January 28, 2025, release of COCO

Harry Dankowicz — Sat, 25 Jan 2025 18:35:29 -0000

A major new release of COCO was posted to the SourceForge site on January 28, 2025.

The following is a summary of changes since the previous release in August 2023 (more details below):

New general-purpose support for analysis of continuation problems defined by constraints that are linearly dependent on the corresponding solution manifold, e.g., analysis of periodic orbits in problems with conserved quantities or bifurcation analysis for dynamical systems with symmetries.
New general-purpose support for generic branch switching that is compatible also with analysis of symmetry-breaking bifurcations in dynamical systems with symmetries.
New general-purpose support for detection, location, and handling of special points for piecewise-constant monitor functions associated with discrete jumps in their values.
Associated updates to existing tutorial documentation and demos, and posting of a preprint on numerical bifurcation analysis for beginner and advanced users with particular interest in generic and non-generic bifurcation analysis of equilibria and periodic orbits.

Additionally, a paper (preprint) authored by Zaid Ahsan, Harry Dankowicz, and Christian Kuehn and titled "Adjoint-Based Projections for Quantifying Statistical Covariance near Stochastically Perturbed Limit Cycles and Tori" is accepted for and pending publication by SIADS. COCO-compatible code (using the August 2023 release) that demonstrates the computational methodology and algorithm is available on Github.

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COCO is the result of software development by Frank Schilder and Harry Dankowicz that began in 2007, with important contributions by Michael Henderson, Erika Fotsch, Mingwu Li, Zaid Ahsan, Jan Sieber, and Yuqing Wang. COCO aims to provide a development platform for advanced toolbox and atlas algorithm development, but also to enable all the functionality of existing continuation packages in a user-accessible format.

The basic philosophy of COCO and the detailed development of template toolboxes and atlas algorithms, as well as explicit code examples, are documented in the book Recipes for Continuation, published by SIAM in 2013. This post describes work done by the developers subsequent to the publication of this book.

New in the January 2025 release:

Support for regularization of singular continuation problems:
Until the January 2025 release, COCO provided support only for continuation in instances where the sum of the dimension, d, of the solution manifold and the number of unknowns, n, was equal to the number of equations, m, i.e., whose dimension was equal to the problem dimensional deficit, n-m. Such problems are characterized by n>=m, where m also equals the rank of the problem Jacobian.

In contrast, the January 2025 release provides general-purpose support also for continuation in instances where d>n-m, including where n<m, provided that d=n-r, where r is the rank of the problem Jacobian. These are problems where rank loss can be accommodated by an appropriate regularization without any other user input than the known value of d. Problems of this kind arise in the bifurcation analysis of dynamical systems with symmetries and/or conserved quantities. Demo encodings of both situations are included in the core/examples folder, see also help/CORE-Tutorial.pdf.

Further examples and discussion of non-generic bifurcation analysis can be found in this preprint.

The code has been tested with Matlab R2024b.

Support for generic branch switching:
Branch points are solutions characterized by membership of more than one solution branch. Isolated branch points are singular points of the continuation problem, where the problem Jacobian is not full rank. Generically, only two solution branches meet at such points and their tangent vectors span a plan that is also spanned by either tangent vector and a second vector obtained from the null vector of the matrix obtained by appending the transpose of the first vector to the problem Jacobian.

Prior to the January 2025 release of COCO, branch switching at a generic branch point relied on the prior storing of such a null vector with the solution file associated with the detection of the branch point in a previous run. COCO did not provide general-purpose support for branch switching for arbitrary problems. Instead, toolboxes such as ep, coll, and po included dedicated branch switching constructors. COCO also did not provide support for branch switching from a point not identified as a branch point in a previous run.

The January 2025 release of COCO turns this paradigm on its head. Given a tangent vector from a previous run, the computation of the desired null vector occurs only at the outset of continuation. As a result, specialized constructors are no longer necessary and branch switching is possible from generic branch points of arbitrary problems, as well as from points not identified as branch points in previous runs. The latter occurs, for example, when the primary and secondary branches contain solutions to one set of constraints, while the primary branch is also a solution to an augmented set of constraints that removes the branch point singularity, as is common in problems with symmetry.

All toolbox demos and tutorials have been updated to reflect this new functionality. The code remains backward compatible, but prior usage is deprecated.

The code has been tested with Matlab R2024b.

Suppose for piecewise-smooth monitor functions and event handling:
Event detection in COCO relies on the crossing of a threshold value of an associated continuation parameter. Prior to the January 2025 release, it was assumed that all such crossings coincided with attainment by the continuation parameter of the threshold value, as the corresponding monitor functions were assumed to be continuous along the solution manifold. Further, event localization used one of three methods, namely,
1. a nonlinear corrector applied to the continuation problem appended with equality between the continuation parameter and the threshold value;
2. a nonlinear corrector applied to the continuation problem without appending equality between the continuation parameter and the threshold value, but coupled with a bisection algorithm;
3. linear interpolation between two solutions on either side of the threshold value.

Method 1. applied to all active continuation parameters, whereas methods 2. and 3. applied to regular and singular continuation parameters, respectively.

The January 2025 release of COCO adds the new function type discrete, which flags a monitor function that is assumed to be piecewise constant along the solution manifold. Events associated with such a function are discrete jumps in the value of an associated continuation parameter. Given a piecewise-constant monitor function, it is possible to embed this in a function call of the form

lim_monitor = @(x) min(mmax,max(mmin,monitor(x)))

in order to only detect discrete jumps between mmin and mmax. Event localization for such events is handled using method 2., albeit without guaranteeing that all discrete jumps between the two solutions on either side of the detection of a jump will be localized (a more refined atlas may be necessary for this). As with previous releases, event handlers may be associated with the localization of discrete jumps of piecewise-constant monitor functions so that appropriate actions may be taken when one is detected.

One possible application for this functionality is the detection and localization of discrete changes to the number of unstable eigenvalues of equilibria or Floquet multipliers of periodic orbits. An event handler may then be defined that takes different forms of action depending on the detailed characteristics of the bifurcation point. Examples of this application can be found in the code accompanying this preprint.

The code has been tested with Matlab R2024b.

COCO for delay-coupled multi-segment boundary-value problems

Harry Dankowicz — Mon, 14 Aug 2023 20:28:03 -0000

The paper Methods of continuation and their implementation in the COCO software platform with application to delay differential equations, published in Nonlinear Dynamics, Vol. 107, 3181- 3243 (2022), was recognized with the Best Paper Award for 2022 during the recent International Nonlinear Dynamics Conference, NODYCON 2023.

This paper reviews the role of continuation methods for bifurcation analysis and optimization, as well as the general construction paradigm of COCO and its compatibility with problems derived from variational principles. It further makes original contributions to a formulation of continuation problems describing suitably discretized delay-coupled multi-segment boundary-value problems with application to continuation of periodic orbits, quasiperiodic invariant tori, connecting orbits, and the solving of optimal control problems.

August 14, 2023, release of COCO

Harry Dankowicz — Mon, 14 Aug 2023 19:36:55 -0000

A major new release of COCO was posted to the SourceForge site on August 14, 2023. NOTE: a bug fix was posted on August 15, 2023

The following is a summary of changes since the previous release in March 2020 (more details below):

New set of tutorial documentation and demos for beginner users with particular interest in basic bifurcation analysis of equilibria and periodic orbits.
Updated handling of higher-order and directional derivatives of vector fields, including default implementations for finite-difference approximations, and upgrades to the COCO-compatible SYMCOCO toolbox for symbolic generation of COCO-compatible encodings of vector fields and their derivatives.
Small change delivering a major punch with the addition of a 'norm' option to the atlas_1d algorithm in order to largely eliminate the sensitivity to mesh size that accompanies the default use of Euclidean norms.
Significant upgrades to the handling of objects and persistent variables to ensure automatic garbage collection and plug memory leaks in the March 2020 release produced by the atlas_kd algorithm.

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The August 2023 release contains:

Getting Started with COCO:
Extensive tutorials and demos has been developed to allow beginners to continuation methods to use COCO for basic bifurcation analysis of equilibria and periodic orbits. See help/GettingStartedwithCOCO.pdf and the tutorials/Getting Started/examples/ folder for fully documented examples.