Lecture

Periodic Solutions in Planar Systems

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In course

COM-502: Dynamical system theory for engineers

Linear and nonlinear dynamical systems are found in all fields of science and engineering. After a short review of linear system theory, the class will explain and develop the main tools for the qualitative analysis of nonlinear systems, both in discrete-t ...

Description

This lecture discusses the concept of periodic solutions in planar systems, focusing on the absence of such solutions in simply connected regions and the conditions for their existence. The instructor explains the criteria for the absence of periodic solutions and provides proofs for the theoretical framework. The lecture also covers the implications of divergence and sign changes in the context of periodic solutions.

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Instructor

Patrick Thiran

Patrick Thiran is a full professor in network and systems theory at the School of Computer and Communication Sciences at EPFL. He holds an electrical engineering degree from the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, an M.Sc. degree in electrical engineering from the University of California at Berkeley, USA, and he received the PhD degree from EPFL, in 1996. He became an adjunct professor in 1998, an assistant professor in 2002, an associate professor in 2006 and a full professor in 2011. He was with Sprint Advanced Technology Labs in Burlingame, California, in 2000-01. His research interests are in communication and social networks, performance analysis and stochastic models. He is currently active in the analysis and design of wireless and PLC networks (scaling laws, medium access control), in network monitoring (network tomography, multi-layer networks), and data-driven network science. He also contributed to network calculus and to the theory of locally coupled neural networks and self-organizing maps. He served as an associate editor for the IEEE Transactions on Circuits and Systems in 1997-99 and for the IEEE/ACM Transactions on Networking in 2006-10. He is currently on the editorial board of the IEEE Journal on Selected Areas in Communication. He is/was on the program committee of different conferences in networking, including ACM Sigcomm, Sigmetrics, IMC, CoNext and IEEE Infocom. He was TPC chair of AMC IMC 2011 and CoNext 2012. He is a Fellow of the Belgian American Educational Foundation and of the IEEE. He received the 1996 EPFL Doctoral Prize and the 2008 Crédit Suisse Teaching Award.

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Related concepts (32)

F-divergence

In probability theory, an -divergence is a function that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of -divergence. These divergences were introduced by Alfréd Rényi in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease in Markov processes.

Divergence

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region.

Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

Planar graph

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

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