20.500.12556/DKUM-44123 Kaos v diskretnih homogenih kvadratičnih sistemih v ravnini Chaos in discrete homogenous quadratic systems in the plane V doktorski disertaciji je obravnavan problem kaotičnosti v diskretnih homogenih kvadratičnih sistemih v ravnini. V uvodnem poglavju je podrobneje opredeljen glavni problem disertacije. V drugem poglavju je obravnavana enolična zveza med diskretnimi homogenimi kvadratičnimi sistemi v ravnini in realnimi dvodimenzionalnimi komutativnimi algebrami. Seznanimo se z osnovnimi pojmi teorije neasociativnih algeber in Markusovo klasifikacijo realnih komutativnih dvodimenzionalnih algeber. Pozornost posvetimo posebnim algebrskim elemetom, kot so nilpotenti reda 2 in projektorji. V tretjem poglavju obravnavamo osnove diskretnih dinamičnih sistemov. Med drugim definiramo (privlačnost) fiksne točke in bazen privlačnosti stabilne fiksne točke. Za opredelitev kaotičnosti uporabimo Devaneyevo definicijo, ki temelji na gostoti periodičnih točk, topološki tranzitivnosti in občutljivosti na začetne pogoje. Obravnavan je tudi pojem topološke konjugacije sistemov in zveza med linearno ekvivalentnimi sistemi ter obstojem izomorfizma med pripadajočimi algebrami. V četrtem poglavju je dokazano, da dinamika v diskretnih homogenih kvadratičnih sistemih v ravnini poteka po žarkih. Ugotovimo, da je koordinatno izhodišče privlačna fiksna točka in da se zanimiva dinamika odvija na robu območja privlačnosti koordinatnega izhodišča. Spoznamo, da se dinamika v diskretnem kvadratičnem sistemu, kjer pripadajoča algebra vsebuje nilpotent(e) reda 2, bistveno razlikuje od dinamike v ostalih sistemih. V petem poglavju je sistematično obravnavana dinamika v kvadratičnem sistemu, kjer pripadajoča algebra vsebuje nilpotent(e) reda 2. Ugotovimo, da bazen privlačnosti koordinatnega izhodišča ni omejena množica in da je, razen v primeru tako imenovane popolnoma periodične dinamike, dinamika dokaj preprosta (in vedno nekaotična). V šestem poglavju obravnavamo dinamiko v kvadratičnih sistemih, kjer pripadajoča algebra vsebuje ideale. Obstoj idealov omogoča redukcijo sistema, kjer je zaradi manjše dimenzije prostora dinamiko lažje obravnavati. V sedmem poglavju obravnavamo dinamiko v kvadratičnih sistemih, kjer pripadajoča algebra premore deljenje. Ugotovimo, da lahko v tem primeru dinamiko diskretnega homogenega kvadratičnega sistema v ravnini enolično povežemo z iteracijo posebnih kvadratnih racionalnih funkcij, kar omogoča dokazati kaotičnost dinamike v vseh teh primerih. V zadnjem poglavju so podani nekateri zaključki. In this doctoral dissertation the problem of chaotic dynamics in discrete homogeneous quadratic systems in the plane is considered. In the introduction the main problem of the thesis is defined in more detail. In the second chapter the 1:1 correspondence between discrete homogeneous quadratic systems and two-dimensional commutative real algebras is considered. The basics of the theory of nonassociative algebras and the Markus classification of two dimensional commutative algebras are summarized. Special algebraic elements, like nilpotents of rank two and idempotents are considered, as well. In chapter three the basics of discrete dynamical systems are summarized. Among others, the (attractive) fixed points and the basin of attraction of a fixed point is considered. The Devaney's definition of chaos which is based on the existence of dense periodic points, topological transitivity and the sensitivity on the initial conditions is adopted. The topological conjugation and the connection between linearly equivalent systems and the existence of an isomorphism of the corresponding algebras are considered, as well. In chapter four it is proven that the dynamics in discrete homogeneous quadratic systems take place on rays. The origin is always an attractive fixed point and interesting dynamics occur on the boundary of the basin of attraction (of the origin). The dynamics of a discrete quadratic system which corresponds to an algebra which contains some nilpotent of rank two differs essentially from the dynamics of the other systems. In chapter five the dynamics in quadratic systems which correspond to algebras containing a nilpotent of rank two are systematically considered. The basin of attraction of the origin is not bounded. The corresponding dynamics are (except in case of the so called completely periodic dynamics) quite simple and always nonchaotic. In chapter six the dynamics in systems where the corresponding algebra contains some ideal are considered. The existence of the ideal allows to solve the system by reduction. The lower dimension allows to better understand the dynamics. In chapter seven the dynamics in systems for which the corresponding algebra is a division algebra is considered . The dynamics of the corresponding discrete quadratic planar system can be associated in a unique way with rational function (of degree 2). The dynamics of the original system are then determined by the iteration of the corresponding rational function which allows to prove the chaotic behavior in these cases. Finally, some conclusions are given. kvadratični sistem diskretni dinamični sistem iteracija kompleksnih funkcij racionalna funkcija stopnje 2 periodične orbite kaos komutativna algebra neasociativna algebra nilpotent reda 2 projektor. quadratic system discrete dynamical system iteration of complex functions rational function of degree 2 periodic orbits chaos commutative algebra nonassociative algebra nilpotent of rank two idempotent. true true false M. Kutnjak] Slovenski jezik Angleški jezik Doktorska disertacija 2014-04-17 11:10:29 2014-05-05 12:11:41 2022-05-12 03:06:50 2021-05-01 03:07:27 2014 0 [Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512.1:517.93(043.3) 20432904 URN:SI:UM:DK:HHJFB2B2 DR_Kutnjak_Milan_2014.pdf DR_Kutnjak_Milan_2014.pdf 1 CBB34008923EB1A8A3CC224D639BC359 a2a8816a0b3e1974d120e50a45ad2bf5a16e1adaea9dd8387174a44918b5c9ff c98676a5-7c0a-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=63163 Fakulteta za naravoslovje in matematiko 0 0 0