University of Bristol

 

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The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. The organisation of the dynamics in the Lorenz system and also how the dynamics depends on the system parameters has long been an object of study. This talk addresses the role of the global stable and unstable manifolds in organising the dynamics. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold and the other two equilibria each have a two-dimensional unstable manifold. The intersections of these two manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent study of these manifolds clarifies not only the creation of these heteroclinic connections, but also helps to explain the dynamics on the attractor by means of symbolic coding in a parameter-dependent way.

This is joint work with Eusebius Doedel (Concordia University, Montreal) and Bernd Krauskopf (University of Bristol).

We consider a queuing system with three queues (nodes) and one server.

The arrival and service rates at each node are such that the system overall

is overloaded, while no individual node is. The service discipline is the

following: once the server is at node j, it stays there until it serves all

customers in the queue.

After this, the server moves to the "more expensive" of the two

queues.

We will show that a.s. there will be a periodicity in the order of

services, as suggested by the behavior of the corresponding

dynamical systems; we also study the cases (of measure 0) when the

dynamical system is chaotic, and prove that then the stochastic one

cannot be periodic either.