Hildeberto Jardón-Kojakhmetov | University of Groningen (original) (raw)

Papers by Hildeberto Jardón-Kojakhmetov

arXiv (Cornell University), Apr 21, 2023

arXiv (Cornell University), Jan 17, 2023

Cornell University - arXiv, Mar 10, 2021

Cornell University - arXiv, Jun 15, 2022

This paper studies a discrete-time time-varying multi-layer networked SIWS (susceptible-infected-... more This paper studies a discrete-time time-varying multi-layer networked SIWS (susceptible-infected-water-susceptible) model with multiple resources under both single-virus and competing multi-virus settings. Besides the human-to-human interaction, we also consider that the disease can diffuse on different types of medium. We use resources to refer to any media through which the pathogen of a virus can spread, and do not restrict the resource only to be water. In the single-virus case, we give a full analysis of the system's behaviour related to its healthy state and endemic equilibrium. In the multi-virus case, we show analytically that different equilibria appear driven by the competition among all viruses. We also show that some analytical results of the time-invariant system can be expanded into time-varying cases. Finally, we illustrate the results through some simulations.

Fast-slow systems with three slow variables and gradient structure in the fast variables have, ge... more Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.

Journal of Dynamical and Control Systems, 2021

Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singu... more Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

2017 American Control Conference (ACC), 2017

Non-hyperbolic points of slow-fast systems (also known as singularly perturbed ordinary different... more Non-hyperbolic points of slow-fast systems (also known as singularly perturbed ordinary differential equations) are responsible for many interesting behavior such as relaxation oscillations, canards, mixed-mode oscillations, etc. Recently, the authors have proposed a control strategy to stabilize non-hyperbolic points of planar slow-fast systems. Such strategy is based on geometric desingularization, which is a well suited technique to analyze the dynamics of slow-fast systems near non-hyperbolic points. This technique transforms the singular perturbation problem to an equivalent regular perturbation problem. This papers treats the nonlinear adaptive stabilization problem of slow-fast systems. The novelty is that the point to be stabilized is non-hyperbolic. The controller is designed by combining geometric desingularization and Lyapunov based techniques. Through the action of the controller, we basically inject a normally hyperbolic behavior to the fast variable. Our results are ex...

IEEE Control Systems Letters, 2018

Through recent research combining the Geometric Desingularization or blow-up method and classical... more Through recent research combining the Geometric Desingularization or blow-up method and classical control tools, it has been possible to locally stabilize non-hyperbolic points of singularly perturbed control systems. In this letter we propose a simple method to enlarge the region of attraction of a nonhyperbolic point in the aforementioned setting by expanding the geometric analysis around the singularity. In this way, we can synthesize improved controllers that stabilize non-hyperbolic points within a large domain of attraction. Our theoretical results are showcased in a couple of numerical examples.

Nonlinear Analysis: Real World Applications, 2021

We study fast-slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time... more We study fast-slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behavior is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast-slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov's method, we are able to not only analyze the stability of the endemic equilibria but also to show that in some of the models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model, for which we provide a detailed description of its dynamics by combining analytic and numerical techniques.

IEEE Control Systems Letters, 2017

Take-down policy If you believe that this document breaches copyright please contact us providing... more Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Automatica, 2019

In this document, we deal with the local asymptotic stabilization problem of a class of slow-fast... more In this document, we deal with the local asymptotic stabilization problem of a class of slow-fast systems (or singularly perturbed Ordinary Differential Equations). The systems studied here have the following properties: (1) they have one fast and an arbitrary number of slow variables, and (2) they have a non-hyperbolic singularity at the origin of arbitrary degeneracy. Our goal is to stabilize such a point. The presence of the aforementioned singularity complicates the analysis and the controller design. In particular, the classical theory of singular perturbations cannot be used. We propose a novel design based on geometric desingularization, which allows the stabilization of a non-hyperbolic point of singularly perturbed control systems. Our results are exemplified on a didactic example and on an electric circuit.

Proceedings. Mathematical, physical, and engineering sciences, 2018

We develop a tool based on bifurcation analysis for parameter-robustness analysis for a class of ... more We develop a tool based on bifurcation analysis for parameter-robustness analysis for a class of oscillators and, in particular, examine a biochemical oscillator that describes the transition phase between social behaviours of myxobacteria. Myxobacteria are a particular group of soil bacteria that have two dogmatically different types of social behaviour: when food is abundant they live fairly isolated forming swarms, but when food is scarce, they aggregate into a multicellular organism. In the transition between the two types of behaviours, spatial wave patterns are produced, which is generally believed to be regulated by a certain biochemical clock that controls the direction of myxobacteria's motion. We provide a detailed analysis of such a clock and show that, for the proposed model, there exists some interval in parameter space where the behaviour is robust, i.e. the system behaves similarly for all parameter values. In more mathematical terms, we show the existence and con...

Applied Mathematics Letters, 2017

For continuously differentiable vector fields, we characterize the ω limit set of a trajectory co... more For continuously differentiable vector fields, we characterize the ω limit set of a trajectory converging to a compact curve Γ ⊂ R n. In particular, the limit set is either a fixed point or a continuum of fixed points if Γ is a simple open curve; otherwise, the limit set can in addition be either a closed orbit or a number of fixed points with compatibly oriented orbits connecting them. An implication of the result is a tightened-up version of the Poincaré-Bendixson theorem.

IFAC-PapersOnLine, 2016

In this paper we explore the methodology of model order reduction based on singular perturbations... more In this paper we explore the methodology of model order reduction based on singular perturbations for a flexible-joint robot within the port-Hamiltonian framework. We show that a flexible-joint robot has a port-Hamiltonian representation which is also a singularly perturbed ordinary differential equation. Moreover, the associated reduced slow subsystem corresponds to a port-Hamiltonian model of a rigid-joint robot. To exploit the usefulness of the reduced models, we provide a numerical example where an existing controller for a rigid robot is implemented.

Applied Mathematics, 2016

This paper presents two contributions to the stability analysis of periodic systems modeled by a ... more This paper presents two contributions to the stability analysis of periodic systems modeled by a Hill equation: The first is a new method for the computation of the Arnold Tongues associated to a given Hill equation which is based on the discretization of the latter. Using the proposed method, a vibrational stabilization is performed by a change in the periodic function which guarantees stability, given that the original equation has unbounded solutions. The results are illustrated by some examples.

Journal of Differential Equations, 2016

This paper studies a slow-fast system whose principal characteristic is that the slow manifold is... more This paper studies a slow-fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results.

Journal of Differential Equations, 2014

We study generic constrained differential equations (CDEs) with three parameters, thereby extendi... more We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal forms of CDEs under topological equivalence. Generic CDEs are important in the study of slow-fast (SF) systems. Many properties and the characteristic behavior of the solutions of SF systems can be inferred from the corresponding CDE. Therefore, the results of this paper show a first approximation of the flow of generic SF systems with three slow variables.

Veel natuurlijke fenomenen spelen zich af op verschillende tijdschalen. Denk bijvoorbeeld aan de ... more Veel natuurlijke fenomenen spelen zich af op verschillende tijdschalen. Denk bijvoorbeeld aan de hartslag, zenuwactiviteit, scheikundige reacties of het weer. Dergelijke fenomenen kunnen daarom worden gemodelleerd door middel van zogenaamde “slow-fast” systemen. Dit zijn gewone differentiaalvergelijkingen die op een singuliere manier afhangen van een kleine parameter. Door deze parameter gelijk aan nul te stellen ontstaat een differentiaalvergelijking met een algebraische beperking. De Groningse wiskundige Floris Takens (1940-2010) heeft in 1975 belangrijke bijdragen geleverd aan de theorie van differentiaalvergelijkingen met algebraische beperkingen en hun relatie tot slow-fast systemen. Zijn resultaten zijn in het bijzonder bruikbaar als men de meer gecompliceerde dynamica van slow-fast systemen wil bestuderen. Dit proefschrift is een studie naar de dynamica en locale eigenschappen van slow-fast systemen en de daaraan gerelateerde differentiaalvergelijkingen met algebraische beper...