- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics
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.
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.
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 $x'(t) \in f(x(t)) + \bar B(0,\delta)$.
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.
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.
We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.
Year of publication
[Back to Top]