{i}}$ are the eigenvalues of the infinitesimal generator of the Koopman operator semigroup, and

$$\begin{aligned} \phi _{\lambda _{k,\mathbf {m}}}(\mathbf {x}):=e^{\mathrm{i}k\theta (\mathbf {x})}\phi _{\lambda _2}^{m_2}(\mathbf {x}) \phi _{\lambda _3}^{m_3}(\mathbf {x}) \cdots \phi _{\lambda _N}^{m_N}(\mathbf {x}) \end{aligned}$$

(15.66)

are their associated eigenfunctions. When the observable can be expanded as (15.65), we can evaluate the value of a Koopman eigenfunction at point $\mathbf {x}$ by using the generalized Laplace average [54] as

$$\begin{aligned}&a_{k,\mathbf {m}}(f)\phi _{\lambda _{k,\mathbf {m}}}(\mathbf {x}) \nonumber \\ =&\lim _{s \rightarrow \infty }{\dfrac{1}{s}\int _0^s{\left[ f\circ \mathbf {S}(t,\mathbf {x}) - \sum _{\mathrm{Re}(\lambda _{k,\mathbf {m}'}) \ge \mathrm{Re}(\lambda _{k,\mathbf {m}})}{a_{k,\mathbf {m}'}(f)\phi _{\lambda _{k,\mathbf {m}'}}(\mathbf {x})e^{\lambda _{k,\mathbf {m}'} t}}\right] e^{-\lambda _{k, \mathbf {m}} t}}\mathrm {d}t }, \end{aligned}$$

(15.67)

where $\mathrm{Re}(\lambda )$ is the real part of $\lambda$.

If we evaluate the value of the Jth principal Koopman eigenfunction associated with the Floquet exponent $\text{ Re } (\lambda _J) \; (> 2 \text{ Re } (\lambda _2))$, we can simplify the generalized Laplace average using an observable $g_J$ defined as

$$\begin{aligned} g_J(\mathbf {x}) = \nabla r_J(\mathbf {x}_0(\theta _*)) \cdot (\mathbf {x} - \mathbf {x}_0(\theta _*)), \end{aligned}$$

(15.68)

where $\theta _* = \theta (\mathbf {x})$. Since this observable converges to zero along a transient orbit, we obtain $a_{k,\mathbf {0}}(g_J)=0$ for all $k\in \mathbb {Z}$, where $\mathbf {0}$ is a zero vector. The gradient of $g_J$ evaluated on the periodic orbit $\chi$ is given by

$$\begin{aligned} \nabla g_J (\mathbf {x}_0(\theta _*))&= (\mathbf {I} - \nabla \theta (\mathbf {x}_0(\theta _*)) \mathbf {F}(\mathbf {x}_0(\theta _*))^\dag ) \nabla r_J(\mathbf {x}_0(\theta _*)) \nonumber \\&= \nabla r_J(\mathbf {x}_0(\theta _*)), \end{aligned}$$

(15.69)

where $\mathbf {I}$ is the identity matrix and we used the fact [41, Sect. 3.4] that $\mathbf {F}(\mathbf {x}_0(\theta _*))$ is parallel to ${\varvec{\gamma }}_1(\mathbf {x}_0(\theta _*))$ and the bi-orthogonal relation (15.51).

Let us denote a vector which has a nonzero element of value q only at the $(p-1)$th component as $\mathbf {m}_{p,q}$. Because $r_p=0$ on $\chi$, the gradient of f evaluated on $\chi$ contains the following terms corresponding to $\mathbf {m}_{p,1}$:

$$\begin{aligned} \sum _{k\in \mathbb {Z}}{a_{k,\mathbf {m}_{p,1}}(f) e^{\mathrm{i}k\theta _*} \nabla r_p(\mathbf {x}_0(\theta _*))}. \end{aligned}$$

(15.70)

It can be easily shown that all other components corresponding to $|\mathbf {m}|_1 \ge 2$, where $|\cdot |_1$ is the $l^1$-norm, do not contribute to the gradient $\nabla f$ on $\chi$ since $r_p=0$ on the periodic orbit. We can then obtain the Koopman modes $a_{k,\mathbf {m}_{p,1}}(f)$ for the generic observable f from its gradient evaluated on the periodic orbit $\chi$ as follows. First, we take a phase-wise inner product of $\nabla f(\mathbf {x}_0(\theta _*))$ and $e^{-\mathrm{i}k \theta _* } {\varvec{\gamma }}_p(\mathbf {x}_0(\theta _*))$, which we denote by $\tilde{z}(\theta _*)$. Then we evaluate the average of $\tilde{z}(\theta _*)$ over the circle. From Eq. (15.69), there is only one nonzero Koopman mode for $|\mathbf {m}|_1 \le 1$, i.e., $a_{0,\mathbf {m}_{J,1}}(g_J)=1$ for the observable $g_J$. In general, the Koopman modes associated with the non-principal eigenfunctions with $|\mathbf {m}|_1 \ge 2$ are nonzero even for the observable $g_J$. However, when $\text{ Re } (\lambda _J) \; > 2 \text{ Re } (\lambda _2)$ holds, all the Koopman modes $a_{k,\mathbf {m}'}(g_J)$ involved in the generalized Laplace average (15.67) are zero.

Thus, we can replace the generalized Laplace average with the following Laplace average when we evaluate the Jth principal Koopman eigenfunction:

$$\begin{aligned} a_{0,\mathbf {m}_{J,1}}(g_J)\phi _{\lambda _J}(\mathbf {x}) = \lim _{s \rightarrow \infty }{\dfrac{1}{s}\int _0^s{g_J\circ \mathbf {S}(t,\mathbf {x})e^{-\lambda _J t}}\mathrm {d}t }. \end{aligned}$$

(15.71)