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.
For a countable quasilinear system of differential equations with diagonal matrix of the coefficients of its linear part, we establish the conditions for the existence of a particular solution ...represented in the form of absolutely and uniformly convergent Fourier series with slowly varying coefficients and frequency in the resonance case.
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.
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