Slow Manifolds, Canard Orbits and the Organization of Mixed-Mode Oscillations

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dc.contributor.advisor Osinga, H en
dc.contributor.advisor Krauskopf, B en
dc.contributor.author Hasan, Ragheb en
dc.date.accessioned 2018-01-29T20:59:55Z en
dc.date.issued 2017 en
dc.identifier.uri http://hdl.handle.net/2292/36887 en
dc.description.abstract A mixed-mode oscillation is a complex waveform with a pattern of alternating smallamplitude oscillations (SAOs) and large-amplitude oscillations (LAOs). MMOs have been observed in many applications, including semiconductor lasers, neuron models and chemical reactions. In this thesis, we aim to understand and elucidate the phenomenological behavior of MMOs inherited in slow-fast systems, which take the form of ordinary differential equations with a group of fast variables and a group of slow variables, with a time-scale separation parameter ". Geometric singular perturbation theory predicts the existence of locally invariant slow manifolds that organize the slow dynamics. The fast dynamics are typically organized by stable and unstable (fast) manifolds of slow manifolds. Interactions of slow manifolds give rise to so-called canard orbits, which create a mechanism for generating SAOs. On the other hand, LAOs require a global return mechanism. In this thesis, we present a framework for studying two-dimensional slow manifolds, canard orbits and their roles in organizing MMOs in parameter regimes where " is too large for applying classical results from the theory. This is achieved by employing advanced numerical methods based on a two-point boundary value problem setup. We first consider an autocatalytic chemical reaction model with one fast and two slow variables. In this system, we find that canard orbits, which occur along intersections of two-dimensional attracting and repelling slow manifolds, are organized in pairs that we call twin canard orbits. Consequently, the extended attracting slow manifold is divided into subsurfaces called ribbons. Ribbons and associated twin canard orbits organize MMOs when " is relatively large. A continuation analysis illustrates how twin canard orbits arise due to generic quadratic tangencies of slow manifolds. In systems with two fast and two slow variables, two-dimensional slow manifolds of saddle type play a key role in organizing MMOs. One goal of this thesis is to introduce a general approach for computing two-dimensional saddle slow manifolds and their threedimensional stable and unstable manifolds, as well as associated canard orbits in R4. We first test and demonstrate our methods for an extended normal form of a folded node. These methods are then reliably implemented for the full four-dimensional Hodgkin- Huxley neuron model, where " is again relatively large. Our results show that MMOs of this model are also organized by ribbons of the attracting slow manifold and bounding twin canard orbits. Overall, we conclude that our approach is suitable for computing two-dimensional slow manifolds in R3 as well as in R4, and in parameter regimes that are beyond what is known from established theory. In particular, we show that it is practical to study fourdimensional slow-fast vector fields without the need for applying any reduction technique. en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA99265042910902091 en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/nz/ en
dc.title Slow Manifolds, Canard Orbits and the Organization of Mixed-Mode Oscillations en
dc.type Thesis en
thesis.degree.discipline Mathematics en
thesis.degree.grantor The University of Auckland en
thesis.degree.level Doctoral en
thesis.degree.name PhD en
dc.rights.holder Copyright: The author en
dc.rights.accessrights http://purl.org/eprint/accessRights/OpenAccess en
pubs.elements-id 722511 en
pubs.org-id Faculty of Science en
pubs.org-id Mathematics en
pubs.record-created-at-source-date 2018-01-30 en


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http://creativecommons.org/licenses/by-nc-sa/3.0/nz/ Except where otherwise noted, this item's license is described as http://creativecommons.org/licenses/by-nc-sa/3.0/nz/

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