On One Class of Systems of Differential Equations and on Retarded Equations

Abstract

We establish a connection between solutions to a broad class of large systems of ordinary differential equations and solutions to retarded differential equations. We prove that solving the Cauchy problem for systems of ordinary differential equations reduces to solving the initial value problem for a retarded differential equation as the number of equations increases unboundedly. In particular, the class of systems under consideration contains a system of differential equations which arises in modeling of multiphase synthesis.In this article we continue to study connection between large systems of ordinary differential equations and retarded differential equations(1.1)The first result in this direction was obtained while studying a system of differential equations which arises in modeling of multiphase synthesis without bifurcation [1,2]. This system has the form(1.2)In [1] it was established that if we let the number n of equations in system (1.2) increase unboundedly and consider only the last components of the solution to the Cauchy problem with the zero initial data x| t=0 = 0 then we obtain a uniformly converging sequencex n n (t) → y(t), t ∈ [0, T ], as n → ∞; moreover, the limit function y(t) satisfies the identityConsequently, the function y(t) is a solution to the retarded differential equation (1.1), where the righthand side has the form f (t, u, v) = −θu + g(v).

On a method of approximation of solutions to delay differential equations

“…In the bunch of the articles [1][2][3][4][5], new connections were established between solutions to delay differential equations of the form dy(t) dt = f (t, y(t), y(t − τ )), t > τ, (1.1) and solutions to some systems of ordinary differential equations of a large dimension n of the form…”

“…In the present article we continue [1][2][3][4][5] and examine the delay differential equation of the form dy(t) dt = −θy(t) + g(t − τ, y(t − τ )), t > τ. (1.3) As shown in [1], this equation is closely related to the system of differential equations (1.2), where …”

“…The methods for solving this "large dimension problem" are studied in [1,3] and based on the theorems about the passage to the limit from the system (1.2), (1.4) to the equation (1.3).…”

“…A new method is justified for approximation of solutions to delay equations.In the bunch of the articles [1-5], new connections were established between solutions to delay differential equations of the formand solutions to some systems of ordinary differential equations of a large dimension n of the formThese connections allow us to study the qualitative properties of solutions to (1.1) by the theoretical results established for (1.2) and to determine approximate solutions to (1.2) by solving boundary value problems for (1.1) numerically.In the present article we continue [1][2][3][4][5] and examine the delay differential equation of the formAs shown in [1], this equation is closely related to the system of differential equations (1.2), where …”

“…In the present article we continue [1][2][3][4][5] and examine the delay differential equation of the form…”