Dinamika Hénonovih preslikavanja

master's thesis

• Author: Ana Belcar
• Mentor: Sonja Štimac (mentor)
• Committee member: Sonja Štimac (predsjednik povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Ana Prlić (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Preslikavanje \(H_{a,b} : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) dano formulom \(H_{a,b}(x,y)= (a-by-x^2, x) \) definirao je Michael Hénon 1976. godine. Pokazalo se da je dinamika tog preslikavanja veoma složena te ju ni danas ne razumijemo. U ovom radu proučavali smo dinamiku Hénonovih preslikavanja. U prvom poglavlju definirani su pojmovi koji se koriste u radu. U drugom poglavlju smo pokazali da je dovoljno proučavati Hénonova preslikavanja za parametre \(0 < |b| <1 \) te smo izračunali parametar \(a_0 (b)\) u kojem Hénonovo preslikavanje prelazi kroz tangencijalnu bifurkaciju i parametar \(a_1(b)\) u kojem prolazi kroz bifurkaciju duplikacije perioda. Izračunali smo i fiksne točke i periodične točke osnovnog perioda dva te pokazali njihov tip (u odnosu na hiperboličnost). U trećem poglavlju smo dokazali da Hénonovo preslikavanje nema periodičnih točaka prije tangencijalne bifurkacije. Za to smo trebali promatrati odvojeno dva slučaja: \(b>0\) i \(b<0\). Takav pristup je bio nužan, jer Hénonovo preslikavanje čuva orijentaciju za \(b>0\), a mijenja orijentaciju za \(b<0\). U četvrtom poglavlju definirali smo određeni skup \(\Lambda\) za koji smo dokazali da je hiperbolički za Hénonovo preslikavanje za određeni skup parametara. To je H-invarijantan skup koji sadrži sve periodične točke i u kojem svaka točka ima stabilan i nestabilan pravac u svojoj tangencijalnoj ravnini. To je, također, onaj skup na kojem je Hénonovo preslikavanje kaotično.

Abstract (english)

The map \(H_{a,b} : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) given by the formula \(H_{a,b}(x,y)= (a-by-x^2, x) \) was defined by Michael Hénon in 1976. It turned out that the dynamics of this map is very complex and we do not understand it even today. In this paper, we studied the dynamics of Hénon maps. The first chapter defines the terms used in the paper. In the second chapter we showed that it is sufficient to study Hénon maps for the parameters \(0 < |b| <1 \) and we calculated the parameter \(a_0 (b)\) in which the Hénon map passes through tangent bifurcation and the parameter \(a_1(b)\) in which it goes through the period doubling bifurcation. We calculated both fixed points and periodic points of the base period two and showed their type (in relation to hyperbolicity). In the third chapter, we proved that the Hénon map has no periodic points before tangent bifurcation. ´For this we had to consider two cases separately: \(b>0\) and \(b<0\). Such an approach was necessary because Hénon map preserves the orientation for \(b>0\)and flips the orientation for \(b<0\). In the fourth chapter, we defined a certain set \(\Lambda\) which we proved to be hyperbolic for Hénon map for a certain set of parameters. It is an H-invariant set containing all periodic points and in which each point has a stable and unstable line in its tangent plane. It is also the set on which the Hénon map shows chaotic behaviour.

• Keywords: Hénonova preslikavanja; tangencijalna bifurkacija; H-invarijantan skup
• Keywords (english): Hénon maps; tangent bifurcation; H-invariant set
• Language: croatian
• URN:NBN: urn:nbn:hr:217:932350
• Study programme: Title: Applied Mathematics; 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: 2020-11-24 11:47:49