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Integral manifolds of impulsive differential equations defined on torus. (English) Zbl 0931.34029

The authors study a system of differential equations with impulses of the form

ϕ ˙=a(ϕ),x ˙=A(ϕ)x+f(ϕ),ϕΓ,Δx=I(ϕ),ϕΓ,(1)

where ϕ=(ϕ 1 ,,ϕ m ) is in the m-dimensional torus T m , x n , Γ is an (m-1)-dimensional closed submanifold of T m , the functions a(ϕ), A(ϕ), f(ϕ) and I(ϕ) are continuous and 2π-periodic in ϕ k , k=1,,m, Δx(t)=x(t+0)-x(t-0). The dynamics of the system (1) is as follows. Between two successive meetings of the point (t,ϕ(t),x(t)) with Γ the motion proceeds along the trajectory of the system ϕ ˙=a(ϕ), x ˙=A(ϕ)x+f(ϕ). When the point (t,ϕ(t),x(t)) meets Γ it is momentarily transfered to the point (t,ϕ(t),x(t)+I(ϕ(t))). For system (1) sufficient conditions are obtained for the existence of an integral manifold of the form J={(ϕ,x):x=u(ϕ), ϕT m , x n }. Some continuous properties of u(ϕ) are investigated.

34C45Invariant manifolds (ODE)
34A37Differential equations with impulses