Analysis Of Large-scale Interconnected Dynamical Systems

Analysis of Large-Scale Interconnected Dynamical Systems

Igor Mezić Department of Mechanical Engineering, University of California, Santa Barbara

Introduction Internet

Systems biology

Power grid Biomolecules

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 I.M. PNAS (2006) Torsional spring

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

G. Gilmore, UCSB (2009)

Morse potential

Immobilized strand

Inverse cascade: small scale

large scale

No scale separation… 200 DOF

Cf. Goedde et al. PRL (1992)

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

Harmonic field approximation Let In normal mode coordinates: Define

Harmonic field approximation Cf. mean field approximation

There is no separation of scales. Yet, there is reduced order representation! P. DuToit, I.M., J. Marsden Physica D (2009)

Operator theory: history and setup Observables on phase space M Koopman operator:

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

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

Statistical Takens Theorem:

I.M. and A. Banaszuk, Physica D (2004)

Invariant sets by Koopman eigenfunctions Quotient space embedding, R2

Trajectories of the Standard Map. Z. Levnajic and I.M., ArXiv (2009)

Invariant sets by Koopman eigenfunctions 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:

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

A Power Grid Model

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

Cf. Susuki Thu 8:45 MS113

A Realistic Power Grid Model NE Power grid model: 10 generators

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

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

Intro to graph-theoretic techniques

x

z

y

z

y x

Graph indicates no chaos

Horizontal-Vertical Decomposition I.M., Proc. CDC(2004)

Skew-product structure

Cf. E. Shea-Brown an L.-S. Young on reliability in neural networks (ArXiv2007) Cf. Alice Hubenko talk Wed 5:15 MS 104

Propagation of uncertainty y

z x 2

3 1

SODE’s:Feynman-Kac

Asymptotically: Lyapunov exponents

A Systems Biology Model 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). Output, execution Trim the network, preserve dynamics!

H-V decomposition

Yueheng Lan and IM (2009)

(node 4 and several connections pruned, with no loss of performance)

Feedback loops Forward, production unit Input, initiator

Additional functional requirements

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

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

Level of output For MFU Level of output with feedback loops

Dynamical graph decomposition B. Eisenhower and I.M. (2009)

Jacobian: H-V decomposition!!! Collective coordinates: actions Cf B. Eisenhower talk, Tue 5:15 CP 13.

Laplace transform and transient modes

Spectral decomposition of a fully nonlinear system uses spectrum of U=PT.

06/19/09

21

I.M., Nonl.Dyn (2005)

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

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

Acknowledgments Students:

Collaborators:

Marko Budisic Bryan Eisenhower George Gilmore Ryan Mohr Blane Rhoads Gunjan Thakur

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)

Postdocs: Alice Hubenko Symeon Grivopoulos Sophie Loire Maud-Alix Mader George Mathew Visiting Professors: Yoshihiko Susuki (Kyoto) Yueheng Lan (Tsinghua)

Sponsors:

Conclusions 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). • Structure of inertial network equations with weak local and strong I. Mezic, "On the dynamics of molecular conformation ". Proceedings of the National Academy coupling terms lead to switching between global equilibria. of Sciences of the USA, (2006).

•

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.