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Global stability analysis using the eigenfunctions of the Koopman operator
Alexandre Mauroy, Igor Mezic
TL;DR
The paper asks how global stability of nonlinear systems can be characterized through Koopman-operator spectra. It develops eigenfunction-based theoretical conditions and numerical approximation methods, obtaining necessary and sufficient criteria for hyperbolic attractors and methods for stability or basin analysis. The approach is primarily limited by polynomial approximations that are typically restricted to low-dimensional systems.
Problem
Koopman spectral properties have received limited attention as a framework for global stability analysis of nonlinear systems.
Method
The paper combines Koopman eigenfunction theory with Taylor- and Bernstein-polynomial numerical methods for global stability and basin-of-attraction analysis.
Results
The paper derives necessary and sufficient stability conditions based on continuously differentiable Koopman eigenfunctions and applies them to fixed points and limit cycles.
Takeaways & Limitations
The framework supports proving or disproving global stability of a fixed point or limit cycle within a specified region of state space.
Takeaways & Limitations
Polynomial-approximation methods are typically limited to low-dimensional systems.
Abstract
from arXiv · showhide
We propose a novel operator-theoretic framework to study global stability of nonlinear systems. Based on the spectral properties of the so-called Koopman operator, our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems. The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles. These results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space.
I. INTRODUCTION
The paper develops a Koopman-operator framework for analyzing global stability in nonlinear systems, addressing limited prior attention to spectral stability analysis. It derives eigenfunction-based conditions and numerical methods for stability analysis and basin estimation.
- The paper addresses the limited theoretical use of Koopman spectral properties for global stability analysis.
- For hyperbolic attractors, necessary and sufficient global-stability conditions are formulated through the existence of specific Koopman eigenfunctions.
- The framework covers hyperbolic fixed points and limit cycles without requiring specific assumptions on the nature of the infinite-dimensional function space.
- New eigenfunction-computation schemes avoid integrating particular trajectories and support systematic global-stability and basin-of-attraction analysis.
- Bernstein-polynomial approximations complement Taylor-polynomial methods when eigenfunctions are not analytic.
- The Koopman operator is linear even for nonlinear dynamical systems, enabling systematic spectral analysis of system behavior.
B. First stability results
This section connects attractor stability with the Koopman operator and establishes a general characterization of global attractivity. The result applies broadly, while the paper then specializes spectral analysis to stability questions.
- The Koopman and Perron-Frobenius operators provide equivalent descriptions of system dynamics, but the paper focuses on the Koopman framework for spectral stability analysis.
- The framework decomposes observables into functions supported outside the attractor and observables restricted to the attractor.
- Proposition 1 states that an attractor is globally attractive in X if and only if the restricted Koopman operator satisfies the stated condition.
- The proposition’s sufficiency follows because the distance to the attractor converges to zero, implying that every limit set lies within the attractor.
- The general result makes no assumption on the attractor’s type and also applies to arbitrary well-defined flows.
III. KOOPMAN EIGENFUNCTIONS AND GLOBAL STABILITY
The paper argues that a small set of Koopman eigenfunctions captures global stability properties that are otherwise difficult to extract from all observables. Eigenfunctions with decaying eigenvalues describe attraction, while purely imaginary eigenfunctions describe dynamics on the attractor.
- Only a few Koopman eigenfunctions are sufficient to capture the system’s stability properties.This addresses the practical difficulty of analyzing all observables in the functional space.
- Koopman eigenfunctions are directly related to system dynamics because knowing them is equivalent to knowing the trajectories.
- For globally stable attractors, eigenfunctions outside the attracting component restrict to eigenfunctions on the attractor with the same eigenvalue.
- Eigenfunctions associated with purely imaginary eigenvalues provide no stability information and instead describe ergodic behavior on the attractor.Their level sets identify initial conditions converging to the same attractor trajectory.
- Eigenfunctions with eigenvalues having nonzero negative real parts capture stability through the restriction of the Koopman operator to the attracting component.Their level sets are related to isostables, which group states converging synchronously toward the attractor.
B. Main results
The main results use zero level sets of Koopman eigenfunctions with negative-real-part eigenvalues to characterize invariant globally asymptotically stable sets. A finite collection of such eigenfunctions can establish global stability and support basin estimation.
- A finite intersection of zero level sets of Koopman eigenfunctions with negative-real-part eigenvalues is invariant and globally asymptotically stable.The result assumes a compact forward invariant state set and continuous eigenfunctions.
- The zero-level-set construction contains the attractor, and equality with the attractor requires enough independent eigenfunctions.
- Only a small finite number of eigenfunctions is sufficient to establish global stability despite the Koopman operator being infinite-dimensional.Typically, N−q eigenfunctions are needed for an attractor of dimension q.
- When the state set is not known to be forward invariant, suitable eigenfunction level sets can define a compact forward invariant region where global stability holds.
- The framework extends local stability of hyperbolic fixed points and limit cycles to global stability through Koopman eigenfunctions.
1) The case of a hyperbolic fixed point:
For hyperbolic fixed points, global stability is characterized by Koopman eigenfunctions associated with stable Jacobian eigenvalues. Their gradients provide the local Jacobian directions, while their global zero-level-set intersection identifies the fixed point.
- The case of a hyperbolic fixed point: The intersection of the zero level sets of the N eigenfunctions is locally and globally reduced to the fixed point under the stated assumptions.Negative real parts then yield global asymptotic stability through the general zero-level-set result.
- The case of a hyperbolic fixed point: Each relevant Koopman eigenvalue is an eigenvalue of the Jacobian, and the eigenfunction gradient is proportional to the corresponding left Jacobian eigenvector.
- The case of a hyperbolic fixed point: Global stability of a hyperbolic fixed point is equivalent to the existence of N C1 Koopman eigenfunctions associated with distinct Jacobian eigenvalues having negative real parts.The gradients at the fixed point must be nonzero and linearly independent.
- The case of a hyperbolic fixed point: The support of these particular eigenfunctions corresponds to the fixed point’s basin of attraction.
- The case of a hyperbolic fixed point: For linear systems, the criterion recovers the usual global stability condition from eigenfunctions formed with left eigenvectors.
- The case of a hyperbolic fixed point: The hyperbolic fixed-point proposition cannot be applied directly when the Jacobian has an eigenvalue with zero real part.In that case, the paper instead invokes the general theorem using continuous or generalized eigenfunctions.
2) The case of a hyperbolic limit cycle:
For hyperbolic limit cycles, global stability is characterized by N−1 smooth Koopman eigenfunctions associated with stable Floquet exponents. Their zero-level sets intersect in the cycle, extending the local Floquet picture globally.
- The case of a hyperbolic limit cycle: Global stability of a hyperbolic limit cycle is equivalent to N−1 C1 Koopman eigenfunctions with negative-real-part eigenvalues and independent transverse gradients.The eigenvalues are the limit cycle’s Floquet exponents.
- The case of a hyperbolic limit cycle: The Koopman eigenvalues associated with the relevant eigenfunctions are Floquet exponents of the limit cycle.Their exponentials over one period are Floquet multipliers.
- The case of a hyperbolic limit cycle: The N−1 eigenfunction zero-level sets intersect locally only on the limit cycle and, under the theorem’s conditions, yield global asymptotic stability.
- The case of a hyperbolic limit cycle: The limit-cycle result is the global counterpart of local stability analysis based on the monodromy matrix and Floquet exponents.
- The case of a hyperbolic limit cycle: The result cannot be applied when the state set contains an unstable fixed point because the relevant Koopman eigenfunctions are not C1 there.The paper therefore restricts the domain to a set excluding a small disk around that point.
IV. NUMERICAL METHODS
The numerical framework computes Koopman eigenfunctions without integrating trajectories, using polynomial bases to estimate attraction basins or establish stability on specified state-space regions.
- The methods compute particular Koopman eigenfunctions to estimate an attractor’s basin of attraction or establish global stability on a given state-space subset.
- Unlike Laplace-average approaches, the proposed numerical schemes do not require trajectory integration.
- The schemes expand the eigenfunctions in Taylor or Bernstein polynomial bases and treat stable fixed points and stable limit cycles separately.
A. Taylor expansion-based method for the fixed point
The Taylor method recursively computes Koopman eigenfunction coefficients near a stable fixed point and uses them to construct Lyapunov functions and inner basin estimates. Its accuracy improves with polynomial degree but is limited by eigenfunction analyticity.
- Taylor coefficient computation: Assuming an analytic vector field and nonresonant Jacobian eigenvalues, Taylor coefficients are solved recursively by increasing expansion order.
- Taylor coefficient computation: The first-order eigenfunction gradient is a left eigenvector of the Jacobian, and the Koopman eigenvalue equals the corresponding Jacobian eigenvalue.
- Basin estimation: A candidate Lyapunov function yields an inner basin approximation through its largest closed level set within the region where it decreases along trajectories.
- Examples: The method accurately estimates basin boundaries, with the best reported approximation using a 14th-order Taylor expansion.
- Limitations: Taylor expansions may fail to capture the complete basin geometry when eigenfunctions are singular at saddle points and analytic only on part of the basin.
- Limitations: For a globally stable system whose eigenfunctions are non-analytic on X, Taylor expansion can diverge and fail to prove global stability there.
B. Bernstein polynomial-based method for the fixed point
The Bernstein method approximates continuous Koopman eigenfunctions over a region through polynomial expansions and least-squares coefficient estimation. It can handle non-analytic eigenfunctions, though it cannot reliably characterize regions containing unstable fixed points or basin boundaries.
- Method: Bernstein polynomials approximate continuous eigenfunctions on the region of attraction, including cases where analytic Taylor expansions perform poorly.
- Method: An affine variable change can place the fixed point and region of interest in [0,1]^N without changing Koopman eigenvalues.
- Method: The finite Bernstein system is overdetermined, so its coefficients are estimated by least squares using the Moore–Penrose pseudoinverse.
- Stability certification: If the system is globally stable in X, the least-squares error tends to zero as s →∞, and an accurate eigenfunction solution proves global stability in X.
- Example: For Example 3, Bernstein polynomials compute non-analytic eigenfunctions throughout X = [−2, 2] × [−2, 2], thereby proving global stability there.
- Limitations: The method provides a dichotomous outcome—global stability on all of X or not—and becomes inaccurate when X contains an unstable fixed point or basin boundary.
C. Bernstein polynomial-based method for the limit cycle
For stable limit cycles, the method expands eigenfunctions in a Fourier basis tangentially and a Bernstein basis transversely. Accurate solutions satisfying the proposition’s conditions certify global stability within the chosen annular region.
- Basis construction: Limit-cycle eigenfunctions are represented with Bernstein polynomials in the transverse coordinate and Fourier modes in the tangential coordinate.
- Coordinate system: The annular coordinates describe the cycle by θ and a transverse variable y, with dynamics expressed as θ̇ = Fθ(θ,y) and ẏ = Fy(θ,y).
- Eigenfunction constraints: For the nonzero Floquet exponent, boundary conditions impose the eigenfunction value and transverse derivative on the limit cycle.
- Numerical solution: Least-squares estimation computes the truncated Fourier–Bernstein coefficients and the resulting eigenfunction approximation.
- Stability certification: An accurate numerical solution satisfying Proposition 3 proves global stability of the limit cycle in the chosen region.
- Examples: The examples certify stability on annular regions, including [0, 2π) × for a cycle with λ = −4.
V. CONCLUSION
The paper develops an operator-theoretic framework linking Koopman-operator eigenfunctions to global stability in nonlinear systems. It establishes theoretical stability conditions, numerical procedures for stability analysis, and directions for extending the approach.
- The framework obtains global stability results for nonlinear systems, providing a global counterpart to classic local stability results.
- Necessary and sufficient stability conditions are derived from the existence of continuously differentiable Koopman eigenfunctions.
- Polynomial-basis numerical methods approximate a fixed point’s region of stability and can prove or disprove stability of a fixed point or limit cycle within a specified region.
- Because Koopman eigenfunctions encode properties such as convergence rate, the framework addresses global behavior beyond stability alone.
- Future work includes cheaper numerical methods, extensions to higher-dimensional and nonhyperbolic systems, other attractors, discrete-time maps, and input-output systems.
- Polynomial-approximation methods are typically limited to low-dimensional systems because of the curse of dimensionality.
APPENDIX
The appendix represents univariate and multivariate polynomials in Bernstein bases and provides matrix operations for differentiation, multiplication, and degree raising. It also describes geometric notation used in the computational development.
- Univariate and multivariate polynomials are represented by coefficient vectors multiplied by Bernstein-polynomial basis vectors.
- Differentiation is implemented through matrices acting on polynomial coefficient vectors, including partial derivatives in the multivariate case.
- Matrix formulations are given for polynomial multiplication and degree raising in both univariate and multivariate Bernstein representations.