Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits

Kvalheim, Matthew D.; Revzen, Shai

doi:10.48550/arxiv.1911.11996

Cited by 5 publications

(16 citation statements)

References 21 publications

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“…where g j (θ) are Floquet eigenfunctions associated with the periodic orbit x γ (θ), and ψ j are isostable coordinates. These isostable coordinates can be defined as level sets of particular eigenfunctions of the Koopman operator [15], [11]. For the most slowly decaying isostable coordinates, an alternative definition can be given that considers the infinite-time decay of initial conditions to the limit cycle [27], [22].…”

Section: Phase-amplitude Reduction Methods and Isostable Coordinatesmentioning

confidence: 99%

“…To implement the adaptive phase-amplitude reduction framework, suppose that ( 9) has a nominal parameter set p 0 that describes the dynamics of an underlying model. It is then possible to rewrite (9) as ẋ = F (x, p) + U e (t, p, x), (11) with the extended input U e (t, p, x) ≡ U (t) + F (x, p 0 ) − F (x, p). Intuitively, the term F (x, p) represents dynamics of the model when using parameter set p, and U e captures both the externally applied input and the mismatch caused when considering the parameter set p as compared to the parameter set p 0 of the underlying model (9).…”

Section: Derivation Of An Adaptive Phase-amplitude Reduction Strategymentioning

confidence: 99%

“…The variable parameter for this model is r 0 with a nominal value of r 0 = 1, consequently the extended input from (11) is…”

Section: Nonradial Isochron Clockmentioning

confidence: 99%

See 2 more Smart Citations

An Adaptive Phase-Amplitude Reduction Framework Without $\mathcal{O}(ε)$ Constraints on Inputs

Wilson¹

2020

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Phase reduction is a well-established technique used to analyze the timing of oscillations in response to weak external inputs. In the preceding decades, a wide variety of results have been obtained for weakly perturbed oscillators that place restrictive limits on the magnitude of the inputs or on the magnitude of the time derivatives of the inputs. By contrast, no general reduction techniques currently exist to analyze oscillatory dynamics in response to arbitrary, large magnitude inputs and comparatively very little is understood about these strongly perturbed limit cycle oscillators. In this work, the theory of isostable reduction is leveraged to develop an adaptive phase-amplitude transformation that does not place any restrictions on the allowable input. Additionally, provided some of the Floquet multipliers of the underlying periodic orbits are near-zero, the proposed method yields a reduction in dimension comparable to that of other phase-amplitude reduction frameworks. Numerical illustrations show that the proposed method accurately reflects synchronization and entrainment of coupled oscillators in regimes where a variety of other phase-amplitude reductions fail.

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Section: Phase-amplitude Reduction Methods and Isostable Coordinatesmentioning

confidence: 99%

Section: Derivation Of An Adaptive Phase-amplitude Reduction Strategymentioning

confidence: 99%

See 1 more Smart Citation

An Adaptive Phase-Amplitude Reduction Framework Without $\mathcal{O}(ε)$ Constraints on Inputs

Wilson¹

2020

Preprint

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“…Proposition 4 Consider the time-varying system (2) with an equilibrium x ⋆ at the origin, 6 and assuming the system is contracting with a time-invariant metric M (x), then there exists a Koopman mapping φ(x, t) satisfying ( 15) and (16).…”

Section: B Extension To Time-varying Systemsmentioning

confidence: 99%

On the equivalence of contraction and Koopman approaches for nonlinear stability and control

Manchester

2021

Preprint

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In this paper we prove new connections between two frameworks for analysis and control of nonlinear systems: the Koopman operator framework and contraction analysis. Each method, in different ways, provides exact and global analyses of nonlinear systems by way of linear systems theory. The main results of this paper show equivalence between contraction and Koopman approaches for a wide class of stability analysis and control design problems. In particular: stability or stablisability in the Koopman framework implies the existence of a contraction metric (resp. control contraction metric) for the nonlinear system. Further in certain cases the converse holds: contraction implies the existence of a set of observables with which stability can verified via the Koopman framework. Furthermore, the converse claims are based on an novel relations between the Koopman method and construction of a Kazantzis-Kravaris-Luenberger observer.

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“…The definition (3) is closely aligned with the one given in [22], however, other definitions that compute Fourier averages of observables [20], [21] can also be used. As a point of emphasis, the constructive definition of isostable coordinates (3) is only possible for a subset of eigenvalues with the smallest magnitude real components [29]. Isostable coordinates associated with larger magnitude eigenvalues must be defined implicitly as level sets of Koopman eigenfunctions with decay rates that are governed by their associated λ j .…”

Section: Background: Isostable Reductionmentioning

confidence: 99%

Data-Driven Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs

Wilson

2021

Preprint

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Isostable reduction is a powerful technique that can be used to characterize behaviors of nonlinear dynamical systems in a basis of slowly decaying eigenfunctions of the Koopman operator. When the underlying dynamical equations are known, previously developed numerical techniques allow for high-order accuracy computation of isostable reduced models. However, in situations where the dynamical equations are unknown, few general techniques are available that provide reliable estimates of the isostable reduced equations, especially in applications where large magnitude inputs are considered. In this work, a purely data-driven inference strategy yielding high-accuracy isostable reduced models is developed for dynamical systems with a fixed point attractor. By analyzing steady state outputs of nonlinear systems in response to sinusoidal forcing, both isostable response functions and isostable-to-output relationships can be estimated to arbitrary accuracy in an expansion performed in the isostable coordinates. Detailed examples are considered for a population of synaptically coupled neurons and for the one-dimensional Burgers' equation. While linear estimates of the isostable response functions are sufficient to characterize the dynamical behavior when small magnitude inputs are considered, the high-accuracy reduced order model inference strategy proposed here is essential when considering large magnitude inputs.

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Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits

Cited by 5 publications

References 21 publications

An Adaptive Phase-Amplitude Reduction Framework Without $\mathcal{O}(ε)$ Constraints on Inputs

An Adaptive Phase-Amplitude Reduction Framework Without $\mathcal{O}(ε)$ Constraints on Inputs

On the equivalence of contraction and Koopman approaches for nonlinear stability and control

Data-Driven Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs

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