Numerical analysis and integration of dynamical systems and the fractal dimension of boundaries

Abstract

The set of Maple routines that comprises the package Ndynamics has been improved. Apart one of the main motivations for its creation, namely, the routines to calculate the fractal dimension of boundaries (via box counting), the package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. Many new Maple-in-built numerical solvers are now programmed and available for the user of the package. The novelty that the package presented at the time of its release, an optional numerical interface, is maintained and updated.

New version program summary

Program Title: Ndynamics - Numerical integration of dynamical systems and the fractal dimension of boundaries

CPC Library link to program files: https://doi.org/10.17632/swkr5w3kx4.1

Licensing provisions: CC by NC 3.0

Programming language: Maple

Journal reference of previous version: Comput. Phys. Commun. 183 (9) (2012) 2019–2020, https://doi.org/10.1016/j.cpc.2012.03.024

Does the new version supersede the previous version?: Yes

Nature of problem: Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries.

Solution method: The default method of integration is a fifth-order Runge–Kutta procedure, but the following numerical methods of integration are programmed and now available for the user of the Ndynamics package: rkf45, ck45, rosenbrock, rkf45_dae, ck45_dae, rosenbrock_dae, dverk78, lsode, gear, taylorseries, mebdfi, and classical. A box counting method is used to calculate the fractal dimension of the boundaries.

Reasons for new version: The Ndynamics package is still being used (as can be seen from the very new citation [1]), so it is worth to update its programming, for instance, by including the new numeric integrators available with the commercial release of Maple (just cited above on the solution method). We have also taken out of usage some such commands that are not available anymore, thus making the package current, more powerful and fixing some aspects. The (at the time of release) great novelty of the package, namely, the possibility of opening an interface for outside maple numerical integration was updated, we have changed the numerical C-integrator suggested and have improved some aspects of the RK45 numerical integration routine available with the present Maple package, the last such upgrade was done more than 12 years ago.

Summary of revisions: Since the first release of our package [2, 3], many years ago, with the implementation of our ideas and methods of how to proceed about the integrating of dynamical system, analyzing graphically such integration and calculating, via a box counting algorithm, the associated fractal dimension, this package has proven to be still in use by the research community. As already mentioned, the most recent citation to our paper (and package) is [1] (in Non-linear Science and Numerical Simulation, Volume 140, Part 1, January 2025), thus very recent and the paper and corresponding computational implementation is still relevant, providing the research community with an useful set of tools. The Ndynamics package was greatly de-phased with new developments on the Maple platform (basin where our program and commands run). The numerical compiler that could be used in conjugation with the Maple programming, one of the innovative ideas we have introduced, at the time of its first release, via the maple command system (allowing for maple to use programs running on the system where it is being used and then return to maple - please see maple help for system [4]) - was somewhat out of date. We have also updated the programming in relation to the new Maple commands for numerical integration now available.

The command Nsolve is the command that the users employ: the surface command that uses a host of others internally. As promised above, since the bulk of the changes that were made now are on those four routines singled out below, let us elaborate a little bit about those. The most relevant are: ‘Nsolve/integrator’; ‘Nsolve/integrator/points’; ‘Nsolve/integrator/C’ and ‘Nsolve/plotter’. Briefly, these four sub-routines get the input from Nsolve and a) determine which points will be used (and how many) via the parameters passed down by Nsolve; b) two of those sub-routines, depending also on those parameters, integrate still “inside” Maple or leaves Maple and perform the numerical integration using the external interface the package manages and finally plots the results. All these processes are controlled (by Nsolve and the global parameters necessary) by the user.

Additional comments including restrictions and unusual features: We are still restricted to working in any Euclidean space. We work with Boxcounting algorithms in order to calculate the fractal dimension. Note that the box-counting dimension can be defined in a metric space, but the definition that allows an easy calculation is only available for Euclidean spaces (see an example in [5] for more recent approaches).

This package provides user-friendly software tools for analyzing the character of a dynamical system, whether it displays chaotic behavior or not. Options within the package allow the user to specify characteristics that separate the trajectories into families of curves. In conjunction with the facilities for altering the user's viewpoint, this provides a graphical interface for the speedy and easy identification of regions with interesting dynamics. An unusual characteristic of the package is its interface for performing the numerical integrations in C++ using a fifth-order Runge–Kutta method (default). This potentially improves the speed of the numerical integration by some orders of magnitude and, in cases where it is necessary to calculate thousands of graphs in regions of difficult integration, this feature is very desirable. Besides that tool, somewhat more experienced users can produce their own C++ integrator and, by using the commands available in the package, use it as the C++ integrator provided with the package as long as the new integrator manages the input and output in the same format as the default one does.

References

• [1]
  Olesia Dogonasheva, Daniil Radushev, Boris Gutkin, Denis Zakharov, Dynamical manifold dimensionality as characterization measure of chimera states in bursting neuronal networks, Commun. Nonlinear Sci. Numer. Simul. 140 (Part 1) (January 2025) 108321, https://doi.org/10.1016/j.cnsns.2024.108321.
• [2]
  L.G.S. Duarte, L.A.C.P. da Mota, H.P. de Oliveira, R.O. Ramos, J.E.F. Skea, Numerical analysis of dynamical systems and the fractal dimension of boundaries, Comput. Phys. Commun. 119 (2–3) (2 June 1999) 256–271, https://doi.org/10.1016/S0010-4655(99)00204-0.
• [3]
  J. Avellar, L.G.S. Duarte, L.A.C.P. da Mota, N. de Melo, J.E.F. Skea, The Ndynamics package—numerical analysis of dynamical systems and the fractal dimension of boundaries, Comput. Phys. Commun. 183 (9) (September 2012) 2019–2020, https://doi.org/10.1016/j.cpc.2012.03.024.
• [4]
  https://www.maplesoft.com/support/help/Maple/view.aspx?path=ssystem.
• [5]
  M. Fernández-Martínez, M.A. Sánchez-Granero, Fractal dimension for fractal structures, Topol. Appl. 163 (15 February 2014) 93–111, https://doi.org/10.1016/j.topol.2013.10.010.