John Guckenheimer

Structural stability of Lorenz attractors

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The dynamics of density dependent population models

SummaryThe dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a “strange attractor” is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).

Sensitive dependence to initial conditions for one-dimensional maps

This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

Isochrons and phaseless sets

SummaryWinfree has developed mathematical models for his phase resetting experiments on biological clocks. These models lead him to ask a number of mathematical questions concerning dynamical systems. This paper deals with these mathematical questions. In Winfrees terminology we show the existence of isochrons and establish some of their properties.

Structurally stable heteroclinic cycles

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of C r vector fields equivariant with respect to a symmetry group. In the space X ( M ) of C r vector fields on a manifold M , there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space X G ( M ) ⊂ X ( M ) of vector fields equivariant with respect to a symmetry group G , the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ 3 equivariant with respect to a particular finite subgroup G ⊂ O (3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

A Dynamical Simulation Facility for Hybrid Systems

This paper establishes a general framework for describing hybrid dynamical systems which is particularly suitable for numerical simulation. In this context, the data structures used to describe the sets and functions which comprise the dynamical system are crucial since they provide the link between a natural mathematical formulation of a problem and the correct application of standard numerical algorithms. We describe a partial implementation of the design methodology and use this simulation tool for a specific control problem in robotics as an illustration of the utility of the approach for practical applications.

A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos

Abstract Many systems exhibit behaviour which can be described by three coupled oscillators. Provided the coupling is not too strong, such systems can be modelled by maps of the two-torus to itself. In this paper we describe and explain those aspects of the bifurcation diagram for two-parameter families of torus maps ƒ that involve change of mode-locking type. We introduce the concepts of partial and full mode-locking . For the coupled oscillators, these notions correspond to the presence of one or two rational relations between the frequencies, respectively. Numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations . In order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time-1 map of a flow on the torus. Then we analyse those condimension-1 and -2 bifurcations for flows on the torus which change the set of frequency ratios. We find a large variety of bifurcation diagrams, in particular many involving homotopically non-trivial saddle connections. Next we ask how the picture changes when the time-1 maps of a family of flows are perturbed to a general family of diffeomorphisms. We find toroidal chaos in the neighbourhood of all these bifurcations, meaning that there exist orbits which perform a pseudo-random sequence of rotations in different directions around the torus. The corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk. Finally, we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps, which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation.

Fixed point theory of iterative excitation schemes in NMR

Iterative schemes for NMR have been developed by several groups. A theoretical framework based on mathematical dynamics is described for such iterative schemes in nonlinear NMR excitation. This is applicable to any system subjected to coherent radiation or other experimentally controllable external forces. The effect of the excitation, usually a pulse sequence, can be summarized by a propagator or superpropagator (U). The iterative scheme (F) is regarded as a map of propagator space into itself, U n+1=F U n . One designs maps for which a particular propagator U or set of propagators {U} is a fixed point or invariant set. The stability of the fixed points along various directions is characterized by linearizing F around the fixed point, in analogy to the evaluation of an average Hamiltonian. Stable directions of fixed points typically give rise to broadband behavior (in parameters such as frequency, rf amplitude, or coupling constants) and unstable directions to narrowband behavior. The dynamics of the maps are illustrated by ‘‘basin images’’ which depict the convergence of points in propagator space to the stable fixed points. The basin images facilitate the optimal selection of initial pulse sequences to ensure convergence to a desired excitation. Extensions to iterative schemes with several fixed points are discussed. Maps are shown for the propagator space S O(3) appropriate to iterative schemes for isolated spins or two‐level systems. Some maps exhibit smooth, continuous dynamics whereas others have basin images with complex and fractal structures. The theory is applied to iterative schemes for broadband and narrowband π (population inversion) and π/2 rotations, MLEV and Waugh spin decoupling sequences, selective n‐quantum pumping, and bistable excitation.