The global-stability problem of equilibria is investigated for coupled systems of differential equations on networks. Using results from graph theory, we develop a systematic approach that allows one ...to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The approach is applied to several classes of coupled systems in engineering, ecology and epidemiology, and is shown to improve existing results.
In this paper, we devote to study a pursuit game described by an infinite three-systems of differential equations in Hilbert space. The game involves transferring of the states as the pursuit is said ...to be completed if the state ζ(・) of the system is shifted to another non zero state ζ1 of the system at some finite time. The control functions of the players are constrained by geometric constraints. We first find the control function that transfers the control system’s state to the state ζ1 at some time. We then extend to solve the pursuit problem where an admissible pursuer’s strategy is constructed and a guaranteed pursuit time is determined
In this paper, we consider the following linear system of second order differential equations (0.1)u′′+A(t)u=0,0≤t<∞where, for each t, A(t) is an n×n matrix with real components, and positive with ...respect to the usual cone K in Rn. Conditions are provided in order that the first conjugate point T of t=0, i.e. the smallest T such that the above equation has a nontrivial solution u:0,T→K satisfying the boundary conditions u(0)=0=u(T)will be a bifurcation point for higher order perturbations of the equation. The paper is mainly motivated by results in Ahmad and Lazer (1997, 1980); Ahmad and Salazar (1981) and Schmitt (1975); Schmitt and Smith (1978). Some additional new consequences are discussed.
In this paper, we propose two new methods to solve large-scale systems of differential equations, which are based on the Krylov method. In the first one, the exact solution with the exponential ...projection technique of the matrix. In the second, we get a new problem of small size, by dropping the initial problem, and then we solve it in ways, such as the Rosenbrock and the BDF. Some theoretical results are presented such as an accurate expression of the remaining criteria. We give an expression of error report and numerical values to compare the two methods in terms of how long each method takes, and we also compare the approaches.
The n-dimensional cyclic systems of first order nonlinear differential equations(A)xi′+pi(t)xi+1αi=0,i=1,…,n(xn+1=x1),(B)xi′=pi(t)xi+1αi,i=1,…,n(xn+1=x1), are analyzed in the framework of regular ...variation. Under the assumption that α1⋯αn<1 and pi(t), i=1,…,n, are regularly varying functions, it is shown that the situation in which system (A) (resp. (B)) possesses decreasing (resp. increasing) regularly varying solutions of negative (resp. positive) indices can be completely characterized, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of decay (resp. growth) precisely. Examples are presented to demonstrate that the main results for (A) and (B) can be applied effectively to some higher order scalar nonlinear differential equations to provide new accurate information about the existence and the asymptotic behavior of their positive strongly monotone solutions.
•We study cyclic systems of differential equations in the framework of regular variation.•Existence and asymptotic behavior of strongly monotone solutions is established.•Applications to scalar differential equations of the higher order are presented.
The article considers the study of methods for numerical solution of systems of differential equations using neural networks. To achieve this goal, thefollowing interdependent tasks were solved: an ...overview of industries that need to solve systems of differential equations, as well as implemented amethod of solving systems of differential equations using neural networks. It is shown that different types of systems of differential equations can besolved by a single method, which requires only the problem of loss function for optimization, which is directly created from differential equations anddoes not require solving equations for the highest derivative. The solution of differential equations’ system using a multilayer neural networks is thefunctions given in analytical form, which can be differentiated or integrated analytically. In the course of this work, an improved form of constructionof a test solution of systems of differential equations was found, which satisfies the initial conditions for construction, but has less impact on thesolution error at a distance from the initial conditions compared to the form of such solution. The way has also been found to modify the calculation ofthe loss function for cases when the solution process stops at the local minimum, which will be caused by the high dependence of the subsequentvalues of the functions on the accuracy of finding the previous values. Among the results, it can be noted that the solution of differential equations’system using artificial neural networks may be more accurate than classical numerical methods for solving differential equations, but usually takesmuch longer to achieve similar results on small problems. The main advantage of using neural networks to solve differential equations` system is thatthe solution is in analytical form and can be found not only for individual values of parameters of equations, but also for all values of parameters in alimited range of values.
Necessary conditions and distinct sufficient conditions are derived for the system x˙=x(1-a20x2-a11xy-a02y2), y˙=y(-q+b20x2+b11xy+b02y2) to admit a first integral of the form Φ(x,y)=xqy+⋯ in a ...neighborhood of the origin, in which case the origin is termed a 1:-q resonant center. Necessary and sufficient conditions are obtained for odd q,q⩽9; necessary conditions, most of which are also sufficient, are obtained for even q,q⩽8. Key ideas in the proofs are computation of focus quantities for the complexified systems and decomposition of the variety of the ideal generated by an initial string of them to obtain necessary conditions, and the theory of Darboux first integrals to show sufficiency.
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