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Realisations of heteroclinic networks in coupled cell systems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Realiseringar av heteroklinanätverk i kopplade dynamiskasystem (Swedish)
Abstract [en]

In the theory of dynamical systems, heteroclinic networks are invariant objects in phase space with network structure, consisting of invariant sets (nodes) and connecting trajectories between them (edges). These are typically not robust dynamical phenomena, but can appear robustly for dynamical systems with network structure - so called coupled cell systems -due to the presence of certain synchrony related invariant subspaces. This link between networks in phase space and networks of dynamical systems is the topic of this thesis. Examples and results from the literature on the existence and construction of heteroclinic networks in coupled cell systems are presented and reviewed, focusing on heteroclinic realisation: how can coupled cell systems be constructed that support a given heteroclinic network? We seek to find explicit vector fields for such realisations, of which there are relatively few examples in the literature, and provide a polynomial vector field for a particular heteroclinic network and coupled system. Finally, we state and prove a theorem on the existence of additional equilibrium points for realisations of this heteroclinic network in such systems.

Abstract [sv]

Inom teorin för dynamiska system är heteroklina nätverk invariant objekt i tillståndsrummet som har nätverksstruktur i den mening att de består av invarianta mängder (noder) och förbindande banor mellan dem (kanter). Dessa objekt är typiskt inte robusta dynamiska fenomen, men kan förekomma robust för dynamiska system med nätverksstruktur - så kallade kopplade dynamiska system - tack vare särskilda invarianta underrum relaterade till synkroni. Det är denna koppling mellan nätverk i tillståndsrummet och nätverk av dynamiska system som är temat för detta examensarbete. En litteraturstudie kring exempel och resultat rörande existens och konstruktion av heteroklina nätverk i kopplade dynamiska system görs med fokus på heteroklin realisering: hur kan kopplade dynamiska system konstrueras att ha ett givet heteroklint nätverk? Vi söker explicita vektorfält för sådana realiseringar, av vilka det finns relativt få i litteraturen, och ger ett polynomt vektorfält för ett särskilt heteroklint nätverk och kopplat dynamiskt system. Slutligen formuleras och bevisas en ny sats rörande existensen av extra jämviktspunkter i realiseringar av detta heteroklina nätverk.

Place, publisher, year, edition, pages
2017.
Series
TRITA-MAT-E ; 2017:25
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-208179OAI: oai:DiVA.org:kth-208179DiVA, id: diva2:1118526
External cooperation
University of Exeter
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2017-06-30 Created: 2017-06-30 Last updated: 2017-06-30Bibliographically approved

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