Questionable, High-Dimensional Dynamical Systems Igor Mezić University of California, Santa Barbara IPAM, UCLA, February 2005Slide 2
Introduction Measure of vulnerability? Vulnerability and unearthly hypothesis of dynamical frameworks. Show approval and information digestion. Deteriorations.Slide 3
Dynamical development of vulnerability: a case x Output measure Input measure Tradeoff: Bifurcation versus contracting elementsSlide 4
Dynamical development of vulnerability: general set-up Skew-item framework.Slide 5
Dynamical advancement of vulnerability: general set-upSlide 6
Dynamical development of instability: general set-up f T F(z) 1 0Slide 7
Dynamical development of vulnerability: straightforward illustrations F w (z) 1 0 Expanding maps: x\'=2xSlide 8
A measure of instability of a detectable 1 0Slide 9
Computation of instability in CDF metricSlide 10
Maximal vulnerability for CDF metric 1 0Slide 11
Variance, Entropy and Uncertainty in CDF metric 0Slide 12
Uncertainty in CDF metric: Pitchfork bifurcation x Output measure Input measureSlide 13
Time-arrived at the midpoint of vulnerability 1 0Slide 14
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 .Slide 16
Introduction When do two dynamical frameworks display comparative conduct ?Slide 17
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.Slide 18
Set-up Time midpoints and invariant measures :Slide 19
Pseudometrics for Dynamical Systems Pseudometric that catches insights : whereSlide 21
Ergodic parcelSlide 22
Ergodic segmentSlide 23
A case: examination of exploratory informationSlide 24
Analysis of trial informationSlide 25
Analysis of test informationSlide 26
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":Slide 27
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 ).Slide 29
Introduction Everson et al., JPO 27 (1997)Slide 30
Introduction 4 modes - 99% of the difference! - no progression ! Everson et al., JPO 27 (1997)Slide 32
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.Slide 33
Example: Factors and consonant examinationSlide 34
Factors and symphonious investigationSlide 35
Harmonic examination: an illustrationSlide 36
Evolution conditions and Koopman administratorSlide 37
Evolution conditions and Koopman administrator Similar:"Wold disintegration"Slide 38
Evolution conditions and Koopman administrator But how to get this from information?Slide 39
Evolution conditions and Koopman administrator is nearly intermittent. - Remainder has nonstop range !Slide 40
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.Slide 41
- 1 0 a 1 a 2 But poor for elements : Irrational a \'s deliver similar insights Invariant measures and time-midpoints Example: Probability histograms !Slide 42
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\'=2xSlide 43
Uncertainty in CDF metric: Examples Uncertainty unequivocally reliant on conveyance of introductory conditions.
"Ayur""veda" – 'Life''Science'. Udai SJC – 3 rd Walk Upbeat Holi Nikhil Rasiwasia. Ayurveda ...
Shuttle at Little NEO. D.J. Scheeres Division of Aviation design The College of Michigan. The Sp ...
You and your understudies can without much of a stretch make Wordles at Wordle.net ... Transfer ...
It is a Pre-University Diploma. An understudy picks branches of knowledge, ... IB Diploma unders ...
Try not to Enter the Computer Lab Until the Teacher is Present. ... On the off chance that somet ...
Note: this could be supplanted with WEBCT where. correspondence is principally between staff ... ...
The fugue is a piece in light of imitative counterpoints. ... that the written work of fugues go ...
Ordinarily lodgings and a great part of the neighborliness business are antiquated in innovation ...
An instructive method that gives the center to class change endeavors ... furthermore, welfare n ...
Settle for the easiest option than 4 yr schools and colleges. ... www.collegeboard.com register ...
The reason for our auto show was to hold an opposition for secondary school ... Sound Works, a n ...
DVTS is an exceptionally valuable and prevalent feature application framework broadly utilized a ...
Paintball would be the game that I am as of now most dynamic in. ... Paintball is an exceptional ...
Stereochemistry. Stereochemistry refers to the 3-dimensional properties and reactions of ...
Mechanomyography (MMG). Non-intrusive method that records and evaluates the motions produced by ...
What are high passages?. Unheated nursery structures.Same as
Review. Spread Spectrum Communications Techniques connected to Steganography and Signal Embeddin ...
Diagram . MotivationMeson-trade model underneath 400 MeVExtension to higher energiesDynamical mo ...