Qualitative Theory Of Differential Equations Research Articles

Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

On infinite spectra of oscillation exponents of third order linear differential equations

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica.

INVERTEBRATES AND CATTLE POPULATION DYNAMICS IN A GRASSLAND ENVIRONMENT: A NONLINEAR INTER-SPECIFIC COMPETITION MODEL

In the proposed study, a nonlinear model is developed to explore the interactive dynamics between cattle and invertebrates when they coexist in a grassland system and compete with one another for the same resource — the grass biomass. The constructed model is theoretically investigated using the qualitative theory of differential equations to show the system’s rich dynamical properties, which are crucial for maintaining the ecosystem’s balance in grasslands. The qualitative findings show that, depending on the parameter combinations, the system not only displays stability of many equilibrium states but also experiences transcritical and Hopf bifurcations. The model results support the idea that inter-specific competition between cattle and invertebrates does not always produce regular dynamic patterns but may also produce periodic and destabilizing patterns. The model’s outputs may assist in striking a balance between pasture and natural grass biomass in grassland with the invertebrates.

First-Stage Dynamics of the Immune System and Cancer

The innate immune system is the first line of defense against pathogens. Its composition includes barriers, mucus, and other substances as well as phagocytic and other cells. The purpose of the present paper is to compare tissues with regard to their immune response to infections and to cancer. Simple ideas and the qualitative theory of differential equations are used along with general principles such as the minimization of the pathogen load and economy of resources. In the simplest linear model, the annihilation rate of pathogens in any tissue should be greater than the pathogen’s average replication rate. When nonlinearities are added, a stability condition emerges, which relates the strength of regular threats, barrier height, and annihilation rate. The stability condition allows for a comparison of immunity in different tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response. The way the tissue responds to an infection shows a correlation with the way it responds to cancer. The results of this paper are formulated in the form of precise statements in such a way that they could be checked by present-day quantitative immunology.

How to combat atmospheric carbon dioxide along with development activities? A mathematical model

Depletion in forest density due to development activities is one of the reasons for the elevated level of carbon dioxide (CO2). In this scenario, the plantation of leafy trees, which absorb more CO2 compared to regular trees, under a planned strategy may help to attain the desired level of atmospheric carbon dioxide. We have formulated a non-linear mathematical model concerning the strategy of maintaining the atmospheric level of CO2 along with development activities and analyzed the proportion of deforested land needed for the plantation of leafy trees. This strategy will ensure that the absorption of carbon dioxide remains at its previous level. To study the long-term behavior of the formulated model system, we employ the qualitative theory of differential equations. We have derived sufficient conditions under which all the considered dynamical variables settle to their equilibrium level. After analyzing the formulated model system, we find that the system undergoes Hopf and transcritical bifurcations with respect to the deforestation rate. Furthermore, numerical simulations have been carried out to support the analytically obtained results. Through simulation, we have determined the proportion of cleared land that should be utilized for the plantation of leafy trees to maintain the level of atmospheric CO2.

A mechanism for detecting normally hyperbolic invariant tori in differential equations

Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the first non-vanishing higher order averaged function. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations.

On Oscillations in a Gene Network with Diffusion

We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems.

Applying Optimal Control Theory to Microeconomic Problems

Nowadays, the theory of consumption, as well as any other field of economic theory, is virtually impossible without applying extensively mathematical modeling. The article explores from microeconomics perspective two models in which optimal distribution of an individual’s income is achieved, thus maximizing total discounted consumption over a certain period of time. The qualitative analysis of the models is carried out with such tools as the classical maximum principle, which corresponds to necessary conditions of optimality of the first order, and the qualitative theory of differential equations. The purpose of the study is to demonstrate efficacy of these tools when solving practical problems of microeconomics. Applying mathematical apparatus for solving such problems has always been relevant. The extremal of the maximum principle with positional control has been found in one of the models considered in the article. Both models have been analyzed and the economic interpretation of the obtained results has been given. In the first model, an individual's choice of either recommended strategies is determined by the ratio of the consumption discount rate and the growth rate of their income. The second model, being a modification of the first one, implies that the optimal strategy is chosen according to the initial level of the subject’s affluence.

On essential values of Sergeev's frequencies and exponents of oscillation for solutions of a third-order linear differential periodic equation

In this paper, we study various types of Sergeev's frequencies and exponents of oscillation for solutions of linear homogeneous differential equations with continuous bounded coefficients. For any preassigned natural number $N$, a periodic third-order linear differential equation is constructively built in this paper, which has the property that its upper and lower Sergeev frequency spectra of strict signs, zeros and roots, as well as the spectra of all upper and lower strong and weak oscillation indices of strict and non-strict signs, zeros, roots and hyperroots contain the same set, consisting of $N$ different essential values, both metrically and topologically. Moreover, all these values are implemented on the same set of solutions of the constructed equation, that is, for each solution from this set, all the frequencies listed above and the oscillation exponents coincide with each other. When constructing the indicated equation and proving the required results, analytical methods of the qualitative theory of differential equations were used, in particular, methods of the theory of perturbations of solutions of linear differential equations, as well as the author's technique for controlling the fundamental system of solutions of such equations in one particular case.

On the application of the qualitative theory of differential equations to a problem of heat and mass transfer

On the application of the qualitative theory of differential equations to a problem of heat and mass transfer

A mathematical model study on plant root pest management

<abstract><p>Unlike conventional methods of pests control, introducing in an appropriate mathematical model can contribute a batter performance on pests control with higher efficiency while lest damage to ecosystem. To fill the research gap on plant root pest control, we propose a plant root pest management model with state pulse feedback control. Firstly, the stability of the equilibrium point of the model (1.3) is analyzed by using the linear approximate equation, given that the only positive equilibrium point of model (1.3) is globally asymptotically stable. Moreover, the existence and uniqueness of order 1 periodic solutions of model (1.3) are discussed in detail according to the geometric theory of semi-continuous dynamical systems, successor functions method and the qualitative theory of differential equations. Finally, with further analysis in different methods, the asymptotic stability of the order 1 periodic solution of model (1.3) is obtained by using Similar Poincare Criterion or interval set theorem. The results show that this model can effectively control the number of pests below the economic level of damage.</p></abstract>

ASYMPTOTIC BEHAVIOR OF Pω(Y0, Y1,±∞)-SOLUTIONS OF THE SECOND ORDER DIFFERENTIAL EQUATIONS WITH THE PRODUCT OF DIFFERENT TYPES OF NONLINEARITIES FROM AN UNKNOWN FUNCTION AND ITS FIRST DERIVATIVE

The task of establishing the conditions of existence, as well as finding asymptotic images of solutions of differential equations, which contain nonlinearities of various types in the righthand side, is one of the most important tasks of the qualitative theory of differential equations. In this work, second-order differential equations, which contain in the right part the product of a regularly varying nonlinearity from an unknown function and a rapidly varying nonlinearity from the derivative of an unknown function when the corresponding arguments are directed to zero or infinity, are considered. Necessary and sufficient conditions for the existence of slowly varying P!(Y0; Y1;±∞) solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives have also been obtained. When additional conditions are imposed on the coefficients of the characteristic equation of the corresponding equivalent system of quasi-linear differential equations, it is established that there is a one-parameter family of such P!(Y0; Y1;±∞)-solutions to the equation. Similar results were obtained earlier when considering second-order equations, which contain on the right-hand side the product of a rapidly varying function from an unknown function and a properly varying function from the derivative of an unknown function when the arguments go to zero or infinity. Results for the equation, considered in this paper, are new.

О континуальных спектрах показателей колеблемости линейных однородных дифференциальных систем

The research subject of this work is at the junction of two sections of the qualitative theory of differential equations, namely: the theory of Lyapunov exponents and the theory of oscillation. In this paper, we study the spectra (i. e., sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential systems with coefficients continuous on the positive semiaxis. For any n≥2, the existence of an n-dimensional differential system with continuum spectra of the oscillation exponents is established. For even n, the spectra of all the oscillation exponents fill the same segment of the numerical axis with predetermined arbitrary positive incommensurable ends, and for odd n, zero is added to the indicated spectra. It turns out that for each solution of the constructed differential system, all the oscillation exponents coincide with each other. When proving the results of this work, the cases of even and odd n are considered separately. The results obtained are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential systems.

Asymptotic representations of the Pω(Y0, Y1, ±∞)-solutions of the second order differential equation, which contains the product of different types of nonlinearities from an unknown function and its derivative

The study of asymptotic representations of solutions of differential equations, in particular, of the second order differential equations, which contain nonlinearities of various types in the right-hand side, play an important role in the development of the qualitative theory of differential equations. This paper considers the type of differential equations of the second order, which contain in the right part the product of a regularly varying function from an unknown function and a rapidly varying function from the derivative of an unknown function when the corresponding arguments tend to zero or infinity. Necessary and sufficient conditions for the existence of slowly varying Pω(Y0, Y1, ±∞)-solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives are also obtained. Note that when additional conditions are imposed on the coefficients of the characteristic equation, such Pω(Y0, Y1, ±∞)-solutions of the equation exist as a one-parameter family. Similar results were obtained earlier when considering second-order equations, which contain in the right-hand side the product of a rapidly varying function from an unknown function and a regularly varying function from the derivative of an unknown function when the arguments tend to zero or infinity. For the equations considered in this paper, similar results are new.

MATHEMATICAL MODELING OF THE QUASI-STATIONARY PROCESSES OF VISCOUS MIXTURE MIXING IN A RECTANGULAR AREA BY THE R-FUNCTIONS METHOD

Mixing processes are found in the chemical, pharmaceutical and food industries. Fluid mixing is one of the fundamental scientific problems associated with modern concepts of regular and chaotic dynamics. The paper considers the problem of mathematical modeling of the quasi-stationary process of mixing a viscous mixture. This problem consists of two sub-problems: determination of the velocity field in the flow region (Eulerian formalism) and investigation of the trajectories of individual fluid particles (Lagrange formalism). To solve the first subproblem, it is proposed to jointly use the principle of superposition, the structural method (method of R-functions) and the Ritz variational method. The methods of nonlinear dynamics and qualitative theory of differential equations are used to solve the second subproblem. A plane quasi-steady flow is considered in a rectangular region and it is assumed that the side walls are at rest, and the upper and lower walls move alternately according to the given laws. According to the method of R-functions, the structures of the solutions were built and the use of the Ritz variational method for the approximation of the uncertain components of the structures was justified. The operation of the proposed method is illustrated by the results of a computational experiment, which was conducted for different modes of wall motion. The practical interest of the considered regimes is due to the fact that they lead to the emergence of chaotic behavior when mixing occurs most efficiently. Using the methods of nonlinear dynamics, the location of periodic (hyperbolic and elliptical) points was investigated and the Poincaré section was constructed. Further research with the help of the method proposed in the work can be related to the consideration of flows in more geometrically complex regions and more complex mixing regimes, as well as in the application to the calculation of industrial problems.

Global dynamics of the Hořava–Lifshitz cosmology in the presence of non-zero cosmological constant in a flat space

Using the qualitative theory of differential equations, the global dynamics of a cosmological model based on Hořava-Lifshitz gravity is studied in the space with zero curvature in the presence of the non-zero cosmological constant. The study shows that there may be three unstable finite equilibrium positions under the Hořava-Lifshitz cosmology model and that the final evolution of the orbits of the cosmological model in the physical region of interest may tend towards some infinite equilibrium position, which may correspond to the late-time state of the universe.

The nonequilibrium potential today: A short review

A brief review is made of the birth and evolution of the “nonequilibrium potential” (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the Jarzynski equality in the Wilson–Cowan model (a population-competition model of the neocortex) and a “thermodynamic uncertainty relation” (TUR) in the KPZ equation (the stochastic field theory of kinetic interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson–Cowan model and relevant aspects of KPZ phenomenology like the EW–KPZ crossover and the memory of initial conditions.

Symbolic computation for the qualitative theory of differential equations

Symbolic computation for the qualitative theory of differential equations

The solution character of capillary retention in porous media at steady state

The saturation solution to the capillary retention phenomenon is critical to understand how much and what type of phase is retained by a heterogeneous medium. Hence, this paper studies the solution qualitatively using methods from the qualitative theory of differential equations. In particular, when the capillary function, defined as the square root of the ratio of permeability to porosity, is differentiable and monotonic is examined. The analysis shows that the saturation solution is bell shaped with a local minimum or maximum point depending on the Leverett curve and the derivative of the capillary function. The case with a discontinuous capillary function is identified here as the limiting case when the regularisation parameter approaches zero. Analytical solution of the linearised differential equation enables the derivation of a parameter that characterises the local flow regime of the solution. The parameter is a function of the capillary number, the viscous limit fractional flow and the diffusivity functions. The results of the analysis are validated and demonstrated numerically.