Comparison of Dynamical Systems

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

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

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Dynamical evolution of uncertainty an example
Output measure
Input measure
Tradeoff Bifurcation
vs. contracting dynamics
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Dynamical evolution of uncertainty general set-up
Skew-product system.
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Dynamical evolution of uncertainty general set-up
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Dynamical evolution of uncertainty general set-up
F(z)
1
0
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Dynamical evolution of uncertainty simple
examples
Expanding maps x2x
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A measure of uncertainty of an observable
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Computation of uncertainty in CDF metric
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Maximal uncertainty for CDF metric
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Variance, Entropy and Uncertainty in CDF metric
0
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Uncertainty in CDF metric Pitchfork bifurcation
Output measure
Input measure
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Time-averaged uncertainty
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Conclusions
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Introduction

• Example thermodynamics from statistical mechanics

PVNkT
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.

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Introduction
When do two dynamical systems exhibit
similar behavior?
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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.

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Set-up
Time averages and invariant measures
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Set-up
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Pseudometrics for Dynamical Systems

• Pseudometric that captures statistics

where
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Ergodic partition
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Ergodic partition
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An example analysis of experimental data
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Analysis of experimental data
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Analysis of experimental data
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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
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Embedding
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Conclusions

• Constructive proof that ergodic partitions and
  invariant
• measures of systems can be compared using a
• single observable deterministicstochastic.
• 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).

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

• 4 modes -99
• of the variance!
• -no dynamics!

Everson et al., JPO 27 (1997)
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(No Transcript)
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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.
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Factors and harmonic analysis
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Factors and harmonic analysis
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Harmonic analysis an example
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Evolution equations and Koopman operator
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Evolution equations and Koopman operator
SimilarWold decomposition
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Evolution equations and Koopman operator
But how to get this from data?
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Evolution equations and Koopman operator
is almost-periodic.
-Remainder has continuous spectrum!
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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.
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Invariant measures and time-averages
Example Probability histograms!
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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 x2x
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Uncertainty in CDF metric Examples
Uncertainty strongly dependent on distribution of
initial conditions.