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

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

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

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

5. Dynamical evolution of uncertainty: general set-up

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

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

8. A measure of uncertainty of an observable1

9. Computation of uncertainty in CDF metric

10. Maximal uncertainty for CDF metric1

11. Variance, Entropy and Uncertainty in CDF metric

12. Uncertainty in CDF metric: Pitchfork bifurcationx
Output measure
Input measure

13. Time-averaged uncertainty1

14. Conclusions

15. 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.

16. Introduction
When do two dynamical systems exhibit similar behavior?

17. 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.

18. Set-up
Time averages and invariant measures:

19. Set-up

20. Pseudometrics for Dynamical Systems
Pseudometric that captures statistics:
where

21. Ergodic partition

22. Ergodic partition

23. An example: analysis of experimental data

24. Analysis of experimental data

25. Analysis of experimental data

26. 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”:

27. Embedding

28. 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).

29. Introduction
Everson et al., JPO 27 (1997)

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

32. 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.

33. Factors and harmonic analysis
Example:

34. Factors and harmonic analysis

35. Harmonic analysis: an example

36. Evolution equations and Koopman operator

37. Evolution equations and Koopman operator
Similar:“Wold decomposition”

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

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

40. 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.

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

42. 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

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