Colin Sparrow

Local and global behavior near homoclinic orbits

We study the local behavior of systems near homoclinic orbits to stationary points of saddle-focus type. We explicitly describe how a periodic orbit approaches homoclinicity and, with the help of numerical examples, discuss how these results relate to global patterns of bifurcations.

T-points: A codimension two heteroclinic bifurcation

The local bifurcation structure of a heteroclinic bifurcation which has been observed in the Lorenz equations is analyzed. The existence of a particular heteroclinic loop at one point in a two-dimensional parameter space (a “T point”) implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits, as well as countably many other topologically more complicatedT points in a small neighborhood in parameter space.

Fictitious play in 3×3 games: The transition between periodic and chaotic behaviour

In the 1960s Shapley provided an example of a two-player fictitious game with periodic behaviour. In this game, player A aims to copy Bs behaviour and player B aims to play one ahead of player A. In this paper we generalise Shapleys example by introducing an external parameter. We show that the periodic behaviour in Shapleys example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again, but now both players aim to play one ahead of the other. In this paper we adopt a geometric (dynamical systems) approach. Here we prove rigorous results on continuity of the dynamics and on the periodic behaviour, while in the sequel to this paper we shall describe the chaotic behaviour.

Extension of order-preserving maps on a cone

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone

An introduction to the Lorenz equations

K_1

A note on periodic points of order preserving subhomogeneous maps

(taking values in another closed cone

SHILNIKOV’S SADDLE-NODE BIFURCATION

K_2

Transitive actions of finite abelian groups of sup-norm isometries

) to the whole of

Continuous Extension of Order-Preserving Homogeneous Maps

K_1

Period Doubling and Stable Orbits

. We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous. Maps