Beyond Slow-Fast: Relaxation Oscillations in Singularly Perturbed Nonsmooth Systems

Abstract

Though it is surely a cliché to say that a PhD thesis is more than the work of an individual, I feel compelled to reiterate the point here. It is true that this thesis is a product of countless hours of work on my part, however, it is also the result of close supervision by my advisor and co-authors, discussions with experts and other students, and the unwavering support of my friends and family.First and foremost, I would like to express my gratitude to my PhD supervisor Prof. Martin Wechselberger. Martin, thank you for the countless hours devoted to me and my work over the past four and a half years (let's not forget the honours year!). Thank you for your patience, and for keeping me on track, which cannot always have been easy. Among other things, you have taught me the importance of keeping 'the big picture' in mind. Above all, thank you for instilling and nurturing my passion for mathematics, which has only grown in these last few years. For this I am forever grateful.Secondly, I would like thank A/Prof. Kristian Uldall Kristiansen, who has in many ways been like a second, unofficial supervisor for me. Thank you for your time and support during my visit to DTU, as well as the countless emails and Skype calls made in an effort to guide me through the details. You have taught me the importance of rigour in mathematics, and inspired me to take on challenges I would not have otherwise thought possible. Truly, this thesis would not have been possible without your guidance and support.I would also like to thank Prof. Peter Szmolyan and Dr. Ilona Kosiuk. Firstly, I thank Peter for valuable feedback and contribution as a co-author on some of the material presented in this thesis. Secondly, I thank both Peter and Ilona for their earlier work which inspired much of this thesis.There are many who deserve thanks for contributing to my mathematical education more generally. I would like to thank A/Prof. Vivien Kirk, Prof. James Sneyd and Dr. Nathan Pages in particular for their hospitality during my visit to Auckland University. This visit (and the following months) taught me the value of collaborative work in mathematics. I would also like to thank several people 'from the office' for helpful discussions, DIY reading groups and the basic support required to get through a PhD.

Singularly perturbed boundary-equilibrium bifurcations

“…The present manuscript provides a classification and detailed dynamical study of singularly perturbed BEBs in the plane. The work can be seen as a (self-contained) continuation of recent work in [19], see also the PhD thesis [17], where the analysis was restricted to a subset of singularly perturbed BF bifurcations, treating three of the total 12 BE unfoldings in detail, and successfully resolving the degeneracies associated with these cases. This manuscript aims to complete the project, by providing a 'complete' description for all 12 unfoldings.…”

Singularly Perturbed Boundary-Equilibrium Bifurcations

“…Setup. The setup is taken from [23], which has also been adopted in [17,19,27]. We consider planar systems (1) ż = Z z, φ y −1 , α , where z = (x, y) ∈ R 2 , φ : R → R, ∈ (0, 0 ], and α ∈ I ⊂ R. The vector field Z : R 2 × R × I → R 2 is assumed to be smooth in all arguments, but generically non-smooth in the limit → 0.…”

“…The present manuscript provides a classification and detailed dynamical study of singularly perturbed BEBs in the plane. The work can be seen as a continuation of recent work in [19], see also the PhD thesis [17], where the analysis was restricted to a subset of singularly perturbed BF bifurcations, treating 3 of the total 12 BE unfoldings in detail, and successfully resolving the degeneracies associated with these cases. This manuscript aims to complete the project, by providing a 'complete' description for all 12 unfoldings.…”

“…Although these bifurcations are degenerate with γ = 0, we nevertheless use this example to illustrate in Figures 11 and 12 Following a suitable parameter-dependent coordinate translation we may assume that system (1) has a nondegenerate BE bifurcation at z bf = (0, 0) when α bf = 0. It follows by arguments analogous to [19, p.38] (see also [17] for further details) that the system…”

“…This manuscript aims to complete the project, by providing a 'complete' description for all 12 unfoldings. Similarly to [17,19], emphasis is placed on understanding the smooth dynamics with 0 < 1. This allows for the treatment of problems arising either naturally or via regularization simultaneously, since the corresponding results for (regularized) PWS systems are easily obtained upon taking the non-smooth singular limit → 0.…”