Globally resonant homoclinic tangencies : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
| dc.confidential | Embargo : No | en_US |
| dc.contributor.advisor | Simpson, David | |
| dc.contributor.author | Muni, Sishu Shankar | |
| dc.date.accessioned | 2022-03-02T02:37:15Z | |
| dc.date.accessioned | 2022-10-20T01:46:29Z | |
| dc.date.available | 2022-03-02T02:37:15Z | |
| dc.date.available | 2022-10-20T01:46:29Z | |
| dc.date.issued | 2022 | |
| dc.description.abstract | The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and 'global resonance', as has been described previously in the area-preserving setting. However, unlike in that setting, local resonant terms also play an important role. To determine how the phenomenon may manifest in bifurcation diagrams, we also study perturbations of a globally resonant homoclinic tangency. We find there exist sequences of saddle-node and period-doubling bifurcations. Interestingly, in different directions of parameter space, the bifurcation values scale differently resulting in a complicated shape for the stability region for each periodic solution. In degenerate directions the bifurcation values scale substantially slower as illustrated in an abstract piecewise-smooth C¹ map. | en_US |
| dc.identifier.uri | http://hdl.handle.net/10179/17633 | |
| dc.publisher | Massey University | en_US |
| dc.rights | The Author | en_US |
| dc.subject | Differentiable dynamical systems | en |
| dc.subject | Bifurcation theory | en |
| dc.subject.anzsrc | 490409 Ordinary differential equations, difference equations and dynamical systems | en |
| dc.title | Globally resonant homoclinic tangencies : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand | en_US |
| dc.type | Thesis | en_US |
| massey.contributor.author | Muni, Sishu Shankar | en_US |
| thesis.degree.discipline | Mathematics | en_US |
| thesis.degree.grantor | Massey University | en_US |
| thesis.degree.level | Doctoral | en_US |
| thesis.degree.name | Doctor of Philosophy (PhD) | en_US |
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