A Lohner-type algorithm for control systems and ordinary differential inclusions
Tomasz Kapela Piotr Zgliczyński
Discrete & Continuous Dynamical Systems - B 2009, 11(2): 365-385 doi: 10.3934/dcdsb.2009.11.365
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.
keywords: control systems Lohner algorithm rigorous numerics differential inclusions
Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds
Maciej J. Capiński Piotr Zgliczyński
Discrete & Continuous Dynamical Systems - A 2011, 30(3): 641-670 doi: 10.3934/dcds.2011.30.641
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.
keywords: Normally hyperbolic manifolds covering relations cone conditions Brouwer degree.
Topological method for symmetric periodic orbits for maps with a reversing symmetry
Daniel Wilczak Piotr Zgliczyński
Discrete & Continuous Dynamical Systems - A 2007, 17(3): 629-652 doi: 10.3934/dcds.2007.17.629
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.
keywords: symmetric periodic orbits reversible systems computer assisted proofs.
Topological horseshoes and delay differential equations
Klaudiusz Wójcik Piotr Zgliczyński
Discrete & Continuous Dynamical Systems - A 2005, 12(5): 827-852 doi: 10.3934/dcds.2005.12.827
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 $x'(t) \in f(x(t)) + \bar B(0,\delta)$.
keywords: Delay differential equations chaos. fixed point index periodic points differential inclusions
Covering relations and non-autonomous perturbations of ODEs
Maciej J. Capiński Piotr Zgliczyński
Discrete & Continuous Dynamical Systems - A 2006, 14(2): 281-293 doi: 10.3934/dcds.2006.14.281
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.
keywords: chaotic behavior. Covering relations non-autonomous ODEs

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