Software

Numerical bifurcation analysis of renewal equations (integrated approach)

Author: Francesca Scarabel

MATLAB codes for the numerical bifurcation analysis of the ODE approximating system obtained by pseudospectral discretization of renewal equations with finite delay using the formulation for the integrated state (main reference [1]). The bifurcation analysis of the ODE system is performed using MatCont.

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  • [1] F. Scarabel, O. Diekmann and R. Vermiglio, Numerical bifurcation analysis of renewal equations via pseudospectral approximation, submitted, arXiv: 2012.05364 [math.NA], 2021.

Numerical bifurcation analysis of nonlinear delay equations

Author: Francesca Scarabel

MATLAB codes for the numerical bifurcation analysis of the ODE approximating system obtained by pseudospectral discretization of delay differential equations and renewal equations with finite delay (main reference [2]), using MatCont. The codes are organized in several folders corresponding to each publication.

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  • [2] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1–23, DOI: 10.1137/15M1040931.

Numerical bifurcation analysis of structured population models formulated as PDE

Author: Francesca Scarabel

MATLAB codes for the numerical bifurcation analysis of the ODE approximating system obtained by pseudospectral discretization of structured population models formulated as PDE [3], using MatCont.

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  • [3] F. Scarabel, D. Breda, O. Diekmann, M. Gyllenberg and R. Vermiglio, Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation, Vietnam J. Math. (2020), DOI: 10.1007/s10013-020-00421-3.

Computation of periodic solutions

Author: Alessia Andò

Demo examples of Python scripts implementing the continuation of periodic solutions for models defined by renewal equations or systems of coupled renewal and delay differential equations, as described in [4].

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  • [4] A. Andò, Collocation methods for complex delay problems of structured populations, PhD thesis, University of Udine, Italy, 2020.

Efficient numerical continuation of equilibria

Author: Alessia Andò

Demo examples of Python scripts implementing the internal continuation method described in [5].

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  • [5] A. Andò, D. Breda and F. Scarabel, Numerical continuation and delay equations: A novel approach for complex models of structured populations, Discrete Contin. Dyn. Syst. Ser. S (2019), DOI: 10.3934/dcdss.2020165.

Software for the CISM advanced school on delay

See the dedicated page.

Information on the school.

Efficient numerical computation of the basic reproduction number for structured populations

The codes relevant to the recently submitted paper D. Breda, F. Florian, J. Ripoll and R. Vermiglio, Efficient numerical computation of the basic reproduction number for structured populations will be available upon publication.

For inquiries please contact Dimitri Breda.

Numerical bifurcation of equations with infinite delay via pseudospectral collocation

Author: Ilaria Fontana

MATLAB codes for discretizing equations with infinite delay via pseudospectral collocation and studying their bifurcations via continuation with MatCont. The codes were developed as part of Ilaria’s MSc thesis.

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Automatic differentiation for equilibria of dynamical systems

Author: Marco Gambone

MATLAB codes based on ADiMat [6] computing equilibria of epidemiological models and studying their stability. The codes were developed as part of Marco’s BSc thesis.

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  • [6] C. H. Bischof, H. M. Bücker, B. Lang, A. Rasch and A. Vehreschild, Combining source transformation and operator overloading techniques to compute derivatives for MATLAB programs, in Proceedings of the Second IEEE International Workshop on Source Code Analysis and Manipulation (SCAM 2002), IEEE Computer Society, Los Alamitos, CA, 2002, pp. 65–72, DOI: 10.1109/SCAM.2002.1134106.

differential

Author: Francesco Florian

C++ library providing interpolatory quadrature and differentiation weights. The library was developed as part of Francesco’s MSc thesis.

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eigTMNc

Authors: Davide Liessi and Dimitri Breda

MATLAB code extending the method of eigTMN to approximate the multipliers of linear coupled renewal equations and retarded functional differential equations. See [7] for renewal equations and [8] for coupled equations.

  • [7] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linear renewal equations, SIAM J. Numer. Anal., 56 (2018), pp. 1456–1481, DOI: 10.1137/17M1140534.
  • [8] D. Liessi, Pseudospectral methods for the stability of periodic solutions of delay models, PhD thesis, University of Udine, Italy, 2018.

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LEs for DDEs

Authors: Dimitri Breda and Sara Della Schiava

MATLAB codes from a recent work inspired by Sara’s MSc thesis, see [9]. Based on the pseudospectral reduction to ODEs, with these codes one can approximate a number of Lyapunov exponents of a DDE. This version is tuned for the Mackey–Glass equation, but modification should be straightforward.

  • [9] D. Breda and S. Della Schiava, Pseudospectral reduction to compute Lyapunov exponents of delay differential equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), pp. 2727–2741, DOI: 10.3934/dcdsb.2018092.

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eigAM/eigTMN

Authors: Dimitri Breda, Stefano Maset, and Rossana Vermiglio

MATLAB codes accompanying the recent book [10]. They are devoted to the computation of either the characteristic roots or multipliers of linear, respectively autonomous and periodic, delay differential equations. Chapter 8 of the book should be enough to learn and experimenting.

  • [10] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB, SpringerBriefs Control Autom. Robot., Springer, New York, 2015, DOI: 10.1007/978-1-4939-2107-2.

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LEVEL – level curves of surfaces

Authors: Dimitri Breda, Stefano Maset, and Rossana Vermiglio

A MATLAB package for the computation of level curves of surfaces. It is a grid-like approach based on adaptive triangulation. The method is described in [11]. The package contains a brief manual and test cases.

  • [11] D. Breda, S. Maset and R. Vermiglio, An adaptive algorithm for efficient computation of level curves of surfaces, Numer. Algorithms, 52 (2009), pp. 605–628, DOI: 10.1007/s11075-009-9303-2.

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TRACE-DDE – Tool for Robust Analysis and Characteristic Equations of Delay Differential Equations

Authors: Dimitri Breda, Stefano Maset, and Rossana Vermiglio

A MATLAB GUI for computing characteristic roots and stability charts of linear autonomous systems of DDEs with several discrete and/or distributed delays, see the book chapter [12]. The GUI structure, developed at UniUD by Daniele Sechi in his BSc thesis, properly works on Mac OS X 10.3 (or superior) with MATLAB 7x, but is no longer maintained.

  • [12] D. Breda, S. Maset and R. Vermiglio, TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations, in J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, eds., Topics in Time Delay Systems: Analysis, Algorithms, and Control, Lect. Notes Control Inf. Sci. 388, Springer, New York, 2009, pp. 145–155, DOI: 10.1007/978-3-642-02897-7.

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