New functionalities in MatCont: delay equations and Lyapunov exponents

TL;DR

This paper presents MatCont version 7p6 enhancements that substantially expand its capabilities to analyze delay differential and renewal equations. The core contributions are a Delay Equation Importer that converts finite-delay DEs into ODEs via pseudospectral collocation, and integrated Lyapunov exponent computations, along with improved homoclinic continuation and Poincaré-event support. Through Mackey–Glass and renewal-equation test cases, the authors demonstrate accuracy, convergence behavior, and performance trade-offs versus specialized DDE tools, highlighting both the broadened applicability and computational costs. The work enables users to study a wider class of dynamical systems within the MatCont GUI, while outlining current limitations (e.g., infinite/state-dependent delays) and future directions for further generalization.

Abstract

MatCont is a powerful toolbox for numerical bifurcation analysis focussing on smooth ODEs. A user can study equilibria, periodic and connecting orbits, and their stability and bifurcations. Here, we report on additional features in version 7p6. The first is a delay equation importer enabling MatCont users to study a much larger class of models, namely delay equations with finite delay (including delay differential and renewal equations). This importer translates the delay equation into a system of ODEs using a pseudospectral approximation with an order specified by the user. We also implemented Lyapunov exponent computations, event functions for Poincaré maps, and enhanced homoclinic continuation. We demonstrate these features with test cases, such as the Mackey-Glass equation and a renewal equation, and provide additional examples in online tutorials.

Figures (8)

• Figure 1: The Delay Equation Importer window filled out with the input for the Mackey--Glass DDE \ref{['Mackey-Glass']}.
• Figure 2: Layout settings for a graphical window, showing the reconstructed coordinates for an RE (left), and the Numeric window for an RE, showing both the auxiliary and the reconstructed coordinates.
• Figure 3: State vectors of the system of ODEs approximating a system of DEs (DDEs, REs, or coupled DDEs/REs) with $d$ coordinates. We denote by $d_{\text{DDE}}$ the number of coordinates described by DDEs, while the remaining $d-d_{\text{DDE}}$ coordinates are described by REs. For a system of DDEs only, $d=d_{\text{DDE}}$; for a system of REs only, $d_{\text{DDE}}=0$. The diagrams show the notation for the collocation of the history functions in the delay interval (first from the left) and of the limit cycles in one period interval normalised to $[0,1]$ (third), and the structure of the columns (one for each continuation point) of the output variable of the function, for equilibria (second) and limit cycles (fourth).
• Figure 4: The window to setup the initial point from a function (left), and the Starter window with the new coordinate array field, showing the result of inserting [2:4,ones(1,3)] in that field for a system of six ODEs in the variables $A$, $B$, $C$, $X$, $Y$, and $Z$ (right).
• Figure 5: Mackey--Glass equation: the orbit obtained from the initial vector (left) and the bifurcation diagram with respect to $\tau$ (right).