Dubious, High-Dimensional Dynamical Systems .


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Questionable, High-Dimensional Dynamical Frameworks. Igor Mezi?. College of California, Santa Clause Barbara. IPAM, UCLA, February 2005. Presentation. Measure of vulnerability? Instability and otherworldly hypothesis of dynamical frameworks. Model acceptance and information osmosis. Disintegrations.
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Questionable, High-Dimensional Dynamical Systems Igor Mezić University of California, Santa Barbara IPAM, UCLA, February 2005

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Introduction Measure of vulnerability? Vulnerability and unearthly hypothesis of dynamical frameworks. Show approval and information digestion. Deteriorations.

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Dynamical development of vulnerability: a case x Output measure Input measure Tradeoff: Bifurcation versus contracting elements

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Dynamical development of vulnerability: general set-up Skew-item framework.

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Dynamical advancement of vulnerability: general set-up

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Dynamical development of instability: general set-up f T F(z) 1 0

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Dynamical development of vulnerability: straightforward illustrations F w (z) 1 0 Expanding maps: x\'=2x

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A measure of instability of a detectable 1 0

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Computation of instability in CDF metric

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Maximal vulnerability for CDF metric 1 0

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Variance, Entropy and Uncertainty in CDF metric 0

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

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Time-arrived at the midpoint of vulnerability 1 0

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Conclusions

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Introduction PV=NkT Example : thermodynamics from factual mechanics "… Any rarified gas will act that way, regardless of how eccentric the flow of its particles… " Goodstein (1985) Example: Galerkin truncation of advancement conditions . Data originates from a solitary discernible time-arrangement .

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Introduction When do two dynamical frameworks display comparative conduct ?

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Introduction Constructive confirmation that ergodic allotments and invariant measures of frameworks can be thought about utilizing a solitary recognizable (" Statistical Takens-Aeyels " hypothesis). A formalism in view of symphonious examination that broadens the idea of looking at the invariant measure.

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Set-up Time midpoints and invariant measures :

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Set-up

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Pseudometrics for Dynamical Systems Pseudometric that catches insights : where

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Ergodic parcel

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Ergodic segment

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A case: examination of exploratory information

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Analysis of trial information

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Analysis of test information

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Koopman administrator, triple deterioration, MOD - Efficient portrayal of the stream field; should be possible with vectors - Lagrangian investigation: FLUCTUATIONS MEAN FLOW PERIODIC APERIODIC Desirable: " Triple disintegration":

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Embedding

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Conclusions Constructive confirmation that ergodic allotments and invariant measures of frameworks can be looked at utilizing a solitary discernible –deterministic+stochastic. A formalism in light of symphonious examination that amplifies the idea of contrasting the invariant measure Pseudometrics on spaces of dynamical frameworks. Insights – based , direct (yet unbounded dimensional ).

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Introduction Everson et al., JPO 27 (1997)

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

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Introduction In this discussion: - Account unequivocally for progression to deliver a disintegration. - Tool: lift to unending dimensional space of capacities on attractor + consider properties of Koopman administrator. - Allows for point by point examination of dynamical properties of the development and held modes. - Split into "deterministic" and "stochastic" parts: helpful for expectation purposes.

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Example: Factors and consonant examination

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Factors and symphonious investigation

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Harmonic examination: an illustration

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Evolution conditions and Koopman administrator

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Evolution conditions and Koopman administrator Similar:"Wold disintegration"

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Evolution conditions and Koopman administrator But how to get this from information?

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Evolution conditions and Koopman administrator is nearly intermittent. - Remainder has nonstop range !

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Conclusions - Use properties of the Koopman administrator to deliver a deterioration - Tool: lift to endless dimensional space of capacities on attractor. - Allows for point by point examination of dynamical properties of the development and held modes. - Split into "deterministic" and "stochastic" parts: valuable for forecast purposes. - Useful for Lagrangian thinks about in liquid mechanics.

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- 1 0 a 1 a 2 But poor for elements : Irrational a \'s deliver similar insights Invariant measures and time-midpoints Example: Probability histograms !

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Dynamical development of vulnerability: straightforward illustrations Types of instability: Epistemic (reducible) Aleatory (irreducible) A-priori (starting conditions, parameters, display structure) A-posteriori (confused flow, perception blunder) Expanding maps: x\'=2x

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Uncertainty in CDF metric: Examples Uncertainty unequivocally reliant on conveyance of introductory conditions.

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