Geometry of jump-induced mixed-mode oscillations and topological horseshoes in three-dimensional slow-fast systems

Abstract

The main goal of this thesis was to investigate two different applications of geometric singular perturbation theory (GSPT), mixed-mode oscillations (MMOs) and topological horseshoes, in three-dimensional slow-fast systems, of two slow and one fast variables. A detailed introduction was made to complement the present analysis. In Chapter 2, we presented basic knowledge towards the general theory of dynamical system, with a specilisation to slow-fast systems. Next, we focused our attention to an introduction to geometric singular perturbation theory (GSPT), see Chapter 3. This served as an excellent framework on which we developed the topic of MMOs, with a swift review of the known triggers of MMOs, see Chapter 4. The chapter also manifests the first of the two main parts of this dissertation and concerns the geometry of jump-induced MMOs. MMOs are complex orbital patterns that involve the alternation of large-amplitude relaxation oscillations (LAOs) with small-amplitude oscillations (SAOs). In Section 4.2, we focused on families of three-dimensional vector fields of the form x 0 = f(x, y, z, , δ), y 0 = g1(x, y, z, , δ), z 0 = g2(x, y, z, , δ), with f, g1, g2 sufficiently smooth functions and two perturbation parameters , δ > 0 ; is responsible for the time-separation. We introduced a novel mechanism, of the so-called “jump type”, that generates MMOs, and established various results regarding the natural connection between said family of three-dimensional slow-fast vector fields, with MMO behaviour, and piecewise-affine maps (PAMs) of one-dimension. Principally, assuming certain conditions, we proved that a reciprocal relation is possible between every PAM that exhibits MMOs with a certain signature and a slow-fast system of the family above. This result intrinsically reduces the study of this type of three-dimensional slow-fast systems into investigating one-dimensional maps, the theory of which has adequately advanced in the recent years. To solidify our theoretical findings, we developed satisfying numerical results, that remain consistent with both ours and those of other relevant works [RPB12]. In addition to the previous, an attempt to generalise the framework was made; see Section 4.3. There, our goal is to reconsider some of the less generic conditions imposed on Section 4.2, thus generalising our point of view. In particular, we question the limiting nature of (4.21), which is ultimately dropped altogether (Section 4.3.1); as a result, we provide with groundwork adequate enough to reproduce the analysis of Section 4.2 without the hindrance of (4.21). The second part realises a different approach and examines the spectrum of slow-fast systems by applying elements of symbolic and chaotic dynamics, in order to investigate the emergence of topological horseshoes in slow-fast theory. A swift introduction takes place in Chapter 5, before the investigation transpires in Chapter 6. Our plan is to explore the chaotic nature of certain slow fast systems, by seeking to satisfy the so-called horseshoe conditions, as were formulated in [KY01; KKY01]. To that end, we individually examine two particular examples: First, inspired by our own work in [Pat+22], we consider a family of slow-fast systems that has the following form x 0 = f0(x)b(z) λ − y =: f(x, y, z, , λ), y 0 = g(x, y, z, , λ), z 0 = h(x, y, z, , λ); (7.1) there, f0(x) = 3 4 x − 1 4 x 3 + 1 2 , b(z) =    1, if z ≤ 0, 1 2 + 1 10 √ −80z 2 + 60z + 25, if 0 < z ≤ 1, 1 2 + 1 10 √ 5, if z > 1, g(x, y, z, , δ) = x, h(x, y, z, , δ) = ( −x(5 − 10y), if x ≤ 0, 0, if x > 0; (7.2) additionally, λ ∈ [0, 1] and 0 < 1. The critical manifold remains a traditional, S-shaped, smooth, three-dimensional surface, i.e. S := {(x, y, z) ∈ R 3 : f(x, y, z, 0, λ) = 0}, that is typically normally hyperbolic, apart from the sets of horizontal tangency, i.e. Li := {(x, y, z) ∈ S : ∂xf(x, y, z, 0, λ) = 0}; these sets traditionally take up the form of straight lines. Despite this being the case in Chapters 4, in Chapter 6 we pursue a different path. We pick one of the fold curves and experiment with a shape that better resembles a logistic curve (see Section 6.3.2), while an attempt to address different cases is also made in the same section. By careful calibration of each piece of the puzzle, i.e. the function b and the construction of a suitable set Q that is going to be our blueprint for concluding chaos, satisfying the horseshoe hypotheses is actually quite direct, and, hence, the chaotic nature of this example becomes evident. The second model is stimulated from the literature of predator-prey systems and inspired by the work of De Maesschalck, [De 15]. We adopt a three-dimensional, slow-fast vector field, of the form x 0 = x(x 2 − 2x + y)Ω(y, z), y 0 = (x − 1 2 ), z 0 = (x − 1 2 )(x 2 − 2x), with two slow and one fast variable. In this example, the critical manifold S is a composite surface that is the union of a plane E := {x = 0}, the surface K := {y = 2x − x 2} and the surface {Ω(y, z) = 0}. As is customary in the study of population dynamics, we restrict to a domain of interest that involves the octants where the relevant parts are the vertical plane E and the parabolic cylinder K. In fact, it is of great interest to consider the intersection of E with K, which arises along a straight line P, after which the stability properties of E and K are swapped. This model involves the complication of considering the so-called slow divergence integral along the normally repelling parts of E, before the eventual canard-induced jump that occurs. Before we complete this part, with a discussion on open topics, it is useful to note that additional material can be found in the postscript of this thesis. Those pieces of complementary theory, however important, ultimately did not play the leading role in this dissertation. Nonetheless, for the sake of rigour and to cover every aspect, instead of simply been omitted altogether, they were chosen to be included merely as an appendix.