Paul Glendinning

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.

From finite to infinite dimensional dynamical systems

Preface. Introduction J.C. Robinson, P.A. Glendinning. Spatial correlations and local fluctuations in host-parasite models M.J. Keeling, D.A. Rand. Lattice dynamical systems L.A. Bunimovich (assisted by C. Giberti). Attractors and dynamics in partial differential equations J.K. Hale. Nonlinear dynamics of extended systems P. Collet. Three lectures on mathematical fluid mechanics P. Constantin. Low-dimensional models of turbulence P.J. Holmes, et al.

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.

Bifurcations near homoclinic orbits with symmetry

Abstract The effect of symmetry on bifurcations associated with homoclinic orbits to saddle-foci is analysed. With symmetry each homoclinic bifurcation contributes three periodic orbits to the global bifurcation picture as opposed to a single orbit in the general case. Bifurcations on these orbits are studied: there are sequences of saddle-node and period-doubling bifurcations, chaos and more complicated homoclinic orbits.

The mathematics of motion camouflage

Motion camouflage is a strategy whereby an aggressor moves towards a target while appearing stationary to the target except for the inevitable change in perceived size of the aggressor as it approaches. The strategy has been observed in insects, and mathematical models using discrete time or neural–network control have been used to simulate the behaviour. Here, the differential equations for motion camouflage are derived and some simple cases are analysed. These equations are easy to simulate numerically, and simulations indicate that motion camouflage is more efficient than the classical pursuit strategy (‘move directly towards the target’).

Modelling human balance using switched systems with linear feedback control

We are interested in understanding the mechanisms behind and the character of the sway motion of healthy human subjects during quiet standing. We assume that a human body can be modelled as a single-link inverted pendulum, and the balance is achieved using linear feedback control. Using these assumptions, we derive a switched model which we then investigate. Stable periodic motions (limit cycles) about an upright position are found. The existence of these limit cycles is studied as a function of system parameters. The exploration of the parameter space leads to the detection of multi-stability and homoclinic bifurcations.

Zeros of the kneading invariant and topological entropy for Lorenz maps

If is a unimodal map, then its topological entropy is related to the smallest positive zero s of a certain power series (the kneading invariant of f) by . Moreover, it is implicit in the results of Jonker and Rand that for each positive entropy basic set in the renormalization decomposition of the non-wandering set of f, there is a real zero of the kneading invariant such that . Here we prove a similar result for Lorenz maps. In contrast to the unimodal case, it is possible for two basic sets in the renormalization decomposition of the non-wandering set of a Lorenz map to have the same entropy, and we show that in this case there is a corresponding double zero of the kneading invariant.

The structure of mode-locked regions in quasi-periodically forced circle maps

Using a mixture of analytic and numerical techniques we show that the mode-locked regions of quasi-periodically forced Arnold circle maps form complicated sets in parameter space. These sets are characterized by ‘pinched-off’ regions, where the width of the mode-locked region becomes very small. By considering general quasi-periodically forced circle maps we show that this pinching occurs in a broad class of such maps having a simple symmetry.

The gluing bifurcation: I. Symbolic dynamics of closed curves

The authors study the periodic orbits which can occur in a neighbourhood of a codimension-two gluing bifurcation involving two trajectories bi-asymptotic to the same stationary point. Provided some simple conditions are satisfied they prove that there are either zero, one or two closed curves and that these have a specific symbolic form which, in particular, allows them to associate a rotation number with each of them. Furthermore, pairs of orbits which can coexist are identified: the two rotation numbers must be Farey neighbours.

Global attractors of pinched skew products

A class of skew products over irrational rotations of the circle is defined which contains some systems which have strange non-chaotic attractors. The global attractor of these systems is characterized: it lies between an upper semi-continuous curve and a lower semi-continuous curve. With additional assumptions on the class of maps considered more detail of the attractor can be given.