Uncertain, High-Dimensional Dynamical Systems Igor Mezić University of California, Santa Barbara IPAM, UCLA, February 2005

Introduction Measure of uncertainty? Uncertainty and spectral theory of dynamical systems. Model validation and data assimilation. Decompositions.

Dynamical evolution of uncertainty: an example Output measure Input measure Bifurcation increases uncertainty, but how? Tradeoff: Bifurcation vs. contracting dynamics

Dynamical evolution of uncertainty: general set-up Skew-product system.

Dynamical evolution of uncertainty: general set-up

Dynamical evolution of uncertainty: general set-up F(z) 1 f T

Dynamical evolution of uncertainty: simple examples Fw(z) 1 Expanding maps: x’=2x

A measure of uncertainty of an observable 1

Computation of uncertainty in CDF metric

Maximal uncertainty for CDF metric 1

Variance, Entropy and Uncertainty in CDF metric

Uncertainty in CDF metric: Pitchfork bifurcation x Output measure Input measure

Time-averaged uncertainty 1

Conclusions

Introduction PV=NkT Example: thermodynamics from statistical mechanics “…Any rarified gas will behave that way, no matter how queer the dynamics of its particles…” Goodstein (1985) Example: Galerkin truncation of evolution equations. Information comes from a single observable time-series.

Introduction When do two dynamical systems exhibit similar behavior?

Introduction Constructive proof that ergodic partitions and invariant measures of systems can be compared using a single observable (“Statistical Takens-Aeyels” theorem). A formalism based on harmonic analysis that extends the concept of comparing the invariant measure.

Set-up Time averages and invariant measures:

Set-up

Pseudometrics for Dynamical Systems Pseudometric that captures statistics: where

Ergodic partition

Ergodic partition

An example: analysis of experimental data

Analysis of experimental data

Analysis of experimental data

Koopman operator, triple decomposition, MOD -Efficient representation of the flow field; can be done with vectors -Lagrangian analysis: FLUCTUATIONS MEAN FLOW PERIODIC APERIODIC Desirable: “Triple decomposition”:

Embedding

Conclusions Constructive proof that ergodic partitions and invariant measures of systems can be compared using a single observable –deterministic+stochastic. A formalism based on harmonic analysis that extends the concept of comparing the invariant measure Pseudometrics on spaces of dynamical systems. Statistics – based, linear (but infinite-dimensional).

Introduction Everson et al., JPO 27 (1997)

Introduction 4 modes -99% of the variance! -no dynamics! Everson et al., JPO 27 (1997)

Introduction In this talk: -Account explicitly for dynamics to produce a decomposition. -Tool: lift to infinite-dimensional space of functions on attractor + consider properties of Koopman operator. -Allows for detailed comparison of dynamical properties of the evolution and retained modes. -Split into “deterministic” and “stochastic” parts: useful for prediction purposes.

Factors and harmonic analysis Example:

Factors and harmonic analysis

Harmonic analysis: an example

Evolution equations and Koopman operator

Evolution equations and Koopman operator Similar:“Wold decomposition”

Evolution equations and Koopman operator But how to get this from data?

Evolution equations and Koopman operator is almost-periodic. -Remainder has continuous spectrum!

Conclusions -Use properties of the Koopman operator to produce a decomposition -Tool: lift to infinite-dimensional space of functions on attractor. -Allows for detailed comparison of dynamical properties of the evolution and retained modes. -Split into “deterministic” and “stochastic” parts: useful for prediction purposes. -Useful for Lagrangian studies in fluid mechanics.

Invariant measures and time-averages Example: Probability histograms! -1 a1 a2 But poor for dynamics: Irrational a’s produce the same statistics

Dynamical evolution of uncertainty: simple examples Types of uncertainty: Epistemic (reducible) Aleatory (irreducible) A-priori (initial conditions, parameters, model structure) A-posteriori (chaotic dynamics, observation error) Expanding maps: x’=2x

Uncertainty in CDF metric: Examples Uncertainty strongly dependent on distribution of initial conditions.