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### Open Access Journals

DCDS-B

We describe a Lohner-type algorithm for the computation of rigorous upper bounds
for reachable set for control systems,
solutions of ordinary differential inclusions
and perturbations of ODEs.

DCDS

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties.
The proof is conducted
in the phase space of the system. In our approach we do not require that the map is a perturbation of some other
map for which we already have an invariant manifold. We provide conditions which imply the existence of
the manifold within an investigated region of the phase space. The required assumptions are formulated in
a way which allows for rigorous computer assisted verification. We apply our method to obtain
an invariant manifold within an explicit range of parameters for the rotating Hénon map.

DCDS

We present a topological method of obtaining the existence of
infinite number of symmetric periodic orbits for systems with
reversing symmetry. The method is based on covering relations. We
apply the method to a four-dimensional reversible map.

DCDS

We show that if an ordinary differential
equation $x'=f(x)$, where $x\in \mathbb R^n$ and $f \in
\mathcal C^1$, has a topological horseshoe, then the
corresponding delay equation $x'(t)=f(x(t-h))$ for small $h >0$
also has a topological horseshoe, i.e. symbolic dynamics and an
infinite number of periodic orbits. A method of computation of $h$
is given in terms of topological properties of solutions of
differential inclusion $x'(t) \in f(x(t)) +
\bar B(0,\delta)$.

DCDS

Covering relations are a topological tool for detecting periodic
orbits, symbolic dynamics and chaotic behavior for autonomous ODE.
We extend the method of the covering relations onto systems with
a time dependent perturbation. As an example we apply the method
to non-autonomous perturbations of the Rössler equations to
show that for small perturbation they possess symbolic dynamics.

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