Differential-algebraic equations in the theory of invariant manifolds for singular equations

Analogs of Grobman-Hartman theorem on stable and unstable manifolds solutions for differential equations in Banach spaces with degenerate Fredholm operator at the derivative are proved. Jordan chains tools and the implicit operator theorem are used. In contrast to the usual evolution equation here the central manifold appears even for the case of spectrum absence on the imaginary axis. If on the imaginary axis there is only a finite number of spectrum points, then the original nonlinear equation is reduced to two differential–algebraic systems on the center manifold.