Search Results (20)

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that $di{m}_K\left(L\right)=\mid L/K\mid <\infty$, the transcendence degree $trd(L/K)<\infty$ and $trd(L/K)=\infty$ with $difftrd(L/K)<\infty$, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension $L=K\left(G\right)$; then, there is a Galois correspondence between the intermediate fields $K\subset K^{\prime }\subset L$ and the subgroups $e\subset G^{\prime }\subset G$, provided that $K^{\prime }$ is stable under a Lie algebra $\Delta$ of invariant derivations of $L/K$. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article

To explore the nonlinear dynamics of a piezoelectric energy harvesting device, we consider the simultaneous parametric and external excitations. Based on Bernoulli–Euler beam theory, a new dynamic model is proposed taking into account the curvature of the beam, geometric and electro-mechanical coupling nonlinearities, and damping nonlinearity, with inextensible deformation. The system is discretized by using the Galerkin–Bubnov procedure and then is investigated by the optimal auxiliary functions method. Explicit analytical expressions of the approximate solutions are presented for a complex problem near the primary resonance. The main novelty of our approach relies on the presence of different auxiliary functions, the involvement of a few convergence-control parameters, the construction of the initial and first iteration, and much freedom in selecting the procedure for obtaining the optimal values of the convergence-control parameters. Our procedure proves to be very efficient, simple, easy to implement, and very accurate to solve a complicated nonlinear dynamical system. To study the stability of equilibrium points, the Routh–Hurwitz criterion is adopted. The Hopf and saddle node bifurcations are studied. Global stability is analyzed by the Lyapunov function, La Salle’s invariance principle, and Pontryagin’s principle with respect to the control variables. Full article

This article studies the general properties of the Stokes drift field. This name is commonly used for the correction added to the mean Eulerian velocity for describing the averaged transport of the material particles by the oscillating fluid flows. Stokes drift is widely known mainly in connection with another feature of oscillating flows known as steady streaming, which has been and remains the focus of a multitude of studies. However, almost nothing is known about Stokes drift in general, e.g., about its energy or helicity (Hopf’s invariant). We address these quantities for acoustic drift driven by simple sound waves with finite discrete Fourier spectra. The results discover that the mean drift energy is partly localized on a certain resonant set, which we have described explicitly. Moreover, the mean drift helicity turns out to be completely localized on the same set. We also present several simple examples to discover the effect of the power spectrum and positioning of the spectral atoms. It is revealed that tuning them can drastically change both resonant and non-resonant energies, zero the helicity, or even increase it unboundedly. Full article

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where $\dot{x}=a(y-x)$, $\dot{y}=cx-\sqrt[3]{x}z$, $\dot{z}=-bz+\sqrt[3]{x}y$, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where $\dot{x}=a(y-x)$, $\dot{y}=cx-xz$, $\dot{z}=-bz+xy$, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. Full article

The purpose of this work is to present a three-species harvesting food web model that takes into account the interactions of susceptible prey, infected prey, and predator species. Prey species are assumed to expand logistically in the absence of predator species. The Crowley–Martin and Beddington–DeAngelis functional responses are used by predators to consume both susceptible and infected prey. Additionally, susceptible prey is consumed by infected prey in the formation of a Holling type II response. Both prey species are considered when prey harvesting is taken into account. Boundedness, positivity, and positive invariance are considered in this study. The investigation covers all the equilibrium points that are biologically feasible. Local stability is evaluated by analyzing the distribution of eigen values, while global stability is evaluated using suitable Lyapunov functions. Also, Hopf bifurcation is analyzed at the harvesting rate ${\mathcal{H}}_1$. At the end, we evaluate the numerical solutions based on our findings. Full article

The paper studies the bifurcations that occur in the T-system, a 3D dynamical system symmetric in respect to the Oz axis. Results concerning some local bifurcations (pitchfork and Hopf bifurcation) are presented and our attention is focused on a special bifurcation, when the system has infinitely many equilibrium points. It is shown that, at the bifurcation limit, the phase space is foliated by infinitely many invariant surfaces, each of them containing two equilibrium points (an attractor and a saddle). For values of the bifurcation parameter close to the bifurcation limit, the study of the system’s dynamics is done according to the singular perturbation theory. The dynamics is characterized by mixed mode oscillations (also called fast-slow oscillations or oscillations-relaxations) and a finite number of equilibrium points. The specific features of the bifurcation are highlighted and explained. The influence of the pitchfork and Hopf bifurcations on the fast-slow dynamics is also pointed out. Full article

In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi–Cartan–Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point. Full article

We consider the general framework of perturbative quantum field theory for the pure Yang–Mills model. We give a more precise version of the Wick theorem using Hopf algebra notations for chronological products and not for Feynman graphs. Next, we prove that the Wick expansion property can be preserved for all cases in order $n=2$. However, gauge invariance is broken for chronological products of Wick submonomials. Full article

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product $A\#H$ is linked to the algebra of H invariants $A^H$ by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for $A/A^H$ to be Galois parallel in the case of H finite dimensional Hopf algebra. Full article

We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor. Full article

This paper considers the time taken for young predators to become adult predators as the delay and constructs a stage-structured predator–prey system with Holling III response and time delay. Using the persistence theory for infinite-dimensional systems and the Hurwitz criterion, the permanent persistence condition of this system and the local stability condition of the system’s coexistence equilibrium are given. Further, it is proven that the system undergoes a Hopf bifurcation at the coexistence equilibrium. By using Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium and the coexistence equilibrium are globally asymptotically stable, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Some numerical simulations are carried out to illustrate the main results. Full article

The ‘fluid mechanical sewing machine’ is a device in which a thin thread of viscous fluid falls onto a horizontal belt moving in its own plane, creating a rich variety of ‘stitch’ patterns depending on the fall height and the belt speed. This review article surveys the complex phenomenology of the patterns, their symmetries, and the mathematical models that have been used to understand them. The various patterns obey different symmetries that include (slightly imperfect) fore–aft symmetry relative to the direction of belt motion and invariance under reflection across a vertical plane containing the velocity vector of the belt, followed by a shift of one-half the wavelength. As the belt speed decreases, the first (Hopf) bifurcation is to a ‘meandering’ state whose frequency is equal to the frequency ${\Omega }_c$ of steady coiling on a motionless surface. More complex patterns can be studied using direct numerical simulation via a novel ‘discrete viscous threads’ algorithm that yields the Fourier spectra of the longitudinal and transverse components of the motion of the contact point of the thread with the belt. The most intriguing case is the ‘alternating loops’ pattern, the spectra of which are dominated by the first five multiples of ${\Omega }_c/3$. A reduced (three-degrees-of-freedom) model succeeds in predicting the sequence of patterns observed as the belt speed decreases for relatively low fall heights for which inertia in the thread is negligible. Patterns that appear at greater fall heights seem to owe their existence to weakly nonlinear interaction between different ‘distributed pendulum’ modes of the quasi-vertical ‘tail’ of the thread. Full article

We construct a class of knot solutions of the time-dependent gravitoelectromagnetic (GEM) equations in vacuum in the linearized gravity approximation by analogy with the Rañada–Hopf fields. For these solutions, the dual metric tensors of the bi-metric geometry of the gravitational vacuum with knot perturbations are given and the geodesic equation as a function of two complex parameters of the time-dependent GEM knots are calculated. By taking stationary potentials, which formally amount to particularizing to time-independent GEM equations, we obtain a set of stationary fields subjected to constraints from the time-dependent GEM knots. Finally, the Landau–Lifshitz pseudo-tensor and a scalar invariant of the static fields are computed. Full article

Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov–Hopf bifurcations at similar parameter values as the original DDE model. Full article