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.
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Acknowledgements
S. S. acknowledges financial support from Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 18H06478. W. K. acknowledges financial support from JSPS KAKENHI Grants No. 16K16125 No. 17H03279. H. N. acknowledges financial support from JSPS KAKENHI Grants No. 16K13847, No. 17H03279, No. 18K03471, No. 18H03287, and JST CREST Grant No. JPMJCR1913.
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Appendix: Laplace Average
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]
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
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
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
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
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 \(\mathbf {m}_{p,1}\):
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:
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Shirasaka, S., Kurebayashi, W., Nakao, H. (2020). Phase-Amplitude Reduction of Limit Cycling Systems. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_15
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