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Igor MezićDepartment of Mechanical Engineering,University of California, Santa Barbara
General background on dynamical systems
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Two descriptions of dynamical systems
operatoracting on a functional space
flowacting on the state space
Trajectory-oriented approachOperator-theoretic approach
(Koopman operator)
SYSTEM ≡SYSTEM ≡
global descriptionlifting pointwise description
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A first comparison…
A trajectory… and a density
[Lasota and Mackey]
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Outline
An operator-theoretic approach to dynamical systems
Spectral properties of the Koopman operator: basic results
Interplay between spectral and geometric properties
Numerical computation
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Outline
An operator-theoretic approach to dynamical systems
Spectral properties of the Koopman operator: basic results
Interplay between spectral and geometric properties
Numerical computation
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A dynamical system is describedby two dual operators
Perron-Frobenius operator
is a density
Koopman operator
is an observable
Duality:
[Liouville 1838, Poincaré, Ulam 1960]
[Koopman 1930]
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A dynamical system is describedby two dual operators
Perron-Frobenius operator
Koopman operator
is an observable
Duality:
[Liouville 1838, Poincaré, Ulam 1960]
[Koopman 1930]
is a measure
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Finite state space
Continuous state space
We focus on the Koopman operator
is the transition matrix of the Markov chain and
Koopman operator = composition operator
semi-group of Koopman operators
Discrete-time system
Continuous-time system
infinitesimal generator
Infinite-dimensional state space
observable = vector
observable = function
observable = functional
dynamical system: PDE
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The Koopman operator is linear
There is a price to pay: the operator is infinite-dimensional
lifting
Finite-dimensional Infinite-dimensional
Dynamical system nonlinear linear
Optimization problem nonconvex convex
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Outline
An operator-theoretic approach to dynamical systems
Spectral properties of the Koopman operator: basic results
Interplay between spectral and geometric properties
Numerical computation
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Eigenvalue
For dissipative systems, we considerthe spectral properties of the Koopman operator
Koopman eigenfunction
eigenfunctions of the Perron-Frobenius operator are Dirac functions
consider Koopman eigenfunctions
Conservative systems
is unitary
Dissipative systems (there is an attractor) or
is not unitary
: analytic functions
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Basic properties of the Koopman eigenfunctions
The constant observable is a Koopman eigenfunction
Multiplication of eigenfunctions
Power of an eigenfunction
Example:
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We can consider the spectral expansion of the Koopman operator
Koopmanmode eigenfunction
: projection of on
Regular part (related to the continuous spectrum)
Example:
if is analytic Taylor expansion
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We can consider the spectral expansion of the Koopman operator
Koopmanmode eigenfunction Regular part (related to the
continuous spectrum)
Example: linear system
Koopman eigenfunctions
Koopman eigenvalues
Consider the identity observable
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Application of the Koopman mode analysis:building energy efficiency
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Outline
An operator-theoretic approach to dynamical systems
Spectral properties of the Koopman operator: basic results
Interplay between spectral and geometric properties
Numerical computation
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Constant eigenfunctions define invariant sets (ergodic partition)
is always an eigenfunction
Two invariant sets and
If is unique the system is ergodic (every state can be reached)
Level sets of are co-dimensional 1 invariant sets (trajectories in 2D)
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Eigenfunctions with purely imaginary eigenvaluesdefine periodic partitions
The level sets of with define a periodic partition of period
Limit cycle: the level sets of the Koopman eigenfunction are the isochrons
[Mezic and Banaszuk 2004, Mezic 2005]
[Mauroy and Mezic, 2012]
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Dissipative systems: the spectrum of the Koopmanoperator can be decomposed in two parts
asymptotic convergence to (Floquet/Lyapunov exponents)
Spectrum of restricted to Spectrum of the Koopman operatoracting on
ergodic motion on(frequencies)
Assume that the system admits an attractor
related to isochrons related to isostables
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The spectrum of the Koopman operatoris related to known notions of stability
: space of analytic functions in a neighborhood of
: eigenvalues of the Jacobian matrix at
is a fixed point
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Action-angle coordinates and global linearization
Rigorous extension of Hartman-Grobman theorem [Lan and Mezic, 2012]
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The spectrum of the Koopman operatoris related to known notions of stability
: space of analytic functions in a neighborhood of
: Floquet exponents of
is a limit cycle
: frequency of
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The spectrum of the Koopman operatoris related to known notions of stability
: space of analytic functions in a neighborhood of
: generalized Floquet exponents of
is a quasiperiodic torus
: basic frequencies of
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The spectrum of the Koopman operatoris related to known notions of stability
: space of analytic functions in a neighborhood of
: Lyapunov exponents of
is a strange attractor
continuousspectrum
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Outline
An operator-theoretic approach to dynamical systems
Spectral properties of the Koopman operator: basic results
Interplay between spectral and geometric properties
Numerical computation
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There exist several methods for computing the Koopman modes and eigenfunctions
We have data: (Extended) Dynamic Mode Decomposition
We know the trajectories: Fourier and Laplace averages
[Mezic 2012, Mohr and Mezic 2014]
We know the vector field: expansion on a finite (polynomial) basis
[Mauroy and Mezic 2013]
[Schmid 2010, Rowley et al. 2009, Williams et al. 2015 ]
related to the problem of moments
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The Koopman eigenfunctions can be computedwith averages along the trajectories
is the projection of on if it is nonzero
The Fourier average
The generalized Laplace average
is a projection of on if it is nonzero
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The Koopman eigenfunctions can be computedwith averages along the trajectories
is the projection of on if it is nonzero
The Fourier average
The generalized Laplace average
is a projection of on if it is nonzero
and if
For with : choose with , i.e.
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The Koopman operator theoretic frameworkis conducive to data analysis
Koopman modes
methods based on SVD
eigenvalues of
right eigenvectors of
left eigenvectorss of
Koopman eigenvalues
Koopman eigenfunctions(linear approximation)
[Schmid 2010, Tu et al. 20014]
[Williams et al. 2015 ] Recent extension(s)
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Applications of dynamic mode decomposition
Proctor and Eckhoff
Rowley et al., 2009
Fluid dynamics
Epidemics
Susuki and Mezic, 2013
Power grid
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Applications of Koopman analysis (to name a few):
• fluid dynamics (Rowley, Mezic, Henningson)
• power grid (Susuki and Mezic)
• energy efficiency (Georgescu and Mezic)
• system estimation (Surana and Banaszuk)
• image processing (Kutz)
• epidemics (Proctor and Eckhoff)
• neural dynamics, cardiac defibrillation (Wilson and Moehlis)
• high dimensional systems (Williams, Kevrekidis, Rowley)
• nonlinear vibration (Cirillo et al.)
• …
And many other applications…
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In summary
The Koopman operator is linear
The spectral properties of the Koopman operator are relatedto (geometric) properties of the dynamical system
The framework yields systematic spectral methods (many applications)
Recent theory in the case of dissipative systems
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Perspectives
Development of efficient numerical methodsto compute the spectral properties of the Koopman operator
relationship to polynomial optimization (and SOS methods)?
Vice-versa: some (new) insight into global optimization?
An invitation to the community:
application of the Koopman framework to control theory
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Bibliography
Koopman operator / Perron-Frobenius operator
• Lasota and Mackey, Chaos, Fractals, and Noise: stochastic aspects of dynamics, Springer-Verlag, 1994
Review Koopman operator
• Mezic, Analysis of Fluid Flows via Spectral Properties of Koopman Operator, Annual Review of Fluid Mechanics, 2013, 45
• Budisic, Mohr, and Mezic, Applied Koopmanism, Chaos, 2012, 22, 047510-047510
Spectral properties (ergodic case)
• Mezic, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynamics, Springer, 2005, 41, 309-325
Dynamic mode decomposition and Koopman operator
• Rowley et al., Spectral analysis of nonlinear flows, Journal of Fluid Mechanics, 2009, 641, 115-127
• Tu et al., On dynamic mode decomposition: Theory and applications, Journal of Computational Dynamics, 2014, 1, 391 - 421