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DDE_Matlab
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README.md

Shift Continuation

This repository holds Matlab and Julia codes for numerical continuations from the article Phase shifts inside Arnold tongues of weakly coupled oscillators by Jakub Záthurecký, Veronika Eclerová, Jan Ševčík, Štěpán Zapadlo, and Lenka Přibylová.

Figures

All of the figures included in our paper may be generated using code from this repository.:

  • Fig. 2: ODE_Matlab/script_shift_Hopf2.m both generates and plots data;
  • Fig. 3: ODE_Matlab/script_shift_VDP2.m both generates and plots data;
  • Fig. 4: DDE_Matlab/script_shift_VDP2.m generates data and DDE_Matlab/script_plot_VDP2.m plots computed data;
  • Fig. 5: ODE_Matlab/script_shift_IN2.m both generates and plots data;
  • Fig. 6: DDE_Matlab/script_shift_Interneuron2.m generates data and DDE_Matlab/script_plot_Interneuron2.m plots computed data.

In addition, we have also shared our computed data files (in ODE_Matlab and DDE_Matlab/data directories). These may be used to simply plot figures from the article without the need to compute continuations from scratch.

Structure

ODE Matlab

Matlab code, based on MatCont (and its programmable interface), which includes scripts and utilities to perform continuation of periodic orbits in coupled neuron models and compute shifts between the coupled models along these curves.

Note that code of the Matcont toolbox is not included in this repository.

We provide several models a user can choose from, which can be found in the MatcontSystems folder. Moreover, scripts to generate figures from the article are script_shift_VDP2.m, script_shift_Hopf2.m, and script_shift_IN2.m.

Last, but not least, we also provide script_shift_VDP2_intervals.m (and similarly for interneuron model) scripts which for a

\lambda
(and other parameters) computes shifts (along with intervals of their stability).

DDE Matlab

Matlab code, based on DDE-BIFTOOL, which includes scripts and utilities to perform continuation of periodic orbits in delayed coupled neuron models and compute shifts between the coupled models along these curves.

We provide coupled van der Pol oscillators (see systems/vdP2.m) and coupled interneuron models (see systems/Interneuron2.m). The primary script to generate figures from the article is split into to (similarly for coupled interneuron models with filename Interneuron2):

  1. script_shift_VDP2.m - computes continuations of periodic orbits and their corresponding shifts and saves this data into the data folder;
  2. script_plot_VDP2.m - plots computed data given a sparse and a dense data files.

Last, but not least, we also provide script_shift_VDP2_intervals.m (and similarly for interneuron model) scripts which for a

\lambda
(and other parameters) computes shifts (along with intervals of their stability).

Julia

Julia code, based on DifferentialEquations.jl, BifurcationKit.jl and DDEBifurcationKit.jl, serves as a truly open-source alternative to our ODE/DDE codes and scripts relying on Matlab (through DDE-BIFTOOL or MatCont).

Here, one should pay most attention to either computation_one.jl (continuation given a fixed

\lambda
) or computation_series.jl (continuation given a range of
\lambda
s).

Important: Unfortunately, DDEBifurcationKit.jl does not yet provide a way to compute Floquet multipliers (thus stability) of a periodic orbit for continuation in DDEs (see this issue).

License

For more information on license, please see LICENSE.