Integration of differential equations by C∞\mathcal{C}^{\infty}-structures

Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated with $\mathcal{C}^{\infty}$-structures is used to integrate to a Lotka-Volterra model and several differential equations that lack sufficient Lie point symmetries and cannot be solved using conventional methods

Algebraic entropy for systems of quad equations

In this work I discuss briefly the calculation of the algebraic entropy for systems of quad equations. In particular, I observe that since systems of multilinear equations can have algebraic solution, in some cases one might need to restrict the direction of evolution only to the pair of vertices yielding a birational evolution. Some examples from the exiting literature are presented and discussed within this framework.Comment: 24 pages (amsart style), 5 figures. This paper is dedicated to the memory of Prof. Decio Lev

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Advances in Fundamental Physics

This Special Issue celebrates the opening of a new section of the journal Foundation: Physical Sciences. Theoretical and experimental studies related to various areas of fundamental physics are presented in this Special Issue. The published papers are related to the following topics: dark matter, electron impact excitation, second flavor of hydrogen atoms, quantum antenna, molecular hydrogen, molecular hydrogen ion, wave pulses, Brans-Dicke theory, hydrogen Rydberg atom, high-frequency laser field, relativistic mean field formalism, nonlocal continuum field theories, parallel universe, charge exchange, van der Waals broadening, greenhouse effect, strange and unipolar electromagnetic pulses, quasicrystals, Wilhelm-Weber’s electromagnetic force law, axions, photoluminescence, neutron stars, gravitational waves, diatomic molecular spectroscopy, information geometric measures of complexity. Among 21 papers published in this Special Issue, there are 5 reviews and 16 original research papers

Réduire la dimension des systèmes complexes : un regard sur l'émergence de la synchronisation

Les systèmes complexes se caractérisent par l’émergence de phénomènes macroscopiques qui ne s’expliquent pas uniquement par les propriétés de leurs composantes de base. La synchronisation est l’un de ces phénomènes par lequel les interactions entre des oscillateurs engendrent des mouvements collectifs coordonnés. Une représentation sous forme de graphe permet de modéliser précisément les interactions, alors que les oscillations se décrivent par des dynamiques non linéaires. Étant donné le lien subtil entre le graphe et la dynamique des oscillateurs, il est difficile de prédire l’émergence de la synchronisation. L’objectif de ce mémoire est de développer de nouvelles méthodes pour simplifier les systèmes complexes d’oscillateurs afin de révéler les mécanismes engendrant la synchronisation. À cette fin, nous introduisons des notions de la théorie des graphes et des systèmes dynamiques pour définir la synchronisation sur des bases mathématiques. Nous présentons ensuite des approches existantes sophistiquées pour réduire la dimension de dynamiques d’oscillateurs. Ces techniques sont toutefois limitées lorsque les dynamiques évoluent sur des graphes plus complexes. Nous développons alors une technique originale—spécialement adaptée pour les graphes aux propriétés plus hétérogènes—pour réduire la dimension de dynamiques non linéaires. En plus de généraliser des approches récentes, notre méthode dévoile plusieurs défis théoriques reliés à la simplification d’un système complexe. Entre autres, la réduction de la dimension du système se bute à une trichotomie : il faut favoriser la conservation des paramètres dynamiques, des propriétés locales du graphe ou des propriétés globales du graphe. Finalement, notre méthode permet de caractériser mathématiquement et numériquement l’émergence d’états exotiques de synchronisation.Complex systems are characterized by the emergence of macroscopic phenomena that cannot be explained by the properties of their basic constituents. Synchronization is one of these phenomena in which the interactions between oscillators generate coordinate collective behaviors. Graphs allow a precise representation of the interactions, while nonlinear dynamics describe the oscillations. Because of the subtle relationship between graphs and dynamics of oscillators, it is challenging to predict the emergence of synchronization. The goal of this master’s thesis is to develop new methods for simplifying complex systems of oscillators in order to reveal the mechanism causing synchronization. To this end, we introduce notions of graph theory and dynamical systems to define synchronization on sound mathematical basis. We then present existing sophisticated approaches for reducing the dimension of oscillator dynamics. Yet, the efficiency of these techniques is limited for dynamics on complex graphs. We thus develop an original method—specially adapted for graphs with heterogeneous properties—for reducing the dimensions of nonlinear dynamics. Our method generalizes previous approaches and uncovers multiple challenges related to the simplification of complex systems. In particular, the dimension reduction of a system comes up against a trichotomy: one must choose to conserve either the dynamical parameters, the local properties of the graph, or the global properties of the graph. We finally use our method to characterize mathematically and numerically the emergence of exotic synchronization states