arXiv subjects

Hildeberto Jardon-Kojakhmetov

Publications and source records attributed to Hildeberto Jardon-Kojakhmetov.

5 recordsLinked to original sources

arXiv2026

Markov Chains and Random Walks with Memory on Hypergraphs: A Tensor-Based Approach

Many complex systems exhibit interactions that depend not only on pairwise connections, but also group structures and memory effects. To capture such effects, we develop a unified tensor framework for modeling higher-order Markov chains with memory. Our formulation introduces an even-order paired tensor that links folded and unfolded dynamics and characterizes their steady states and convergence. We further show that a Markov chain with memory can be approximated by a low-dimensional nonlinear tensor-based system and then provide a full system analysis. As an application, we define random walks on hypergraphs where memory naturally arises from the hyperedge structure, providing new tools for analyzing higher-order networks with time-dependent effects.

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arXiv2025

On Tensor-based Polynomial Hamiltonian Systems

It is known that a linear system with a system matrix A constitutes a Hamiltonian system with a quadratic Hamiltonian if and only if A is a Hamiltonian matrix. This provides a straightforward method to verify whether a linear system is Hamiltonian or whether a given Hamiltonian function corresponds to a linear system. These techniques fundamentally rely on the properties of Hamiltonian matrices. Building on recent advances in tensor algebra, this paper generalizes such results to a broad class of polynomial systems. As the systems of interest can be naturally represented in tensor forms, we name them tensor-based polynomial systems. Our main contribution is that we formally define Hamiltonian cubical tensors and characterize their properties. Crucially, we demonstrate that a tensor-based polynomial system is a Hamiltonian system with a polynomial Hamiltonian if and only if all associated system tensors are Hamiltonian cubical tensors-a direct parallel to the linear case. Additionally, we establish a computationally tractable stability criterion for tensor-based polynomial Hamiltonian systems. Finally, we validate all theoretical results through numerical examples and provide a further intuitive discussion.

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arXiv2021

Controlling Canard Cycles

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.

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arXiv2019

Geometric analysis of Oscillations in the Frzilator model

A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a strongly attracting limit cycle. This cycle corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model. In addition, we give a detailed description of the structure of the limit cycle, and the timescales along it.

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arXiv2019

A survey on the blow-up method for fast-slow systems

In this document we review a geometric technique, called \emph{the blow-up method}, as it has been used to analyze and understand the dynamics of fast-slow systems around non-hyperbolic points. The blow-up method, having its origins in algebraic geometry, was introduced in 1996 to the study of fast-slow systems in the seminal work by Dumortier and Roussarie \cite{dumortier1996canard}, whose aim was to give a geometric approach and interpretation of canards in the van der Pol oscillator. Following \cite{dumortier1996canard}, many efforts have been performed to expand the capabilities of the method and to use it in a wide range of scenarios. Our goal is to present in a concise and compact form those results that, based on the blow-up method, are now the foundation of the geometric theory of fast-slow systems with non-hyperbolic singularities. We cover fold points due to their great importance in the theory of fast-slow systems as one of the main topics. Furthermore, we also present several other singularities such as Hopf, pitchfork, transcritical, cusp, and Bogdanov-Takens, in which the blow-up method has been proved to be extremely useful. Finally, we survey further directions as well as examples of specific applied models, where the blow-up method has been used successfully.

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