The current spectacular interest in dynamical systems has gradually built up over the last forty years. Poincaré [13] set the change for the subject at the turn of the century by moving away from the analytical approach to differential equations with an emphasis on the topology of the orbit structure of the phase plane. His initial impetus was reinforced in the first half of this century by Birkhoff [7], Andronov and Pontryagin [2], and others. However, it can be argued that the real explosion of interest for dynamical systems in the mathematics community was triggered by the great contributions of Smale and Arnold in the 1960s. The global theory of dynamical systems was developed by Smale [15] throughout the decade. His ability to focus on the essential ingredients of hyperbolic dynamics was unparalleled. That his insights would feed the frenzy of activity on chaotic behaviour and strange attractors in the following decades would have been difficult to predict. By contrast, Arnold, alongside Moser, was developing the tools to understand the special qualitative features of Hamiltonian systems and area-preserving maps. The work involved fine diophantine estimates for the persistence of invariant tori or circles under perturbation of integrable systems. The global approach of the sixties and the attempt by the Smale school to classify a generic set of systems, which in some sense could be described simply, led to the recognition of many special types of dynamics which had degenerate structure. This in turn gave a new impetus to the work of Andronov et al. [1], Neimark, Sacker, and Hopf on families of dynamical systems. The occurrence of change of structure, or bifurcation, in families was a major route for applied workers to predict global phenomena by making local calculations. For example, the Neimark-Sacker-Hopf bifurcations predict the growth of a limit cycle or invariant circle around an equilibrium point which undergoes a change of stability. There are nondegeneracy conditions associated with the creation of this circle, but they are all calculated at the fixed point. The applied modeller may have no idea which of the parameters are critical to unfolding the degenerate behaviour of the defining equations. Of course, some parameters may have no structural effect on the system whatsoever. Often a dy-namical system modelling a real-life phenomenon comes prepackaged with too many parameters for the applied mathematician to handle-it is very daunting to search in large-dimensional parameter spaces! The growth and classification of bifurca-tion has been extremely valuable in allowing modellers to focus their search for key dynamical behaviour in the parameter space. Moreover, suppose the bifurcational behaviour of a system has been understood, and a complete set of topological types of phase portrait that occur as the degenerate system unfolds by varying parameters has been found. The collection of pictures for a bifurcation can then be an immensely useful guide to the behaviour

We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970's) it was also used in the modelling process, in particular to determine the values of critical parameters. More recently, it is used in a systematic way in the design of dynamical models and to determine which parameters are relevant. MATCONT and CL_MATCONT are freely available MATLAB numerical continuation packages for the interactive study of dynamical systems and bifurcations. MATCONT is the GUI-version, CL_MATCONT is the command-line version. The work started in 2000 and the first publications appeared in 2003. Since that time many new functionalities were added. Some of these are fairly simple but were never before implemented in continuation codes, e.g. Poincare´maps. Others were only available as toolboxes that can be used by experts, e.g. continuation of homoclinic orbits. Several others were never implemented at all, such as periodic normal forms for codimension 1 bifurcations of limit cycles, normal forms for codimension 2 bifurcations of equilibria, detection of codimension 2 bifurcations of limit cycles, automatic computation of phase response curves and their derivatives, continuation of branch points of equilibria and limit cycles. New numerical algorithms for these computations have been published or will appear elsewhere; here we restrict to their software implementation. We further discuss software issues that are in practice important for many users, e.g. how to define a new system starting from an existing one, how to import and export data, system descriptions, and computed results.

MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold bifurcation points of limit cycles. All curves are computed by the same function that implements a prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo-inverse. The continuation of bifurcation points of equilibria and limit cycles is based on bordering methods and minimally extended systems. Hence no additional unknowns such as singular vectors and eigenvectors are used and no artificial sparsity in the systems is created. The sparsity of the discretized systems for the computation of limit cycles and their bifurcation points is exploited by using the standard Matlab sparse matrix methods. The MATLAB environment makes the standard MATLAB Ordinary Differential Equations (ODE) Suite interactively available and provides computational and visualization tools; it also eliminates the compilation stage and so makes installation straightforward. Compared to other packages such as AUTO and CONTENT, adding a new type of curves is easy in the MATLAB environment. We illustrate this by a detailed description of the limit point curve type.

We describe the results of Monte Carlo simulations for kinetics at the collapse transition of a homopolymer in a lattice model. We find the kinetic laws corresponding to the three kinetic stages of the process: R g 2 (t)ϭR g 2 (0)ϪAt 7/11 at the early stage corresponding to formation and growth of locally collapsed clusters, the coarsening stage is characterized by growth of clusters according to the law S ϰ t 1/2 , where S is the average number of Kuhn units per cluster, and the final relaxation stage is described by the law R g 2 (t)ϭR g 2 (ϱ)ϩA 1 (1) e Ϫt/ 1(1) with 1 (1) ϰ N 2 . We also present preliminary results on the equilibrium properties and ''collapse'' transition of a random copolymer. The transition curve is determined as a function of hydrophobic bead concentration n a . We discuss the different collapsed copolymer states as a function of the composition. At low hydrophilicity we believe the critical value of the interaction parameter is governed by the law c (n a ) ϰ n a Ϫ2/3 . In the kinetics we see unusual phenomena such as the appearance of a metastable long-lived states with few clusters and nontrivial loop structure.

No abstract

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

No abstract