Qualitative Theory Of Differential Equations Research Articles

Phase-plane analysis to an “anisotropic” higher-order traffic flow model

The qualitative theory of differential equations is applied to investigate the traveling wave solution to an “anisotropic” higher-order viscous traffic flow model under the Lagrange coordinate system. The types and stabilities of the equilibrium points are discussed in the phase plane. Through the numerical simulation, the overall distribution structures of trajectories are drawn to analyze the relation between the phase diagram and the selected conservative solution variables, and the influences of the parameters on the system are studied. The limit-circle, limit circle-spiral point, saddle-spiral point and saddle-nodal point solutions are obtained. These steady-state solutions provide good explanation for the phenomena of the oscillatory and homogeneous congestions in real-world traffic.

Chaotic dynamics of two coaxially-nested cylindrical shells reinforced by two beams

Non-linear dynamics and contact interactions of beam-shell structures composed of two closed cylindrical shells which are coaxially nested and reinforced by two beams located symmetrically on the shell external perimeter is studied. In the present work, clearances between the mentioned structural members are taken into account, two beams are subjected to harmonic loads, and the dissipation factors are neglected.3D PDEs governing non-linear dynamics of beams and shells within the geometric theory of Novozhilov are employed, whereas the contact pressure is defined through Kantor's model. PDEs are reduced to ODEs by means of the FEM (finite element method), and the solution convergence is validated through different numbers of finite elements located along the structural members thickness and by employment of the Runge principle with respect to spatial coordinates. The Cauchy problem is solved by the explicit integration (Euler method), which allows one to carry out the computation without the need to define solutions in a few initial points.Analysis of vibrations, including contact interactions, is realized with the use of methods of non-linear dynamics and the qualitative theory of differential equations, time histories/signals, phase portraits, Poincarè sections, Fourier spectra, wavelet-based analysis as well as the Lyapunov exponents.

Oscillation criteria for generalized Liénard type system

‎In this work we use qualitative theory of differential equations to study the qualitative behavior of the solutions of a generalized Liénard‎‎ system‎. ‎Under quite general assumptions we present some sharp conditions under which the‎‎solutions of the system are oscillatory‎. ‎Some examples are presented to illustrate our results‎.

Mathematical encounters

This evocative essay focuses on mathematical activities witnessed by the author along 1962-64 at IMPA. The list of research problems proposed in September 1962 by Mauricio Peixoto at the Seminar on the Qualitative Theory of Differential Equations is pointed out as a landmark for the beginning of the systematic research in the Qualitative Theory of Differential Equations and Dynamical Systems in Brazil.

Contact interaction of flexible Timoshenko beams with small deflections

In this work chaotic dynamics contact interaction of two flexible Tymoshenko beams, under the action of a transversal alternating load is investigated. The contact interaction of the beams is taken into account by the Kantor model. The geometric nonlinearity is taken into account by the model of T. von Karman. The system of partial differential equations of the twelfth order reduces to the system of ordinary differential equations by the method of finite differences of the second order. The resulting system by methods of Runge-Kutta type of the second, fourth and eighth orders was solved. Our theoretical/numerical analysis is supported by methods of nonlinear dynamics and the qualitative theory of differential equations. Chaotic vibrations of two flexible beams of Timoshenko were investigated and the optimal step values over the spatial coordinate and the time steps for the numerical experiment were found. Convergence for all applicable numerical methods have been achieved and shown that chaotic signals are true.

CFD Analysis of 2D Unsteady Flow Past a Square Cylinder at Low Reynolds Numbers

A study of the behaviour of flow past a square cylinder for Reynolds numbers 10 and 20 is presented. Open source software Navier2d in Matlab is used in this study. The investigation starts from a uniform initial mesh and then refine the initial mesh using a mesh refinement method which was proposed based on both qualitative theory of differential equations and the finite volume method implemented in Navier2d. The horizontal and vertical velocity component profiles and pressures are shown on the once refined meshes. The comparisons between the profiles and pressures are conducted to show the variations from Reynolds number 10 to 20. The twice refined meshes are also presented and these refined meshes provide the information where the behaviour of flow is complex.

Bifurcations and Exact Traveling Wave Solutions of the Celebrated Green–Naghdi Equations

In this paper, we study the bifurcations and exact traveling wave solutions of the celebrated Green–Naghdi equations by using the qualitative theory of differential equations and the bifurcation theory of dynamical systems. We obtain all possible phase portraits of bifurcations of the system under various conditions about the parameters associated with the planar dynamical system. Then we show the existence of traveling wave solutions including solitary wave solutions, blow-up solutions, periodic wave solutions and periodic blow-up solutions, and give their exact explicit expressions. These results can help to understand the dynamical behavior of the traveling wave solutions of the system.

Chaotic interaction dynamics of three structures: Two cylindrical shells nested into each other and their reinforcing local rib

This paper studies the chaotic dynamics of two cylindrical shells nested into each other with a gap and their reinforcing beam, also with a gap, which is subjected to a distributed alternating load. The problem is solved using methods of nonlinear dynamics and the qualitative theory of differential equations. The Novozhilov equations for geometrically nonlinear structures are used as the governing equations. Contact pressure is determined by Kantor’s method. Using finite elements in spatial variables, the partial differential equations for the beam and shells are reduced to the Cauchy problem, which is solved by explicit integration (Euler’s method). The chaotic synchronization of this system is studied.

Chaotic dynamics of size dependent Timoshenko beams with functionally graded properties along their thickness

Chaotic dynamics of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Kármán geometric nonlinearity. We assume that the beam properties are graded along the thickness direction. The influence of size-dependent and functionally graded coefficients on the vibration characteristics, scenarios of transition from regular to chaotic vibrations as well as a series of static problems with an emphasis put on the load-deflection behavior are studied. Our theoretical/numerical analysis is supported by methods of nonlinear dynamics and the qualitative theory of differential equations supplemented by Fourier and wavelet spectra, phase portraits, and Lyapunov exponents spectra estimated by different algorithms, including Wolf’s, Rosenstein’s, Kantz’s, and neural networks. We have also detected and numerically validated a general scenario governing transition into chaotic vibrations, which follows the classical Ruelle-Takens-Newhouse scenario for the considered values of the size-dependent and grading parameters.

Elipsoid Monge’a i konfiguracje linii krzywiznowych na powierzchniach i przestrzeniach euklidesowych – uwagi historyczne

This is an essay on the historical landmarks leading to the study of principal confgurations on surfaces in R^3 , their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations, founded by Poincare in 1881. Recent development concerning the space R^4 are mentioned. Two open problems are proposed at the end.

On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

This paper studies the nonnegativity and local and global stability properties of the solutions of a newly proposed SEIADR model which incorporates asymptomatic and dead-infective subpopulations into the standard SEIR model and, in parallel, it incorporates feedback vaccination plus a constant term on the susceptible and feedback antiviral treatment controls on the symptomatic infectious subpopulation. A third control action of impulsive type (or “culling”) consists of the periodic retirement of all or a fraction of the lying corpses which can become infective in certain diseases, for instance, the Ebola infection. The three controls are allowed to be eventually time varying and contain a total of four design control gains. The local stability analysis around both the disease-free and endemic equilibrium points is performed by the investigation of the eigenvalues of the corresponding Jacobian matrices. The global stability is formally discussed by using tools of qualitative theory of differential equations by using Gauss-Stokes and Bendixson theorems so that neither Lyapunov equation candidates nor the explicit solutions are used. It is proved that stability holds as a parallel property to positivity and that disease-free and the endemic equilibrium states cannot be simultaneously either stable or unstable. The periodic limit solution trajectories and equilibrium points are analyzed in a combined fashion in the sense that the endemic periodic solutions become, in particular, equilibrium points if the control gains converge to constant values and the control gain for culling the infective corpses is asymptotically zeroed.

Dynamical behaviours and exact travelling wave solutions of modified generalized Vakhnenko equation

By using the bifurcation theory of planar dynamical systems and the qualitative theory of differential equations, we studied the dynamical behaviours and exact travelling wave solutions of the modified generalized Vakhnenko equation (mGVE). As a result, we obtained all possible bifurcation parametric sets and many explicit formulas of smooth and non-smooth travelling waves such as cusped solitons, loop solitons, periodic cusp waves, pseudopeakon solitons, smooth periodic waves and smooth solitons. Moreover, we provided some numerical simulations of these solutions.

Accuracy verification of a 2D adaptive mesh refinement method for incompressible or steady flow

Locating accurate centres of vortices is one of the accuracy measures for computational methods in fluid flow and the lid-driven cavity flows are widely used as benchmarks. This paper analyses the accuracy of an adaptive mesh refinement method using 2D steady incompressible lid-driven cavity flows. The adaptive mesh refinement method performs mesh refinement based on the numerical solutions of Navier–Stokes equations solved by Navier2D, a vertex centred Finite Volume code that uses the median dual mesh to form the Control Volumes (CVs) about each vertex. The accuracy of the refined meshes is demonstrated by the centres of vortices obtained in the benchmarks being contained in the twice refined cells. The adaptive mesh refinement method investigated in this paper is proposed based on the qualitative theory of differential equations. Theoretically infinite refinements can be performed on an initial mesh. Practically we can stop the process of refinement based on tolerance conditions. The method can be applied to find accurate numerical solutions of any mathematical models containing the continuity equation for incompressible fluid or steady-state fluid flow.

Hopf Bifurcation in Coastal Ecogeosystems

The qualitative theory of differential equations (catastrophe theory) is used to analyze the optimal preservation of the ecological state of an object. As the object, we consider a seacoast that is subject to permanent destruction by waves and is fed to maintain its initial state. We model the maintenance of the equilibrium state of ecological system in some optimal mode. Unlike the well-known approaches, such model takes into account essentially nonlinear effects and control by means of beach feed, which can be interpreted as feedback. The analysis is carried out by the methods from stability theory. Characteristics of the limit cycle are obtained and its stability is analyzed.

To the History of the Appearance of the Notion of the ε-Entropy of an Authomorphism of a Lebesgue Space and (ε,T)-Entropy of a Dynamical System with Continuous Time

The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.

Noisy contact interactions of multi-layer mechanical structures coupled by boundary conditions

In this work mathematical models of temporal part of chaos at chosen spatial locations of a plate locally reinforced by ribs taking into account an interplay of their interactions are derived and studied numerically for the most relevant dynamical parameters. In addition, an influence of the additive external noise on chaotic vibrations of multi-layer beam–plate structures coupled only by boundary conditions is investigated. We illustrate and discuss novel nonlinear phenomena of the temporal regular and chaotic contact/no-contact dynamics with the help of Morlet wavelets and Fourier analysis. We show how the additive white noise cancels deterministic chaos close to the boundary of chaotic region in the space of parameters, and we present windows of on/off switching of the frequencies during the contact dynamics between structural members.In order to solve the mentioned design type nonlinear problem we apply methods of qualitative theory of differential equations, the Bubnov–Galerkin method in higher approximations, the Runge–Kutta methods of 4th, 6th and 8th order, as well as the computation and analysis of the largest Lyapunov exponent (Benettin׳s and Wolf׳s algorithms are used). The agreement of outcomes of all applied qualitatively different numerical approaches validate our simulation results. In particular, we have illustrated that the Fourier analysis of the studied mechanical structures may yield erroneous results, and hence the wavelet-based analysis is used to investigate chaotic dynamics in the system parameter space.

Lyapunov functions and strict stability of Caputo fractional differential equations

One of the main properties studied in the qualitative theory of differential equations is the stability of solutions. The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to fractional differential equations causes many difficulties. In this paper a new definition (based on the Caputo fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo fractional differential equation is introduced. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory.

Optimal weight based on energy imbalance and utility maximization

This paper investigates the optimal weight for both male and female using energy imbalance and utility maximization. Based on the difference of energy intake and expenditure, we develop a state equation that reveals the weight gain from this energy gap. We ​construct an objective function considering food consumption, eating habits and survival rate to measure utility. Through applying mathematical tools from optimal control methods and qualitative theory of differential equations, we obtain some results. For both male and female, the optimal weight is larger than the physiologically optimal weight calculated by the Body Mass Index (BMI). We also study the corresponding trajectories to steady state weight respectively. Depending on the value of a few parameters, the steady state can either be a saddle point with a monotonic trajectory or a focus with dampened oscillations.

Nonlocal Cauchy problems close to an asymptotically stable equilibrium point

In this work we investigate the existence of solutions for semilinear Cauchy problems with nonlocal initial conditions in the neighborhood of an asymptotically stable equilibrium point of the evolution equation. Using Granas' continuation principle for contractive maps and the qualitative theory of differential equations in Banach spaces, under mild assumptions, we prove the existence of a unique solution. We also show that the main abstract result can be applied to nonlocal initial boundary value problems for reaction–diffusion equations with non-convex nonlinearities.

A new qualitative proof of a result on the real Jacobian conjecture.

Let F= (f, g) : R2 → R2 be a polynomial map such that det DF(x) is different from zero for all x ∈ R2. We assume that the degrees of f and g are equal. We denote by f and G the homogeneous part of higher degree of f and g, respectively. In this note we provide a proof relied on qualitative theory of differential equations of the following result: If f and g do not have real linear factors in common, then F is injective.