A finite atlas for solution manifolds of differential systems with discrete state-dependent delays

Abstract: Let

. Consider the delay differential equation

x'(t)=g(x(t-d_1(Lx_t)),ldots,x(t-d_{{�f k}}(Lx_t)))

for

continuously differentiable,

L

a continuous linear map from

C ([- r, 0], m a t h b b R^{n})

into a finite-dimensional vectorspace

F

, each

d_{k} : F s u p s e t W o [0, r]

,

, continuously differentiable, and

x_{t} (s) = x (t + s)

. The solutions define a semiflow of continuously differentiable solution operators on the submanifold

X_{f} s u b s e t C^{1} ([- r, 0], m a t h b b R^{n})

which is given by the compatibility condition

p h i^{'} (0) = f (p h i)

with

f(phi)=g(phi(-d_1(Lphi)),ldots,phi(-d_{{�f k}}(Lphi))).

We prove that

X_{f}

has a finite atlas of at most

manifold charts, whose domains are almost graphs over

X_{0}

. The size of the atlas depends solely on the zerosets of the delay functions

d_{k}

.

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