Matlab code to reproduce computations in [Dodson and Sandstede 2019].
This repository provides Matlab code to solve for spiral wave patterns, spectra, and additional related tasks from S Dodson and B Sandstede. Determining the source of period-doubling instabilities in spiral waves. SIAM Journal on Applied Dynamical Systems 18 (2019) 2202-2226. Details about the methods can be found in the paper, here we provide instructions on how to run the codes.
In general, the codes are organized into folders by task, and the contents of each folder are described in the relevant sections. The folder Util contains utility functions that one or more scripts may call. The scripts work for the Karma and Rössler models by calling the model specific functions, which lie in the Util folder, and much of this code can be adapted for other reaction-diffusion systems by modifying these model specific functions.
Contains files for Rössler and Karma model asymptotic wave trains, spirals on bounded disks, and boundary sinks which can be used as initial conditions for solving or continuing systems. Files are labeled with model and key system or numerical parameters.
All patterns are set up as root finding problems using the built-in Matlab solver fsolve. The Matlab scripts used to solve each of the patterns is listed below.
Asymptotic Wave Trains: solve_1D_wave_train
Spiral Wave on Disks: solve_spiral_pattern
Works for spiral on disks with Neumann boundary conditions or non-reflecting boundary conditions, depending on functions specified inthe variable file_names.solver_fcn and the phase condition given by
file_names.set_up_phase_condition.
Boundary Sink: solve_boundary_sink
Methods based on those in Lloyd and Scheel, 2017 and Goh and Scheel, 2018.
The continuation code uses a mixture of secant and simple continuation methods.
Asymptotic Wave Trains: continue_wave_train
Secant continuation.
Spiral Waves on Disks: continue_spiral
Secant continuation that works for spiral waves on bounded disks with Neumann or non-reflecting boundary conditions based on solver files. Continue from a Neumann boundary condition to non-reflecting using the non-reflecting options (file_names.problem ='Rossler_2D_spiral_nonreflecting' and
file_names.set_up_phase_condition ='spiral_non_reflecting_phase_condition') with continuation parameter κ (contPar.Name = 'kappa').
Point Eigenfunction: continue_non_reflecting_bc_eigenfunction
Simple continuation that starts with spiral and eigenfucntion with Neumann boundary conditions and continues to non-reflecting boundary conditions using the two-step procees of (1) solving for spiral wave and (2) computing new eigenfunction. The two-step process is necessary because the linearization in the eigenvalue problem requires a spiral solution with the same parameter values.
Functions to compute the spectra of spirals on bounded disks and boundary sink.
Essential Spectrum: compute_essential_spectrum
Simple continuation to compute the essential spectrum. Methods follow those outlined in Rademacher et al 2007.
Absolute Spectrum: continue_absolute_spectrum
Simple continuation along absolute spectrum curve. An initial starting point is provided for the Karma and Rossler systems in the data_files/ folder. Methods follow those outlined in [](Rademacher et al 2007). The function defined by file_names.spatial_evals_fcn = 'spatial_evals_fcn_karma' computes the spatial eigenvalues ν for the given value of the temporal eigenvalue λ. By operating independently from the continuation this function provides a second check of the spatial eigenvalue distribution.
Spiral Point Spectrum: compute_point_spectra
Uses the sparse eigenvalue solver eigs to compute eigenvalues of spiral on a bounded disk near the user defined seed points (seed_real and seed_imag). Works for Neumann or non-reflecting boundary conditions based on defined jacobian fuction. Warning: do not use eigs if the system is highly non-normal, especially if one of the diffusion coefficients is 0.
Boundary Sink Point Spectrum: compute_point_spectra_boundary_sink
Uses the sparse eigenvalue solver eigs to compute eigenvalues of spiral on a bounded disk near the user defined seed points (seed_real and seed_imag).