On some classes of systems of differential equations

Abstract

We consider an autonomous system of differential equations x ̇=f(x), where x∈R^n, the vector function f(x) and its derivatives ∂f_i/∂x_i (i,j=1,…,n) are continuous. Three classes of autonomous systems are identified and the properties that systems of each class possess are described. We will assume that the system belongs to the first class on the set D⊆R^n, if the right parts of this system do not depend on varibles x_1,…,x_n, that is this system has the form x ̇=C, where C∈R^n, x∈D. We will assign to the second class the systems that are not included in the first class, for which the next condition is met “each of the function f_i is increasing on the set D⊆R^n with respect to all variables on which it explicitly depends, with the exception of variable x_i, i=1,…,n”. Solutions of systems of the first and second classes have the property of monotonicity with respect to initial conditions. We will assign to the third class the systems that are not included in the first class, for which the condition is met “each of the function f_i is decreasing on the set D⊆R^n with respect to all variables on which it explicitly depends, with the exception of variable x_i, i=1,…,n”. The conditions for the absence of periodic solutions for autonomous systems of the second order are obtained, complementing the known Bendikson conditions. It is proved that systems of two differential equations of all three specified classes cannot have periodic solutions.