Analysis of Large-Scale Interconnected Dynamical Systems

Analysis of Large-Scale Interconnected Dynamical Systems

Analysis of Large-Scale Interconnected Dynamical Systems

Igor MezićIgor MezićDepartment of Mechanical Engineering, Department of Mechanical Engineering, University of California, Santa BarbaraUniversity of California, Santa Barbara

Introduction

Internet

Systems biology

Biomolecules

Power grid

Introduction

Issues:

-Complex node topology-(Nonlinear) Dynamics at nodes-Extremely large number of degrees of freedom-Uncertainty in parameters describing dynamics-Stochastic effects-Mixture of discrete and continuous dynamics

This talk:-Coupled oscillator models with switching dynamics-An operator theoretic framework.-Geometric concepts; visualization of invariant sets.-Elements of graph theoretic analysis.-A systems biology model.-BUT OF COURSE, turbulence!

A coupled oscillator system

Englander et al (1980)Peyrard, Bishop and collaborators.

Morse potential

Torsional spring

Immobilized strand

I.M. PNAS (2006)

G. Gilmore, UCSB (2009) Inverse cascade: small scale large scale

No scale separation…

P. DuToit, I.M., J. Marsden Physica D (2009)

Cf. Goedde et al. PRL (1992)

200 DOF

Let

Harmonic field approximation

In normal mode coordinates:

Harmonic field approximation

P. DuToit, I.M., J. Marsden Physica D (2009)

There is no separation of scales. Yet, there is reduced order representation!

Define

Cf. mean field approximation

Operator theory: history and setup

Vector field case:

Koopman operator:

Observables on phase space M

B.O. Koopman “Hamiltonian Systems and Transformations in Hilbert Space”, PNAS (1931)

Operator theory: history and setup

B.O. Koopman and J. von Neumann “Dynamical Systems of Continuous Spectra”, PNAS (1932)

Methods based on analysis of the Perron-Frobenius operator:

Lasota and Mackey, “Chaos, fractals, and noise: stochastic aspects of dynamics”,David Ruelle, Lai-Sang Young, , Vivian Baladi,Michael Dellnitz, Oliver Junge, Erik Bollt, Gary Froyland…

Koopman and Von Neumann on chaos

B.O. Koopman and J. von Neumann “Dynamical Systems of Continuous Spectra”, PNAS (1932)

Operator theory and harmonic analysis

Of importance in study of design of search algoritms

(c.f. G. Mathew work, Mon AM)

And characterizing ergodicity in ocean flows

(c.f. S. Scott talk, Monday)

Cf. M. Dellnitz, O. Junge,, SIAM J. Numer. Anal.) (1999).

Ergodic partition

I.M. and A. Banaszuk, Physica D (2004)Statistical Takens Theorem:

Rokhlin( 1940;s), Oxtoby, Ulam, Yosida, Mane,

Invariant sets by Koopman eigenfunctions

Z. Levnajic and I.M., ArXiv (2009)

Quotient space embedding, R2

Trajectories of the Standard Map.

Invariant sets by Koopman eigenfunctions

Cf. M. Budisic talk, CP31 Thu 3-4

Quotient space embedding, R3

-Use spectral technique of Belkin, Lafon, Coifman and collaborators,-Replace Euclidean distance (L^2 norm) with a negative Sobolev space-type modification:

A Power Grid Model

Y. Susuki, T. Hikihara (Kyoto)And I.M. (2009) Cf. Susuki Thu 8:45 MS113

A Realistic Power Grid Model

Y. Susuki, T. Hikihara (Kyoto)And I.M. (2009)

NE Power grid model: 10 generators

Cf. Y. Susuki talk, MS 113 Thu 8:45

x

yz

x

z y

Intro to graph-theoretic techniquesIntro to graph-theoretic techniques

Graph indicates no chaos

Horizontal-Vertical Decomposition

I.M., Proc. CDC(2004)

Cf. E. Shea-Brown an L.-S. Youngon reliability in neural networks (ArXiv2007)

Skew-product structure

Cf. Alice Hubenko talk Wed 5:15 MS 104

Propagation of uncertaintyPropagation of uncertainty

SODE’s:Feynman-Kac Asymptotically: Lyapunov exponentsSODE’s:Feynman-Kac Asymptotically: Lyapunov exponents

3322

11x

z y

A Systems Biology Model

(node 4 and several connections pruned, with no loss of performance)H-V decomposition

Output, execution

Forward, production unit

Feedback loops

Trim the network,preserve dynamics!

Input, initiator

Additional functional requirements

Minimal functional units: sensitive edges (leading to lack of production)

easily identifiable

Level of outputFor MFU

Level of output with feedback loops

T. Lipniacki, P. Paszek, A. R. Brasier, B. Luxon, M. Kimmel, Biophys. J. 228, 195 (2004). A. Hoffmann, A. Levchenko, M. L. Scott, D. Baltimore, Science 298, 1241 (2002).

Yueheng Lan and IM (2009)

Cf. Yueheng Lan talk Thu 8:15 MS 113Alice Hubenko talk Wed 5:15 MS 104

Dynamical graph decomposition

Collective coordinates: actions

Jacobian: H-V decomposition!!!

Cf B. Eisenhower talk, Tue 5:15 CP 13.

B. Eisenhower and I.M. (2009)

04/08/23 21

Laplace transform and transient modes

Spectral decomposition of a fully nonlinear systemuses spectrum of U=PT.

C.W. Rowley, I. Mezic, S. Bagheri, P. Schlatter, and D.S. Henningson (just submitted )

•C. Rowley (Princeton): Arnoldi iteration reveals Koopman modes!

Koopman modes

I.M., Nonl.Dyn (2005)

Acknowledgments

Students:

Marko BudisicBryan EisenhowerGeorge GilmoreRyan MohrBlane RhoadsGunjan Thakur

Postdocs:

Alice HubenkoSymeon GrivopoulosSophie LoireMaud-Alix MaderGeorge Mathew

Visiting Professors:

Yoshihiko Susuki (Kyoto)Yueheng Lan (Tsinghua)

Sponsors:Collaborators:

S. Bagheri (KTH)Andrzej Banaszuk (UTRC)Takashi Hikihara (Kyoto)D.S. Henningson (KTH)Jerry Marsden (Caltech)Clancy Rowley (Princeton)P. Schlatter (KTH)Phillip du Toit (Caltech)

Conclusions

• Structure of inertial network equations with weak local and strong coupling terms lead to switching between global equilibria.

• Koopman operator formalism enables study of invariant partitions (fixed, periodic, quasiperiodic) despite the large interconnected and nonsmooth nature of the systems.

• The same (spectral formalism enables extraction of quasiperiodic, stable and unstable modes for large systems. This is a dynamically consistent (as opposed to energy-based, POD) decomposition.

• Graph theoretic methods for decomposition and uncertainty propagation are coupled to operator formalism.

• Much more work is needed on the operator theoretic/ geometric/probabilistic front.

I. Mezic and A. Banaszuk, "Comparison of systems with complex behavior". Physica D (2004).I. Mezic, “Coupled Nonlinear Dynamical Systems:Asymptotic Behavior and Uncertainty

Propagation,” Proc. CDC (2004).I. Mezic, "Spectral properties of dynamical systems, model reduction and decompositions".Nonlinear Dynamics (2005). I. Mezic, "On the dynamics of molecular conformation ". Proceedings of the National Academyof Sciences of the USA, (2006).