ARL-TR-7959 MAR 2017

US Army Research Laboratory

Global Stability and Dynamics of Strongly 

Nonlinear Systems Using Koopman Operator 

Theory

by Bryan Glaz and Adam Svenkeson

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ARL-TR-7959 ● MAR 2017

US Army Research Laboratory

Global Stability and Dynamics of Strongly 

Nonlinear Systems Using Koopman Operator 

Theory

by Bryan Glaz and Adam Svenkeson

Vehicle Technology Directorate, ARL

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1. REPORT DATE (DD-MM-YYYY)

March 2017

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DRI Report

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October 2014–September 2016

4. TITLE AND SUBTITLE

Global Stability and Dynamics of Strongly Nonlinear Systems Using Koopman 

Operator Theory

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6. AUTHOR(S)

Bryan Glaz and Adam Svenkeson

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DRI-14-VT-004

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US Army Research Laboratory

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Aberdeen Proving Ground, MD 21005-5066

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ARL-TR-7959

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14. ABSTRACT

A spectral operator theoretic perspective was taken for modeling nonlinear dynamics subjected to external inputs, due to 

prescribed forcing or an environment. Under this Director’s Research Initiative (DRI), we focused on spectral properties 

(eigenfunctions, eigenvalues) of the Koopman operator (i.e., an infinite dimensional linear operator that can describe the full 

dynamics of a nonlinear system). We showed how external forcing could be treated in a Koopman decomposition-based 

framework. Furthermore, during the course of our study using Koopman decompositions of periodically excited Hopf 

bifurcation systems, we characterized a new attractor that had never been described in the literature or texts. We called this 

attractor the "quasi-periodic intermittency" attractor and showed that it may be a fundamental building block of a variety of 

multi-scale dynamics systems. Perhaps the most far reaching impact of this DRI will be a contribution that was not planned in 

the original proposal. This contribution has to do with the generalization of Koopman decompositions using a fractional 

calculus perspective on complexity. By using a combination of Koopman operator theory and fractional calculus, we showed 

that our generalized spectral decompositions are better suited for complex systems influenced by an external environment in 

which long-term memory is introduced. 

15. SUBJECT TERMS

nonlinear dynamics, spectral decompositions, fractional calculus, applied mathematics, Director’s Research Initiative

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Bryan Glaz

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(410) 278-2421

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Contents

List of Figures iv

1. Introduction 1

2. Accomplishments 1

2.1 Prescribed External Forcing 1

2.2 Spectral Decompositions of Nonlinear Systems with Memory 4

3. Conclusion 5

4. References 7

Distribution List 8
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List of Figures

Fig. 1 Spectrum of nonlinear observable; computational fluid dynamics 

simulation (blue) are for the full-order computational fluid

dynamics simulations of the forced system, while normal form (red) 

corresponds to Eq. 4.............................................................................. 3

Fig. 2 The quasi-periodic intermittency attractor (yellow) plotted on the 

torus. The toroidal grid (grey) is shown for reference. Rotation around 

the large diameter of the torus corresponds to the radial growth/decay 

that occurs over the slow forcing frequency, while rotation around the 

smaller diameter (i.e., phase velocity as the attractor spins around 

cross sections of the torus) corresponds to the faster natural frequency 

of the system. ........................................................................................ 4
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1. Introduction

Complex, nonlinear dynamical systems are pervasive across many Army relevant 

scientific disciplines. Typically, our engineering objectives are to control the 

dynamics (e.g., fluids, polymers), understand underlying phenomena (e.g.,

neuroscience models, turbulence, networks), and/or synthetically mimic natural 

phenomena (e.g., swarm dynamics, self-assemblies, active matter/fluids). 

However, our inability to achieve these objectives for a variety of high-dimensional 

dynamical systems is due to a lack of mathematical tools to describe lowdimensional (simple) features that may underpin such complex systems.

In this 3-year effort, we focused on one perspective for extracting simplicity in the 

form of low-dimensional features that otherwise would be hidden by complexity 

(high dimensionality and multi-scale dynamics). This perspective is based on the 

theoretical constructs developed in the 1930s by mathematician Bernard 

Koopman,

1 who showed that finite dimensional nonlinear dynamics can be fully 

described by an infinite dimensional linear operator. This operator is referred to as 

the Koopman operator and the resulting projections as Koopman decompositions.

2

Furthermore, as described in Budisic and Mezic’s research,

2 the Koopman operator 

approach entails tracking dynamics of observables (i.e., measureable quantities), 

rather than dynamics of states. This perspective makes Koopman operator 

approaches ideal for dealing with ill-described systems in only measurement data 

available rather than access to model equations.

The contributions of this effort focused on the broad area of dynamical systems 

subjected to external forcing. We considered 2 scenarios: the first in which an 

external forcing is applied to a nonlinear Hopf bifurcation, and the second in which 

the external environment induces long-time memory in the nonlinear system. 

Details of our work in the first scenario are described by Glaz et al.,

3 while details 

of the second are provided in the research of Svenkeson et al.

4 Section 2 is 

Accomplishments and Section 3 is the Conclusion.

2. Accomplishments

2.1 Prescribed External Forcing

To study the impact of external forcing, we studied a canonical Hopf bifurcation 

system with external forcing. Fluid dynamics induced by periodically forced flow 

around a cylinder was analyzed computationally for the case when the forcing 
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frequency is much lower than the von Karman vortex shedding frequency (i.e.,

Hopf bifurcation frequency) corresponding to the constant flow velocity condition. 

Beginning from the generic dynamical system,

zż = FF(zz), (1)

we introduced a prescribed forcing U(t) by oscillating the cylinder in the 

streamwise direction.

3 This created a simple canonical situation for studying the 

forced bifurcation system under potentially high-dimensional conditions. The 

forced system is then

zż = FF(zz) + UU(tt). (2)

However, in the dynamics of observables perspective we are interested in tracking 

a measurable quantity g that may be a nonlinear function of state z (i.e., g(z)). In 

operator form, we showed that the forced system in observable space is described 

by a bi-linear equation,

gġ = LLgg + (UU ∙ ∇∇)gg, (3)

where LL is a linear operator that describes the dynamics due to the unforced system, 

Eq. 1, while the forcing appears as a bi-linear term involving UU and g.

By using the Koopman mode decomposition approach, we found a new normal 

form equation that extends the classical Hopf bifurcation normal form by a timedependent term for Reynolds numbers close to the Hopf bifurcation value. This was 

done by projecting onto the first Koopman mode associated with the Hopf 

bifurcation and obtaining a reduced order set of equations.

3 The resulting 2-

dimensional (2-D) normal form equation was

ηη = λλ + ββηη2ηη̅ ̇ − iiii(ζζ − ζζ)̅ηη, (4)

where ηη is a complex valued variable representing the observable, ζζ is a complex 

valued variable representing the simple harmonic prescribed forcing term, and the 

remaining terms parameterize the dynamical system and the forcing amplitude. As 

was shown by Glaz et al.3 using normal form theory, Eq. 4 is only necessary when

the forcing frequency is much less than the Hopf bifurcation frequency. Otherwise, 

nonlinear interactions between the observable and forcing do not result in firstorder effects. The validity of the simple 2-D model in Eq. 4 is shown in Fig. 1. In 

Fig. 1, there is rich spectral content even though there are only 2 underlying 

frequencies, the Hopf bifurcation frequency (26 Hz) and the prescribed forcing 

frequency (0.5 Hz). However, the muli-scale nonlinear interaction between the 

2 scales is captured by the Koopman operator-based approach. Furthermore, the 
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Koopman operator approach correctly reproduces the underlying simplicity of the 

problem in that only one mode is relevant. However, from simply looking at the 

spectrum of the forced system, one may incorrectly conclude that multiple modes 

have been introduced by forcing this nonlinear system.

Fig. 1 Spectrum of nonlinear observable; computational fluid dynamics simulation (blue) 

are for the full-order computational fluid dynamics simulations of the forced system, while 

normal form (red) corresponds to Eq. 4

Furthermore, we found that the dynamics of the flow in this regime are 

characterized by alternating instances of quiescent and strong oscillatory behavior, 

and that this pattern persists indefinitely. We establish the theoretical underpinnings 

of this phenomenon, that we name quasi-periodic intermittency, using the new 

normal form model in Eq. 4 and showed that the dynamics are caused by the 

tendency of the flow to oscillate between the unstable fixed point and the stable 

limit cycle of the unforced flow. The quasi-periodic intermittency phenomena is

also characterized by positive Finite-Time Lyapunov Exponents that, over a long 

period of time, asymptotically approach zero.

3 The quasi-periodic intermittency 

attractor is shown in Fig. 2 for the same forced Hopf bifurcation case associated 

with Fig. 1. Because of the pervasiveness of the underlying ingredients of quasiperiodic intermittency, the attractor is a potentially fundamental building block that 

may underpin a variety of multi-scale dynamical systems.
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Fig. 2 The quasi-periodic intermittency attractor (yellow) plotted on the torus. The

toroidal grid (grey) is shown for reference. Rotation around the large diameter of the torus

corresponds to the radial growth/decay that occurs over the slow forcing frequency, while

rotation around the smaller diameter (i.e., phase velocity as the attractor spins around cross

sections of the torus) corresponds to the faster natural frequency of the system.

2.2 Spectral Decompositions of Nonlinear Systems with 

Memory

Following the extension of Koopman operator theory to systems with memory in 

time made during Year 1, during Year 2 we worked toward developing a test 

capable of determining whether an ill-described (black-box) system is better 

represented by ordinary differential equations or fractional differential equations. 

The physical interpretation of the fractional order system is that of a surrounding 

environment acting on a system.

5 However, the theory in Stanislavsky5 was limited 

to that of a linear fractional order oscillator. An important contribution of our work 

in4 is that the analogy of a fractional order oscillator to a physical system 

surrounded by an environment can be extended to nonlinear systems by utilizing 

the Koopman operator perspective as a starting point. 

Traditional modeling approaches assume the memoryless system

dd

dddd

xx = FF(xx). (5)

However, in some instances (e.g., when the system of interest is coupled to a 

complex environment) it may be advantageous to assume a system with long-term 

memory in time, where memory effects are represented by fractional order 

differential operators,

ddαα

dd αα xx = FF(xx). (6)
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The Koopman spectral decomposition of the memoryless system has the form

gg(xx(tt)) = ∑ VVkkee ∞ λλkk kk=1 (7)

with exponential expansion functions. The spectral decomposition of a system with 

memory has the form 

gg(xx(tt)) = ∑ VVkkEEαα(λλkkttαα) ∞

kk=1 (8)

with Mittag-Leffler expansion functions; we previously derived Eq. 8 through an 

extension of the Koopman operator theory. Given a data set originating from 

observations of a black-box system, we now have 2 tools in Eqs. 7 and 8 that can 

potentially be used to construct 2 different models of the system. Methods exist for 

approximating the eigenvalues λλkk and modes VVkk of the memoryless spectral 

decomposition, such as generalized Laplace analysis and dynamic mode 

decomposition. 

The analysis leads to the decomposition of a nonlinear system with memory into

modes whose temporal behavior is anomalous and lacks a characteristic scale. On 

average, the time evolution of a mode follows a Mittag-Leffler function, and the 

system can be described using the fractional calculus. When analyzing data from 

an ill-defined (black-box) system, the spectral decomposition in terms of MittagLeffler functions that we proposed may uncover inherent memory effects through 

identification of a small set of dynamically relevant structures that would otherwise 

be obscured by conventional spectral methods. Consequently, the theoretical 

concepts we present may be useful for developing more general methods for 

numerical modeling that are able to determine whether observables of a dynamical 

system are better represented by memoryless operators, or operators with long-term 

memory in time, when model details are unknown.

3. Conclusion

While we made significant progress in accounting for external forcing within a 

Koopman operator-based framework, much work remains to develop a controloriented methodology. The addition of a prescribed external force can be 

considered an open loop control example. In this, we showed that the forcing in a

Koopman-mode-based reduced order model must appear as a bi-linear term. Future 

work should address the inclusion of feedback control capabilities using this term. 
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Furthermore, a key element of an approach that is fully capable of dealing with 

nonlinear systems is the ability to detect and include new modes that may be 

introduced by forcing (or far from equilibrium dynamics). 

No methods exist yet for approximating the eigenvalues and modes in the spectral 

decomposition for systems with memory involving Mittag-Leffler functions. 

However, in principle we can compare the spectral properties of both types of 

decompositions, memoryless and with memory. The caveat is that we still need to 

develop numerical methods for approximating the eigenvalues and modes in 

Eq. 8 to be able to test a data set for memory effects. One direction that could be 

pursued is to extend the nonlinear optimization methods to be compatible with 

Mittag-Leffler functions rather than the exponential time dependence. 

Properly extracting the modes derived from a mechanistic process, though 

seemingly without pattern or chaotic, is a key contribution of this work. While the 

introduction of fractional calculus to obtain a more general spectral decomposition 

approach may be more effective in properly extracting coherency from complex 

systems, we have yet to investigate systems with stochastic features. In many data 

driven system, randomness will be present in the measurements. Future work 

should address the effects of stochastic/random noise in the system such that 

spectral decompositions identify coherent features (modes) and appropriate 

interactions/effects of the noise. Once this is done, one could develop controloriented reduced-order models that focus on the deterministic dynamics, the 

random effects, or both. 
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4. References

1. Koopman BO. Hamiltonian systems and transformation in Hilbert space. 

Proceedings of the National Academy of Sciences of the United States of 

America. 1931;17(5):315.

2. Budisic M, Mezic I. Applied Koopmanism. Chaos. 2012;22. doi: 

10.1063/1.4772195.

3. Glaz B, Mezić I, Fonoberova M, Loire S. Quasi-periodic intermittency in 

oscillating cylinder flow. arXiv. 2016;1609:06267.

4. Svenkeson A, Glaz B, Stanton S, West B. Spectral decompositions of 

nonlinear systems with memory. Physical Review E. 2016;93:022211.

5. Stanislavsky AA. Fractional oscillator. Phys. Rev. E. 2004;70:051103.
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1 DEFENSE TECHNICAL

(PDF) INFORMATION CTR

DTIC OCA

2 DIRECTOR

(PDF) US ARMY RESEARCH LAB

RDRL CIO L

IMAL HRA MAIL & RECORDS

MGMT

1 GOVT PRINTG OFC

(PDF) A MALHOTRA

2 DIR USARL

(PDF) RDRL VTV

B GLAZ

A SVENKESON