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

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Papers by Hildeberto Jardón-Kojakhmetov

Research paper thumbnail of A topological perspective on singular canards for critical sets with transverse intersections

arXiv (Cornell University), Apr 21, 2023

Research paper thumbnail of Stable Chimera States: A Geometric Singular Perturbation Approach

arXiv (Cornell University), Jan 17, 2023

Research paper thumbnail of Control of a flexible-joint manipulator with only position measurements

Research paper thumbnail of Slow-Fast Torus Knots

Cornell University - arXiv, Mar 10, 2021

Research paper thumbnail of Discrete-time Layered-network Epidemics Model with Time-varying Transition Rates and Multiple Resources

Cornell University - arXiv, Jun 15, 2022

Research paper thumbnail of The hyperbolic umbilic singularity in fast-slow systems

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.

Research paper thumbnail of Controlling Canard Cycles

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.

Research paper thumbnail of Slow-fast port-Hamiltonian mechanical systems

Research paper thumbnail of Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

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...

Research paper thumbnail of Improving the Region of Attraction of a Non-Hyperbolic Point in Slow-Fast Systems With One Fast Direction

IEEE Control Systems Letters, 2018

Research paper thumbnail of A geometric analysis of the SIR, SIRS and SIRWS epidemiological models

Nonlinear Analysis: Real World Applications, 2021

Research paper thumbnail of Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

IEEE Control Systems Letters, 2017

Research paper thumbnail of Stabilization of a class of slow–fast control systems at non-hyperbolic points

Research paper thumbnail of Parameter-robustness analysis for a biochemical oscillator model describing the social-behaviour transition phase of myxobacteria

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...

Research paper thumbnail of Limit sets within curves where trajectories converge to

Applied Mathematics Letters, 2017

Research paper thumbnail of Model reduction of a flexible-joint robot: a port-Hamiltonian approach

Research paper thumbnail of Vibrational Stabilization by Reshaping Arnold Tongues: A Numerical Approach

Applied Mathematics, 2016

Research paper thumbnail of Analysis of a slow–fast system near a cusp singularity

Journal of Differential Equations, 2016

Research paper thumbnail of Polynomial normal forms of constrained differential equations with three parameters

Journal of Differential Equations, 2014

Research paper thumbnail of Classification of constrained differential equations embedded in the theory of slow fast systems: Ak singularities and geometric desingularization

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...

Research paper thumbnail of A topological perspective on singular canards for critical sets with transverse intersections

arXiv (Cornell University), Apr 21, 2023

Research paper thumbnail of Stable Chimera States: A Geometric Singular Perturbation Approach

arXiv (Cornell University), Jan 17, 2023

Research paper thumbnail of Control of a flexible-joint manipulator with only position measurements

Research paper thumbnail of Slow-Fast Torus Knots

Cornell University - arXiv, Mar 10, 2021

Research paper thumbnail of Discrete-time Layered-network Epidemics Model with Time-varying Transition Rates and Multiple Resources

Cornell University - arXiv, Jun 15, 2022

Research paper thumbnail of The hyperbolic umbilic singularity in fast-slow systems

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.

Research paper thumbnail of Controlling Canard Cycles

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.

Research paper thumbnail of Slow-fast port-Hamiltonian mechanical systems

Research paper thumbnail of Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

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...

Research paper thumbnail of Improving the Region of Attraction of a Non-Hyperbolic Point in Slow-Fast Systems With One Fast Direction

IEEE Control Systems Letters, 2018

Research paper thumbnail of A geometric analysis of the SIR, SIRS and SIRWS epidemiological models

Nonlinear Analysis: Real World Applications, 2021

Research paper thumbnail of Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

IEEE Control Systems Letters, 2017

Research paper thumbnail of Stabilization of a class of slow–fast control systems at non-hyperbolic points

Research paper thumbnail of Parameter-robustness analysis for a biochemical oscillator model describing the social-behaviour transition phase of myxobacteria

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...

Research paper thumbnail of Limit sets within curves where trajectories converge to

Applied Mathematics Letters, 2017

Research paper thumbnail of Model reduction of a flexible-joint robot: a port-Hamiltonian approach

Research paper thumbnail of Vibrational Stabilization by Reshaping Arnold Tongues: A Numerical Approach

Applied Mathematics, 2016

Research paper thumbnail of Analysis of a slow–fast system near a cusp singularity

Journal of Differential Equations, 2016

Research paper thumbnail of Polynomial normal forms of constrained differential equations with three parameters

Journal of Differential Equations, 2014

Research paper thumbnail of Classification of constrained differential equations embedded in the theory of slow fast systems: Ak singularities and geometric desingularization

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...