Invariant manifold theory for impulsive functional differential equations with applications

The primary contribution of this thesis is a development of invariant manifold theory for impulsive functional differential equations. We begin with an in-depth analysis of linear systems, immersed in a nonautonomous dynamical systems framework. We prove a variation-of-constants formula, introduce appropriate generalizations of stable, centre and unstable subspaces, and develop a Floquet theory for periodic systems. Using the Lyapunov-Perron method, we prove the existence of local centre manifolds at a nonhyperbolic equilibrium of nonlinear impulsive functional differential equations. Using a formal differentiation procedure in conjunction with machinery from functional analysis -- specifically, contraction mappings on scales of Banach spaces -- we prove that the centre manifold is smooth in the state space. By introducing a coordinate system, we are able to prove that the coefficients of any Taylor expansion of the local centre manifold are unique and sufficiently regular in the time and lag arguments that they can be computed by solving an impulsive boundary-value problem. After proving a reduction principle, this leads naturally to explorations into bifurcation theory, where we establish generalizations of the classical fold and Hopf bifurcations for impulsive delay differential equations. Aside from the centre manifold, we demonstrate the existence and smoothness of stable and unstable manifolds and prove a linearized stability theorem. One of the applications of the theory above is an analysis of a SIR model with pulsed vaccination and finite temporary immunity modeled by a discrete delay. We determine an analytical stability criteria for the disease-free equilibrium and prove the existence of a transcritical bifurcation of periodic solutions at some critical vaccination coverage level for generic system parameters. Then, using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution until a Hopf point is identifed. A cylinder bifurcation is observed; the periodic orbit expands into a cylinder in the extended phase space before eventually contracting onto a periodic orbit as the vaccination coverage vanishes. The other application is an impulsive stabilization method based on centre manifold reduction and optimization principles. Assuming a cost structure on the impulsive controller and a desired convergence rate target, we prove that under certain conditions there is always an impulsive controller that can stabilize a nonhyperbolic equilibrium with a trivial unstable subspace, robustly with respect to parameter perturbation, while guaranteeing a minimal cost. We then exploit the low-dimensionality of the centre manifold to develop a two-stage program that can be implemented to compute the optimal controller. To demonstrate the effectiveness of the two-stage program, which we call the centre probe method, we use the method to stabilize a complex network of 100 diffusively coupled nodes at a Hopf point. The cost structure is one that assigns higher cost to controlling of nodes that have more neighbours, while the jump functionals are required to be diagonal -- that is, they do not introduce further coupling. We also introduce a secondary goal, which is that the number of nodes that are controlled is minimized

Worrisome Properties of Neural Network Controllers and Their Symbolic Representations

We raise concerns about controllers' robustness in simple reinforcement learning benchmark problems. We focus on neural network controllers and their low neuron and symbolic abstractions. A typical controller reaching high mean return values still generates an abundance of persistent low-return solutions, which is a highly undesirable property, easily exploitable by an adversary. We find that the simpler controllers admit more persistent bad solutions. We provide an algorithm for a systematic robustness study and prove existence of persistent solutions and, in some cases, periodic orbits, using a computer-assisted proof methodology.Comment: accepted to ECAI2

Bifurcation of Bounded Solutions of Impulsive Differential Equations

Electronic version of an article published as International Journal of Bifurcation and Chaos, Volume 26, No. 14, 2016, 1-20 doi:10.1142/S0218127416502424 © copyright World Scientific Publishing Company, http://dx.doi.org/10.1142/S0218127416502424In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results

Smooth centre manifolds for impulsive delay differential equations

Preprint of an article published in Journal of Differential Equations, available at: https://doi.org/10.1016/j.jde.2018.04.021The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation and smoothness of the nonautonomous process, and a Floquet theorem for periodic systems. Three examples are provided to illustrate the results.National Sciences and Engineering Research Council of Canada, Alexander Graham Bell Canada Graduate Scholarships-Doctoral Progra

Continuous approximation of linear impulsive systems and a new form of robust stability

The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.jmaa.2017.08.026 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/The time-scale tolerance for linear ordinary impulsive differential equations is introduced. How large the time-scale tolerance is directly reflects the degree to which the qualitative dynamics of the linear impulsive system can be affected by replacing the impulse effect with a continuous (as opposed to discontinuous, impulsive) perturbation, producing what is known as an impulse extension equation. Theoretical properties related to the existence of the time-scale tolerance are given for periodic systems, as are algorithms to compute them. Some methods are presented for general, aperiodic systems. Additionally, sufficient conditions for the convergence of solutions of impulse extension equations to the solutions of their associated impulsive differential equation are proven. Counterexamples are provided.NSERC Discovery Gran

Comparing malaria surveillance with periodic spraying in the presence of insecticide-resistant mosquitoes: Should we spray regularly or based on human infections?

The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.mbs.2016.03.009 © 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/There is an urgent need for more understanding of the effects of surveillance on malaria control. Indoor residual spraying has had beneficial effects on global malaria reduction, but resistance to the insecticide poses a threat to eradication. We develop a model of impulsive differential equations to account for a resistant strain of mosquitoes that is entirely immune to the insecticide. The impulse is triggered either due to periodic spraying or when a critical number of malaria cases are detected. For small mutation rates, the mosquito-only submodel exhibits either a single mutant-only equilibrium, a mutant-only equilibrium and a single coexistence equilibrium, or a mutant-only equilibrium and a pair of coexistence equilibria. Bistability is a likely outcome, while the effect of impulses is to introduce a saddle-node bifurcation, resulting in persistence of malaria in the form of impulsive periodic orbits. If certain parameters are small, triggering the insecticide based on number of malaria cases is asymptotically equivalent to spraying periodically.Ontario Graduate ScholarshipNSERC Discovery Gran

Periodic orbits in Hořava–Lifshitz cosmologies

We consider spatially homogeneous Hořava–Lifshitz models that perturb General Relativity (GR) by a parameter v∈(0,1) such that GR occurs at v=1/2. We describe the dynamics for the extremal case v=0, which possess the usual Bianchi hierarchy: type I (Kasner circle of equilibria), type II (heteroclinics that induce the Kasner map) and type VI0,VII0 (further heteroclinics). For type VIII and IX, we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor. Therefore we obtain a new behaviour which is not described by the BKL picture of bouncing Kasner-like states

Heritable genome editing in C. elegans via a CRISPR-Cas9 system

CRISPR-Cas systems have been used with single-guide RNAs for accurate gene disruption and conversion in multiple biological systems. Here we report the use of the endonuclease Cas9 to target genomic sequences in the C. elegans germline, utilizing single-guide RNAs that are expressed from a U6 small nuclear RNA promoter. Our results demonstrate that targeted, heritable genetic alterations can be achieved in C. elegans, providing a convenient and effective approach for generating loss-of-function mutants

A pilot study of basal ganglia and thalamus structure by high dimensional mapping in children with Tourette syndrome

Background: Prior brain imaging and autopsy studies have suggested that structural abnormalities of the basal ganglia (BG) nuclei may be present in Tourette Syndrome (TS). These studies have focused mainly on the volume differences of the BG structures and not their anatomical shapes.  Shape differences of various brain structures have been demonstrated in other neuropsychiatric disorders using large-deformation, high dimensional brain mapping (HDBM-LD).  A previous study of a small sample of adult TS patients demonstrated the validity of the method, but did not find significant differences compared to controls. Since TS usually begins in childhood and adult studies may show structure differences due to adaptations, we hypothesized that differences in BG and thalamus structure geometry and volume due to etiological changes in TS might be better characterized in children. Objective: Pilot the HDBM-LD method in children and estimate effect sizes. Methods: In this pilot study, T1-weighted MRIs were collected in 13 children with TS and 16 healthy, tic-free, control children. The groups were well matched for age.  The primary outcome measures were the first 10 eigenvectors which are derived using HDBM-LD methods and represent the majority of the geometric shape of each structure, and the volumes of each structure adjusted for whole brain volume. We also compared hemispheric right/left asymmetry and estimated effect sizes for both volume and shape differences between groups. Results: We found no statistically significant differences between the TS subjects and controls in volume, shape, or right/left asymmetry.  Effect sizes were greater for shape analysis than for volume. Conclusion: This study represents one of the first efforts to study the shape as opposed to the volume of the BG in TS, but power was limited by sample size. Shape analysis by the HDBM-LD method may prove more sensitive to group differences