Koopmanov operator i numerička spektralna analiza dinamičkih sustava

master's thesis

• Author: Vlatka Vazdar
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Committee member: Željka Milin Šipuš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-02-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatramo različite aspekte teorije Koopmanovih i Perron-Frobeniusovih operatora pridruženih dinamičkom sustavu u diskretnom ili neprekidnom vremenu. Za dani dinamički sustav Perron-Frobeniusov operator opisuje evolucije vjerojatnosnih gustoća pod djelovanjem sustava, dok Koopmanov operator opisuje evolucije funkcija prostora stanja. Navedeni su operatori linearni za sve dinamičke sustave, što nam omogućuje opis nelinearnih dinamičkih sustava linearnim, ali beskonačnodimenzionalnim operatorom. Perron-Frobeniusov i Koopmanov operator imaju lijepa svojstva: oba su parcijalne izometrije na svojim prirodnim domenama. Također, Koopmanov operator pridružen preslikavanju \(\Phi\) dualan je Perron-Frobeniusovom operatoru pridruženom istom preslikavanju, iz čega slijede bitna svojstva spektra jednog i drugog operatora. Za dinamičke sustave s neprekidnim vremenom promatraju se familije, odnosno polugrupe, Perron-Frobeniusovih i Koopmanovih operatora. Ako je sustav zadan običnim diferencijalnim jednadžbama, za određene se klase funkcija djelovanje svake od tih familija može opisati jednim operatorom - generatorom polugrupe. Najpopularnija metoda za aproksimaciju Perron-Frobeniusovog operatora je Ulamova metoda, kojoj je ideja podijeliti prostor stanja na konačno mnogo disjunktnih skupova i promatrati danu transformaciju kao Markovljev lanac s konačnim prostorom stanja. Stacionarne distribucije tog Markovljevog lanca aproksimiraju stacionarne gustoće Perron-Frobeniusovog operatora. Za aproksimaciju Koopmanovog operatora koristi se proširena dinamička modalna dekompozicija (EDMD), koja za dani rječnik funkcija aproksimira projekciju Koopmanovog operatora na potprostor razapet funkcijama iz rječnnika. Slična se metoda, poznata kao gEDMD, može iskoristiti za aproksimaciju generatora Koopmanove familije. Efikasnost ovih metoda provjeravamo na nekoliko primjera diskretnih i neprekidnih dinamičkih sustava.

Abstract (english)

In this thesis we have studied several aspects of the theory of Koopman and Perron-Frobenius operators associated with both discrete-time and continuous-time dynamical systems. The Perron-Frobenius operator describes the evolution of probability density governed by the underlying dynamical system, whereas the Koopman operator describes the evolution of observables, functions of the state space. Those operators are linear for any dynamical system, which allows us to describe nonlinear dynamical systems using a linear, yet infinite dimensional operator. The Perron-Frobenius operator and the Koopman operator have nice operator-theoretic properties: they are both partially isometric on their respective natural domains. Additionally, the Koopman operator associated with a transformation \(\Phi\) is the dual of the Perron-Frobenius operator associated with \(\Phi\), from which some spectral properties of the operators can be derived. For continuous-time dynamical systems we have the corresponding families, more specifically semigroups, of Koopman and Perron-Frobenius operators. If the dynamical system can be modeled by ordinary differential equations, the action of each family to certain classes of functions can be modeled using a single operator - the semigroup generator. The most popular method for approximating the Perron-Frobenius operator is Ulam’s method. The idea behind Ulam’s method is to split the state space into a finite number of disjoint sets (boxes) and model the system as a finite state Markov chain. Stationary distributions of that Markov chain approximate stationary densities of the Perron-Frobenius operator. The most widely used method for approximating the Koopman operator is Extended Dynamic Mode Decomposition (EDMD), which, given a dictionary of observables, approximates the projection of the Koopman operator onto the subspace spanned by those observables. A similar method, known as generator EDMD (gEDMD), can be employed to approximate the generator of the Koopman family. We have tested the efficiency of these methods on several examples of discrete-time and continuous-time dynamical systems.

• Keywords: Perron-Frobeniusov operator; Koopmanov operator; Ulamova metoda
• Keywords (english): Perron-Frobenius operator; Koopman operator; Ulam’s method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:950330
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-10-27 18:57:44