Rigorous Numerics in Dynamics

Rigorous Numerics in Dynamics

Motivation Nonlinear dynamics shape the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these beautiful phenomena are modeled by nonlinear dynamical systems, mainly in the form of ordinary differential equations (ODEs), partial differential equations (PDEs) and delay d...

Rigorous Numerics of Chaotic Dynamical Systems

1. Introduction The importance of computers in the present day research on dynamical systems cannot be questioned. Todays physics or chemistry would not function without access to permanently faster and larger computers. However, from the mathematical point of view, the role of computers is still very limited in this process. Of course, mathematics provides numerical algorithms for solving diie...

The Conley Index and Rigorous Numerics

Multivalued Maps As a Tool in Modeling and Rigorous Numerics

Applications of the fixed point theory of multivalued maps can be classified into several areas: (1) Game theory and mathematical economics; (2) Discontinuous differential equations and differential inclusions; (3) Computing homology of maps; (4) Conley index methods in chaotic dynamics; (5) Digital imaging and computer vision. We briefly recall the history of the most classical and well develo...

Rigorous numerics for symmetric homoclinic orbits in reversible dynamical systems

We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and ...