Preface This is a guide to the software package AUTO for continuation and bifurcation problems in ordinary differential equations. graphics program PLAUT and the pendula animation program. An earlier graphical user interface for AUTO on SGI machines was written by Taylor & Kevrekidis (1989). Special thanks are due to Sheila Shull, California Institute of Technology, for her cheerful assistance in the distribution of AUTO over a long period of time. Over the years, the development of AUTO has been supported by various agencies through the California Institute of Technology. Work on this updated version was supported by a general research grant from NSERC (Canada). The development of HomCont has much benefitted from various pieces of help and advice from, among others, W. This manual uses the following conventions. command This font is used for commands which you can type in. PAR This font is used for AUTO parameters. filename This font is used for file and directory names. variable This font is used for environment variable. site This font is used for world wide web and ftp sites. function This font is used for function names.

AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont)

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.

/pdf/mixed-mode-oscillations-with-multiple-time-scales-2gobomm5x9.pdf

Mixed-Mode Oscillations with Multiple Time Scales

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.

https://www.tandfonline.com/doi/pdf/10.1080/13873950701742754

New features of the software MatCont for bifurcation analysis of dynamical systems

The oxidation of formic acid on thin, epitaxially-grown Pd overlayers on Au and Pt single crystal electrodes was studied as a function of overlayer thickness ranging from submonolayer coverages up to the equivalent of 10 monolayers. The results were compared with those for massive Pd single crystal surfaces. The electrocatalytic properties of the Pd overlayers were found to depend markedly on their thickness and on their crystallographic orientation. Pd(100) showed the highest catalytic activity of all three low-index faces, and Pd films on Pt(hkl), even when two or three layers thick, generally exhibited a much higher activity than Pd films on Au(hkl) or massive Pd(hkl). In all cases a high resistivity of the Pd surfaces against poisoning by CO was observed, which makes Pd an interesting alternative to the often quickly deactivated Pt as catalyst for organic fuel cell reactions.

Formic Acid Oxidation on Ultrathin Pd Films on Au(hkl) and Pt(hkl) Electrodes

A green, facile, one-pot synthesis of well-defined Au NPs@POMGNSs tricomponent nanohybrids is reported (POM stands for polyoxometalate and GNSs for graphene nanosheets). The synthesis is convenient, rapid and environmentally friendly. The POMs serve as both reducing, encapsulating molecules, and bridging molecules; this avoids the introduction of other organic toxic molecules. Characterization using transmission electron microscopy, X-ray diffraction, X-ray photoelectron spectroscopy, and Raman spectroscopy analysis is performed, and the structure of the prepared nanohybrids of Au NPs@POMGNSs is verified. Most importantly, the amperometric measurements show the Au NPs@POMGNSs nanohybrids have high catalytic activity with good sensitivity, good long-term stability, wide linear range, low detection limit, and fast response towards H2O2 detection for application as an enzyme-free biosensor. Transformation of the POMs during H2O2 detection does not affect the catalytic activities of the nanohybrids. Thus, the synergistic effect of Au NPs and GNSs in the nanohybrids leads to the enhanced catalytic property.

Marc T.M. Koper

Papers

Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram

Stability study and categorization of electrochemical oscillators by impedance spectroscopy

Oscillations and Complex Dynamical Bifurcations in Electrochemical Systems

Bursting and mixed-mode oscillations during the hydrogen peroxide reduction on a platinum electrode

A simplified approach to the modeling of wave propagation at electrode/electrolyte interfaces