M. Budisic: Analysis of Dynamical Systems Using the Koopman Operator Formalism

16th USNC of TAM, July 2, 2010State College, PA, USA

Marko Budišić (presenter)Igor Mezić

ANALYSIS OF DYNAMICAL SYSTEMS

USING THE KOOPMAN OPERATOR

FORMALISM

Igor had unexpected obligations that he had to give priority to.

First of all, I would like to thank AFOSR for their support to the research presented in this talk.

I will present an overview of two different methods for analyzing the behavior of dynamical systems, which have theoretical grounding in operator theory of dynamical systems, and, in particular, the Koopman operator.

Research was done by our group, in collaboration with Clarence Rowley, Andrzej Banaszuk, Yoshihiko Susuki, Umesh Vaidya, and others - full list of references at the end.

Friday, July 2, 2010 Marko Budišić: Analysis of Dynamical Systems Using the Koopman Operator Formalism 2

Operator Theory and Dynamical Systems

Operator-theoreticApproach

Geometric Approach Change of objects in state space: trajectories, invariant manifolds, etc.

Change of objects on state space: observables, measures

Focus: Spectral Analysis of linear, infinite-dimensional operators,

induced by dynamical systems.

Perron-Frobenius Op.

• Evolution of measures • “Eulerian” picture

Koopman Op.

• Evolution of observables • “Lagrangian” picture

The more common, geometric, analysis of dynamical systems focuses on change of objects IN the state space. The operator-theoretic approach focuses on change of objects ON the state space.

We use spectral features of operators to: A) Make conclusions about objects in state space, B) analyze evolution of observables.

[Uf ](x) = (f ! T )(x)

U =n!

k=1

ei2!"kP"kT + Uc

T : M !M

f : M ! C

Friday, July 2, 2010 Marko Budišić: Analysis of Dynamical Systems Using the Koopman Operator Formalism

The Koopman Operator

3

• infinite-dimensional• linear (but NOT a linearization)• bounded • unitary on the attractor

1

i

[Mezić, Banaszuk, 2004]

Two types of analyses:• Eigenfunction analysis• Expansions of observables into

eigenfunctions

Dyn. system:

Observable:

Koopman operator tells us what happens to an observable one step in future – composition of observable with the action of the dynamical system.Unitarity on attractor enables the study of the operator through spectral decomposition.The methods are based on: 1) Analysis of eigenfunctions 2) Analysis of expansions of observables into eigenfunctions

f(x) := limN!"

1N

N#1!

n=0

f(Tn(x))

Uf = fP 0T f = f

(x, y) ! T2, ! ! [0, 1)

x+ = x + ! sin 2"y

y+ = x + y + ! sin 2"y


Time-averaging and invariant sets

4

Level sets of a TAF form a stationary partition of the state space.

Time-averaged Functions (TAF)

[Mezić, Wiggins, 1999]

To analyze eigenfunctions we will first have to construct them.Eigenfunctions at 1 can be constructed by TIME AVERAGING – projection of an observable to 1-eigenspace.Level sets of eigenfunctions at 1 are invariant sets for dynamics.Plotting of level sets in different colors reveals an invariant partition.Different choices of observables result in different invariant partitions revealed.

f(x) := limN!"

1N

N#1!

n=0

f(Tn(x))

Uf = fP 0T f = f

(x, y) ! T2, ! ! [0, 1)

x+ = x + ! sin 2"y

y+ = x + y + ! sin 2"y


Time-averaging and invariant sets

4

Level sets of a TAF form a stationary partition of the state space.

Averaging different observables reveals different invariant sets in the state space.

Time-averaged Functions (TAF)

[Mezić, Wiggins, 1999]

To analyze eigenfunctions we will first have to construct them.Eigenfunctions at 1 can be constructed by TIME AVERAGING – projection of an observable to 1-eigenspace.Level sets of eigenfunctions at 1 are invariant sets for dynamics.Plotting of level sets in different colors reveals an invariant partition.Different choices of observables result in different invariant partitions revealed.

fLevel sets of

! = "

" = pm !b

N

N!

i=1

sin

"

#!

j!Jeijcj cos !jt + !

$

%


Application: New England Power Grid

5

• A modal perturbation could trigger Coherent Swing Instability.• Analysis of spatially-averaged dynamics through a single-observable.

f(!, ") = sin 2!Observable:

[Susuki, Mezić, 2009]

Reduced order (mean angle) model:

Perturbation:

Dynamical system model of a ring of power generators and their deviation from the rest of the power grid (the infinite bus).Delta is the average deviation of the rotor position from the infinite bus. Blue dot is the normal operating condition, the system is hit by a perturbation, which might induce the instability in the system.Size of invariant set denotes the size of deviation resulting from perturbation.This analysis could be performed on a full scale model (or experimental data), just using the particular observable.Segue: In this case, a single observable sufficed for the analysis. What happens when we have multiple observables?

f!

1

f! 2

-0.5 0 0.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Mesochronic Scatter Plots (MSP)

6

f1(x, y) = cos(2!x + 7!y)

f2(x, y) = cos(9!x + !y) [Levnajić, Mezić, 2010]f!1

f!2

Explain how the MSP is constructed.Coloring based on a single observable corresponds to chosing color based on vertical or horizontal strips in MSP. Product of invariant partitions: divide MSP into boxes and color each box differently.Takeaway: 1.trajectories that stay close together, end up close together in MSP2.if trajectories are so close that they are in the same inv. set. for both observables, they get mapped to the same point3.dimension of MSP does not depend on the dimension of the state spaceSegue: We could continue adding variables, and you can imagine this process of refinement: a) getting unwieldy to visualize, b) limiting to something, so we turn to theory.

f!

1

f! 2

-0.5 0 0.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6



6

f1(x, y) = cos(2!x + 7!y)


f!2


f!

1

f! 2

-0.5 0 0.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6



6

f1(x, y) = cos(2!x + 7!y)


f!2


f!

1

f! 2

-0.5 0 0.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6



6

f1(x, y) = cos(2!x + 7!y)


f!2


f!

1

f! 2

-0.5 0 0.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6



6

f1(x, y) = cos(2!x + 7!y)


f!2


! : M ! hs(Zd, "."s)

!!(x)!2s =

!

k!Zd

|!k(x)|2"1 + (2"!k!2)

2#s ,

x !"!. . . !k(x) . . .

"T

!k(x) =1

(2")d/2e2!i k·x, k ! Zd


Ergodic Partition and Quotient Space

7

Ergodic Quotient Space: Limiting structure of the MSP.Distinct ergodic sets mapped to distinct points in EQS.

Observables span the space of functions.

Initial conditions are mapped to sequences of time averages.

Negative-Sobolev norm (low-pass filter!)

[Budišić, Mezić, 2009]

Refining partitions by averaging more observables limits to the Ergodic Partition.

Properties of Ergodic Partition:•Dynamics is ergodic on elements.•The finest invariant partition.

Metrization of EQS:

If we would construct the MSP for a countably dense set of observables, we would obtain the Ergodic Quotient Space.A point in Ergodic Quotient Space corresponds to an ergodic set in the state space.Collection of ergodic sets is the Ergodic Partition - the finest invariant partition, i.e., every other invariant partition can be constructed by unions of ergodic sets.If we start with a Fourier basis of observables, the EQS is isomorphic to a multisequence space. We can endow this space with a weighted L2 norm, which corresponds to a negative Sobolev space norm - it has a dynamical meaning.NSN metric is equivalent to Empirical Distance that measures how long two trajectories stay close, on average.Segue: This is an infinite-dim space, however, as we have seen, structures might be remarkably low-dimensional.

! : M ! hs(Zd, "."s)

!!(x)!2s =

!

k!Zd

|!k(x)|2"1 + (2"!k!2)

2#s ,

x !"!. . . !k(x) . . .

"T

DS(x, y) ! "!(x)# !(y)"s

!k(x) =1

(2")d/2e2!i k·x, k ! Zd


Ergodic Partition and Quotient Space

7

x1(t)x2(t)

Ergodic Quotient Space: Limiting structure of the MSP.Distinct ergodic sets mapped to distinct points in EQS.

Empirical Distancecompares average residence times of trajectories in spherical sets.

[Mathew, Mezić, 2010][Budišić, Mezić, 2009]

DS

Refining partitions by averaging more observables limits to the Ergodic Partition.

Properties of Ergodic Partition:•Dynamics is ergodic on elements.•The finest invariant partition.

Metrization of EQS:

If we would construct the MSP for a countably dense set of observables, we would obtain the Ergodic Quotient Space.A point in Ergodic Quotient Space corresponds to an ergodic set in the state space.Collection of ergodic sets is the Ergodic Partition - the finest invariant partition, i.e., every other invariant partition can be constructed by unions of ergodic sets.If we start with a Fourier basis of observables, the EQS is isomorphic to a multisequence space. We can endow this space with a weighted L2 norm, which corresponds to a negative Sobolev space norm - it has a dynamical meaning.NSN metric is equivalent to Empirical Distance that measures how long two trajectories stay close, on average.Segue: This is an infinite-dim space, however, as we have seen, structures might be remarkably low-dimensional.

Friday, July 2, 2010 Marko Budišić: Analysis of Dynamical Systems Using the Koopman Operator Formalism 8

Points on EQS as samples on a (lo-dim) structure lying in hi-dim space.

[Coifman, Lafon, 2006]

Ergodic Coordinates:Eigenfunctions of the heat equation on the structure.L2 distance is the Empirical Distance between ergodic sets.

Approximate EQS by a neighborhood graph, and compute eigenvectors of a discrete Laplacian on it.

Diffusion Maps: Multiscale Analysis of the EQS, from pairwise distances between points.

Heat Diffusion as Surveying on EQS

Ergodic Coordinates unravel the Ergodic Quotient Space.Each ergodic set is assigned coordinates based on heat flow directions along EQS.Heat flow directions are computed using Diffusion Maps algorithm - a spectral computation.The L2 distance in Ergodic Coordinates is the diffusion distance.

!(1)

!(2

)

-0.03 -0.02 -0.01 0 0.01 0.02

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


Ergodic Coordinate Embedding

9

First ergodic coordinate

Second ergodic coordinateErgodic Coordinate Embedding

Cf. MSP:

Goal for visualization of invariant partition is to achieve large variation of color in nearby ergodic sets.Ergodic coordinate show multiscale property - first coordinates vary across large-scale features, while later coordinates vary across smaller scale features. The more similar two coordinates are on larger scales, the more similar first N coordinates they will have.They have an effect of “unraveling” and straightening the EQS/MSP.Potential application:a) analysis of hi-dim systems with lo-dim ergodic partitionb) control design?c) observable design?Segue: Ergodic Coordinates have a particularly strong significance in “white box” cases -- we have access to (a lot) of states, but don’t know what observables are relevant.

!(1)

!(2

)

-0.03 -0.02 -0.01 0 0.01 0.02

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


Ergodic Coordinate Embedding

9

First ergodic coordinate

Second ergodic coordinateErgodic Coordinate Embedding

Cf. MSP:

Goal for visualization of invariant partition is to achieve large variation of color in nearby ergodic sets.Ergodic coordinate show multiscale property - first coordinates vary across large-scale features, while later coordinates vary across smaller scale features. The more similar two coordinates are on larger scales, the more similar first N coordinates they will have.They have an effect of “unraveling” and straightening the EQS/MSP.Potential application:a) analysis of hi-dim systems with lo-dim ergodic partitionb) control design?c) observable design?Segue: Ergodic Coordinates have a particularly strong significance in “white box” cases -- we have access to (a lot) of states, but don’t know what observables are relevant.

R = 2Rz + ! log R sin " sin 2#t

z = 1! 4R! z2 + !z

Rsin " sin 2#t

" =!

R+

2!

Rcos " sin 2#t

! = 0.01 ! = 0.35


Application: Perturbed Spherical Hill Vortex

10

• Hamiltonian perturbation of an action-action-angle map (KAM type system).

• Ergodic Coordinates reveal invariant tori that exist at high perturbation values.

[Vaidya, Mezić, preprint 2010]

State space contains a positively-invariant torus. Images show Theta=0 section through that torus.System was studied as an example of a KAM-type system, but at higher perturbations new invariant structures were uncovered, without particular design of observable.Segue: For the final part of this talk, we will shift our attention from analysis of eigenfunctions to analysis of expansions INTO eigenfunctions.

g : M ! CJ

g(xn) = g(Tnx) = Ung(x)

U!k = "k!k

g(xn) =

!

""#

...gj(xn)

...

$

%%& =!'

k=1

!

""#

...vj...

$

%%&

k

!nk"k(x)

g(x) =

!

""#

...gj(x)

...

$

%%& =!'

k=1

!

""#

...vj...

$

%%&

k

!k(x)


Koopman Modes

11

Evolution of a v.v. observable

Efunction expansion

Koopman modespreserved during the time evolution

Computation ofKoopman modes

•Record observable as system evolves.•Use observables’ snapshots as a basis for the Krylov subspace in Arnoldi algorithm.

[Rowley, Mezić, 2009]

Koopman Modes are a part of analysis of the observables. Understand the evolution of the observable through coefficients of expansion in eigenfunctions of the Koopman.Each KM corresponds to one eigenfunction of the Koopman operator.For a single observable, KM is just the coefficient of expansion.For a vector-valued observable, KM is a vector of such coefficients.KM can be computed using Arnoldi algorithm.


Application: Jet in the Crossflow

12

Spectral analysis of nonlinear flows 9

(a) (b)

Figure 3. Positive (red) and negative (blue) contour levels of the streamwise velocity compo-nents of two Koopman modes. The wall is shown in gray. (a) Mode 2, with !v2! = 400 andSt2 = 0.141. (b) Mode 6, with !v6! = 218 and St6 = 0.0175.

are compared to the frequencies obtained directly from the Ritz eigenvalues (red verticallines). The shedding frequencies and a number of higher harmonics are in very goodagreement with the frequencies of the Koopman modes. In particular, the dominantKoopman eigenvalues match the frequencies for the wall mode (St = 0.017) and theshear-layer mode (St = 0.14). Note that the probe signals are local measures of thefrequencies at one spatial point, whereas the Koopman eigenvalues correspond to globalmodes in the flow with time-periodic motion.

The streamwise velocity component u of Koopman modes 2 and 6 are shown in Fig-ure 3. Each mode represents a flow structure that beats with one single frequency, andthe superposition of several of these modes results in the quasiperiodic global system.The high-frequency mode 2 (Figure 3(a)) can be associated with the shear layer vortices;along the jet trajectory there is first a formation of ring-like vortices that eventuallydissolve into smaller scales due to viscous dissipation. Also visible are upright vortices:on the leeward side of the jet, there is a significant structure extending towards the wall.This indicates that the shear-layer vortices and the upright vortices are coupled and os-cillate with the same frequency. The spatial structures of modes 4 and 8 are very similarto those of mode 2, as one expects, since the frequencies are very close.

On the other hand, the low-frequency mode 6 shown in figure 3(b) features large-scale positive and negative streamwise velocity near the wall, which can be associatedwith shedding of the wall vortices. However, this mode also has structures along the jettrajectory further away from the wall. This indicates that the shedding of wall vorticesis coupled to the jet body, i.e. the low frequency can be detected nearly anywhere in thevicinity of the jet since the whole jet is oscillating with that frequency.

4.2. Comparison with linear global modes and POD modesThe linear global eigenmodes of the Navier-Stokes equations linearized about an unstablesteady state solution were computed by Bagheri et al. (2009) for the same flow parametersas the current study. They computed 22 complex conjugate unstable modes using theArnoldi method combined with a time-stepper approach. The frequency of the mostunstable (anti-symmetric) mode associated with the shear-layer instability was St =0.169, not far from the value St = 0.14 observed for the DNS. However, the mode withthe lowest frequency associated with the wall vortices was St = 0.043, quite far fromthe observed frequency of St = 0.017. These linear frequencies can capture the dynamicsonly in an neighborhood of the unstable fixed point, while the Koopman modes correctlycapture the behavior on the attractor.

We also compared the Koopman modes with modes determined by Proper OrthogonalDecomposition (POD) of the same dataset, and although we do not show the resultsexplicitly, we comment on the main similarities and di!erences. The POD modes capturesimilar spatial structures, but the most striking distinction is in the time coe"cients:


(a) (b)







8 C.W. Rowley, I. Mezic, S. Bagheri, P. Schlatter, and D.S. Henningson

!1 !0.5 0 0.5 1!1

!0.5

0

0.5

1

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.240

100

200

300

400

(a) (b)

St

!vj!

Re{!j}

Im{!j}

Figure 2. (a) The empirical Ritz values !j . The value corresponding to the first Koopman modeis shown with the blue symbol. (b) The magnitudes of the Koopman modes (expect the firstone) at each frequency. In both figures, the colors vary smoothly from red to white, dependingon the magnitude of the corresponding mode.

spectrum shows the frequency content u1(!) of u1(t). The peak frequency correspondsto a vortex shedding of wake vortices with the Strouhal number St ! fD/Vjet = 0.0174.

In figure 1(d,f), a second probe located a few jet diameters along the jet trajectoryx2

P = (12, 6, 2), shows a second oscillation that can be identified with the shedding ofthe shear-layer vortices. The peak frequency beats with St = 0.141 which is nearly oneorder of magniture larger than the low-frequency mode. Note that the peak frequenciesof the power spectra vary slightly depending on the location of the probe.

4.1. Koopman modes and frequenciesIn this section we compute the Koopman modes and show that they directly allow anidentification of the various shedding frequencies. The empirical Ritz values "j and theempirical vectors vj of a sequence of flow-fields {u0,u1, . . . ,um!1} = {u(t = 200),u(t =202), . . . ,u(t = 700)} with m = 251 are computed using the algorithm described earlier.Thus, the transient time (t < 200) is not sampled and only the asymptotic motion inphase space is considered.

Figure 2(a) shows that nearly all the Ritz values are on the unit circle |"j | = 1indicating that the sample points ui lie on or near an attracting set. The Koopmaneigenvalue corresponding to the first Koopman mode is the time-averaged flow and isdepicted with blue symbol in figure 2(a). This mode, shown in figure 1(b), captures thesteady flow structures as discussed previously. In figure 2(a), the other (unsteady) Ritzvalues vary smoothly in color from red to white, depending on the magnitude of thecorresponding Koopman mode. The magnitudes defined by the global energy norm "vj",and are shown in figure 2(b) with the same coloring as the spectrum. In figure 2(b) eachmode is displayed with a vertical line scaled with its magnitude at its correspondingfrequency !j = Im{log("j)}/!t (with !t = 2 in our case). Only the !j # 0 are shown,since the eigenvalues come in complex conjugate pairs. Ordering the modes with respectto their magnitude, the first (2-3) and second (4-5) pair of modes oscillate with St2 =0.141 and St4 = 0.136 respectively, whereas the third pair of modes (6-7) oscillate withSt6 = 0.017. All linear combinations of the frequencies excite higher modes, for instance,the nonlinear interaction of the first and third pair results in the fourth pair, i.e. St8 =0.157 and so on.

In figures 1(e) and (f) the power spectra of the two DNS time signals (black lines)

Mixing by injection of fluid through a hole into a steady cross-flow.

Ritz values: approx. Koopman spectrum8 C.W. Rowley, I. Mezic, S. Bagheri, P. Schlatter, and D.S. Henningson

!1 !0.5 0 0.5 1!1

!0.5

0

0.5

1

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.240

100

200

300

400

(a) (b)

St

!vj!

Re{!j}

Im{!j}








•DNS•Flow vel. field is the observable•256 X 201 X 144 gridpoints•250 time snapshots

Strouhal No. ～ Mode freq.

Koopman Modemagnitudes

Extraction of modes for a DNS fluid model in steady state.Level sets of two highly energetic Koopman modes are shown.Ritz values approximate the spectrum - indication that system was close to attractor.Upper mode: coupling of the shear layer ring-vortices and upright vorticesLower mode: shedding of the wall vortices and coupling to the jet body


Application: Jet in the Crossflow

12


(a) (b)








(a) (b)







8 C.W. Rowley, I. Mezic, S. Bagheri, P. Schlatter, and D.S. Henningson

!1 !0.5 0 0.5 1!1

!0.5

0

0.5

1

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.240

100

200

300

400

(a) (b)

St

!vj!

Re{!j}

Im{!j}








Mixing by injection of fluid through a hole into a steady cross-flow.

Ritz values: approx. Koopman spectrum8 C.W. Rowley, I. Mezic, S. Bagheri, P. Schlatter, and D.S. Henningson

!1 !0.5 0 0.5 1!1

!0.5

0

0.5

1

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.240

100

200

300

400

(a) (b)

St

!vj!

Re{!j}

Im{!j}








•DNS•Flow vel. field is the observable•256 X 201 X 144 gridpoints•250 time snapshots

Strouhal No. ～ Mode freq.

Koopman Modemagnitudes

Extraction of modes for a DNS fluid model in steady state.Level sets of two highly energetic Koopman modes are shown.Ritz values approximate the spectrum - indication that system was close to attractor.Upper mode: coupling of the shear layer ring-vortices and upright vorticesLower mode: shedding of the wall vortices and coupling to the jet body


Summary

13

Methods

Applications

•identifying invariant sets by averaging of observables•Ergodic Quotient Space and ergodic coordinates•Koopman mode analysis

•swing instability of New England power grid•jet in a crossflow•perturbation of Hill spherical vortex•...?

Koopman AnalysisFamiliar linear algorithms can be harnessed

to uncover features in full nonlinear dynamics.


References

14

Mezić and Banaszuk. Comparison of systems with complex behavior. Physica D (2004)

Levnajić and Mezić. Ergodic Theory and Visualization I: Visualization of Ergodic Partition and Invariant Sets. Chaos (to be published in 2010)

Susuki and Mezić. Ergodic Partition of Phase Space in Continuous Dynamical Systems. Proc. 48th IEEE CDC, Shanghai, China (2009)

Budišić and Mezić. An Approximate Parametrization of the Ergodic Partition using Time Averaged Observables. Proc. 48th IEEE CDC, Shanghai, China (2009)

Vaidya and Mezić. Existence of invariant tori in action-angle-angle maps with degeneracy. (to be published)

Rowley, Mezić et al. Spectral analysis of nonlinear flows. Journal of Fluid Mechanics (2009)

M. Budisic: Analysis of Dynamical Systems Using the Koopman Operator Formalism

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