Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

Periodicity and Chaos Amidst Twisting and Folding in Two-Dimensional Maps

We study the dynamics of three planar, noninvertible maps which rotate and fold the plane. Two maps are inspired by real-world applications whereas the third map is constructed to serve as a toy model for the other two maps. The dynamics of the three maps are remarkably similar. A stable fixed point bifurcates through a Hopf-Neimark-Sacker which leads to a countably infinite set of resonance tongues in the parameter plane of the map. Within a resonance tongue a periodic point can bifurcate through a period-doubling cascade. At the end of the cascade we detect Henon-like attractors which are conjectured to be the closure of the unstable manifold of a saddle periodic point. These attractors have a folded structure which can be explained by means of the concept of critical lines. We also detect snap-back repellers which can either coexist with Henon-like attractors or which can be formed when the saddle-point of a Henon-like attractor becomes a source

Dynamics amidst folding and twisting in 2-dimensional maps

Dynamica van het vouwen en draaien bij 2-dimensionale afbeeldingen Een prooi-roofdier model uit de biodynamica was de inspiratie tot een wiskundig model waarbij een vlak wordt dubbel gevouwen en daarna gedraaid. (fold and twist) Deze vouw en draai worden steeds herhaald. Dan blijkt dat een aantal belangrijke wiskundige eigenschappen, waaronder het chaotisch gedrag, in beide modellen op vergelijkbare manier terug te vinden zijn. De vouw en draai is een fundamentele eigenschap van het prooi-roofdier model. Echter niet alleen in het prooi-roofdier model is de vouw en draai fundamenteel maar ook een ander model, het discrete Lorenz-63 model afkomstig uit de weerkunde, blijkt beter te worden begrepen als het vergeleken wordt met de vouw en draai afbeelding. Daarmee is duidelijk geworden dat de vouw en draai afbeelding geschikt is als een fundamenteel model voor de bestudering van 2-dimensionale afbeeldingen met de eigenschap dat verschillende toestanden op een bepaald moment leiden tot een dezelfde toestand op het volgende moment

Traveling Salesman Problem

The idea behind TSP was conceived by Austrian mathematician Karl Menger in mid 1930s who invited the research community to consider a problem from the everyday life from a mathematical point of view. A traveling salesman has to visit exactly once each one of a list of m cities and then return to the home city. He knows the cost of traveling from any city i to any other city j. Thus, which is the tour of least possible cost the salesman can take? In this book the problem of finding algorithmic technique leading to good/optimal solutions for TSP (or for some other strictly related problems) is considered. TSP is a very attractive problem for the research community because it arises as a natural subproblem in many applications concerning the every day life. Indeed, each application, in which an optimal ordering of a number of items has to be chosen in a way that the total cost of a solution is determined by adding up the costs arising from two successively items, can be modelled as a TSP instance. Thus, studying TSP can never be considered as an abstract research with no real importance