Applied Mathematics Seminars in 2020

Abstract: Multi-scale phenomena in e.g., biology, engineering, and neuroscience are frequently described by singularly perturbed ordinary differential equations with solutions varying over vastly separated timescales, making their analysis a challenging problem. In recent decades, significant progress has been made via the development of Geometric Singular Perturbation Theory (GSPT), which provides a powerful theoretical framework for the analysis of such problems. When combined with a geometric method for the desingularisation of singularities known as blow-up, GSPT can provide a remarkably detailed and geometrically informative understanding of the dynamics.
However, GSPT in its standard form has several limitations which restrict the scope of its applicability. Discontinuity, exponential nonlinearities and 'hidden scales' all present significant challenges to the theory. In this talk, we will explore these limitations in the context of a simple electrical oscillator model - the Le Corbeiller oscillator - and show how they can be overcome using a combination of tools that are adapted from GSPT and the theory of piecewise-smooth (PWS) systems. Our aim will be to understand the onset of multi-scale 'relaxation oscillations' which converge to PWS cycles as a singular perturbation parameter tends to zero. Our main analytical tool is the blow-up method, which must be adapted to resolve degeneracy stemming from (i) the loss of smoothness and (ii) the presence of an essential singularity.
The talk will conclude with a brief discussion on the systematic implementation of these ideas, which have so far been developed only in the context of applications.

Abstract: Invariant tori are prominent features of Hamiltonian systems. In particular, integrable systems are foliated by tori with half the dimension of the phase space. KAM theory implies that many of these tori persist under smooth perturbations, but as the perturbations grow, only the ``robust'’ tori persist in the face of increasing chaos. Perhaps the simplest example of such dynamics is the case of area-preserving maps. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. For higher dimensional cases, the number theoretic properties of the robust tori are still unknown, though it has been conjectured that those with frequency vectors in cubic irrational field should be most robust.

We develop a method based on a weighted Birkhoff average (an idea due to, Das, Saiki, Sander, and Yorke) to identify chaotic orbits, resonances, and rotational invariant tori. It can quickly compute frequency vectors of the latter to machine precision. Variants of Chirikov’s standard map are used as 2D test cases. As a higher-dimensional test, we study a ``standard’' family of 3D, volume-preserving maps---to which KAM theory applies, and attempt to identify the most robust two-tori.

(This research is in collaboration with Evelyn Sander of George Mason University)

Abstract: We review a few applications of extreme value theory to:
(i) open systems;
(ii) give the distribution of observables defined along the temporal evolution of a dynamical system.
Applications are given for the class of prevalent observables.

Abstract: The study of lattice dynamics, i.e., the motion of a spatially discrete system governed by a system of differential-difference equations, is a classical subject. Of particular interest are lattices that support the propagation of solitary waves. In this talk, we will compare the properties of two kinds of lattices, one integrable and one non-integrable: the Toda lattice and the Hertzian chain. As is well known, the Toda lattice is an integrable system and has exact soliton solutions. In contrast, the Hertzian chain, which has many physical and engineering applications, is a non-integrable system and no exact solitary-wave solutions are known. Here we will analyze the similarities and differences between the solitary waves in these two systems, we will discuss how each of these systems respond to a velocity perturbation, and we will compare the interaction dynamics of solitary waves. This is a joint work with Dr. Gino Biondini and Dr. Surajit Sen.

Abstract: Cell degeneration, including that resulting in retinal diseases such as retinitis pigmentosa and AMD, is linked to metabolic issues. In the retina, photoreceptor degeneration can result in disturbances of glucose levels and metabolic processes. To identify the key mechanisms in metabolism that may be culprits of this degeneration, we develop and investigate two mathematical models of the metabolic pathway of aerobic glycolysis in a healthy cone photoreceptor. We develop two different nonlinear systems of enzymatic functions and differential equations. In one case, we mathematically model cone molecular and photoreceptor cellular interactions. In the other case, within a single cone cell, we consolidate some of the metabolic processes in the glycolytic pathway and focus on the glucose, lactate, and pyruvate levels. We perform numerical simulations, use available metabolic data to estimate parameters and fit the models to this data. We conduct uncertainty and sensitivity analysis to identify the processes that have the largest impact on each system.

In the molecular and cellular level model, we consider the case of a healthy cone, a cone with low levels of glucose, and a cone with low and no rod-derived cone viability factor (RdCVF). The three key processes identified are metabolism of fructose-1,6-bisphosphate, production of glycerol-3-phosphate and competition that rods exert on cone resources. The first two processes are proportional to the partition of the carbon flux between glycolysis and the pentose phosphate pathway or the Kennedy pathway, respectively. The last process is the rods’ competition for glucose, which may explain why rods also provide the RdCVF signal to compensate.

In the other model, we use bifurcation techniques and identify a bistable regime, biologically corresponding to a healthy versus a pathological state. The system exhibits a saddle node bifurcation and hysteresis. Model simulations reveal the modulating effect of external lactate in bringing the system to steady state; the bigger the difference between external lactate and initial internal lactate concentrations, the longer the system takes to achieve steady state. Sensitivity analysis reveals that the rate of b-oxidation of ingested outer segment fatty acids consistently plays an important role in the concentration of glucose, G3P, and pyruvate, whereas the extracellular lactate level consistently plays an important role in the concentration of lactate and acetyl coenzyme A.

The ability of these mechanisms to affect key metabolites’ levels (as revealed in our analyses) signifies the importance of inter-dependent and inter-connected feedback processes modulated by and affecting cone’s metabolism.

Abstract: Banded vegetation patterns are surprisingly common in drylands. In these ecosystems, water is a limiting nutrient, and ecohydrological processes are considered to be relevant to the formation and maintenance of patterns. Modeling challenges for capturing and predicting the evolution of these patterns include a lack of first principles mechanisms around which to build a model, and the range of timescales that are relevant to the system. Timescales include the rapid timescales of rainfall and runoff (minutes to hours), the seasonal timescales of plant growth and death (days to months), and the longer time scales associated with pattern evolution (years to decades). In this talk, I will discuss results from studies based on satellite data, I will provide a brief overview of reaction-advection-diffusion modeling approaches, and I will present models where we have worked to incorporate topographic variation and fast-slow switching to capture the dynamics of the system.

Abstract:  We consider deterministic fast-slow systems where the fast dynamics is assumed to be (non)uniform hyperbolic. The aim is to prove that the slow dynamics converges to a stochastic differential equation (with the correct interpretation of the stochastic integrals). Various results in this direction will be described.

Abstract:  Networks of coupled nonlinear dynamical systems arise as models throughout the sciences. Such network systems may display unexpected collective behaviour even if the individual dynamical systems that it is made of are quite simple. This behaviour includes (partial) synchronisation, where (some of) the agents of the network evolve in unison (think of the simultaneous firing of neurons). It has also been observed that synchrony in networks often emerges and breaks through unusual bifurcation scenarios.
This talk is motivated by the question how we can predict and compute these synchrony breaking bifurcation scenarios from intrinsic geometric properties of the network. One of these geometric properties is an algebraic structure that we called "hidden network symmetry". This structure lets us invoke representation theory to determine the generic local behaviour of network dynamical systems.
 This is joint work with Jan Sanders and Eddie Nijholt.