Some Extremal Properties of the Solutions of Ordinary Differential Equations Systems

Let f = (f1, . . . , fm) ⊂ U(x0) : Em → Em be a C1-vector field, i. e. f ∈ C(U(x0)) and moreover f ∈ Lip(U(x0) is a function being an endomorphism of the neighbourhood U(x0) on Em. A system of differential equations ẋ = f(x) is canonic iff div f(x) = 0 (Liouville’s condition) [1]. It is evident that in general this condition is not fulfilled. In what follows we will consider a system with additive perturbation y of the vector field f(x) of the form ẋ = f(x+ y) where x+ y ∈ U(x0) ⊂ Em.