Demo - System with cusp and Takens-Bogdanov bifurcations

Demo contributed by Maikel Bosschaert

from Giannakopoulos, F. and Zapp, A. (2001). Bifurcations in a planar system of differential delay equations modeling neural activity. Physica D: Nonlinear Phenomena, 159(3):215-232.

Contents

Differential equations

$$\mu\dot{u}_{1}(t)=-u_{1}(t)+q_{11}\alpha(u_{1}(t-T))-q_{12}u_{2}(t-T)+e_{1}$$

$$\mu\dot{u}_{2}(t)=-u_{2}(t)+q_{21}\alpha(u_{1}(t-T))-q_{22}u_{2}(t-T)+e_{2}$$

(c) DDE-BIFTOOL v. 3.1.1(120), 02/09/2015 

clear % clear variables
close all; % close figures
addpath('../../ddebiftool',...
    '../../ddebiftool_extra_psol',...
    '../../ddebiftool_extra_nmfm',...
    '../../ddebiftool_utilities');

disp('Cusp demo');

funcs=set_funcs(...
    'sys_rhs', @cusp_rhs,...
    'sys_tau', @()6,...
    'sys_deri', @cusp_deri,...
    'sys_mfderi',@(xx,par,varargin)cusp_mfderi(xx,par,varargin{:}));

Cusp demo

continue stst in E

Q=0;
E=0;

q11=2.6; q12=Q; q21=1; e1=E; e2=0; T=1;
par = [q11,q12,q21,e1,e2,T];

stst_br = SetupStst(funcs,'x',[0;0],'parameter',par,...
    'contpar',4,'max_step',[4 0.02],'max_bound',[4 1],'min_bound',[4 0],...
    'newheuristics_tests',0);

stst_br.method.continuation.plot = 1;
[stst_br,s,f,r] = br_contn(funcs,stst_br,300);

stst_br.method.bifurcation.plot_testfunctions=1;

stst_br.method.stability.minimal_real_part=-5;
stst_br=br_stabl(funcs,stst_br,0,0);
stst_br=br_bifdet(funcs,stst_br);

BR_CONTN warning: boundary hit.
BR_BIFDET: Fold detected near par(4) = 0.4899601374.
BR_BIFDET: Fold located at  par(4) = 0.4913668450.
BR_BIFDET: Normal form coefficient: b = 0.7321765321

Current plot held

continue fold point in (E,Q)

FPI=br_getflags(stst_br);
start_ind = FPI(bif2num('fold'),1);

[fold_branch, suc] = SetupFold(funcs, stst_br, start_ind, 'contpar', [2 4], 'dir', 4, 'step', 0.02);

Qmin=-0.1; Qmax=2; Emin=-0.6; Emax=0.6;
fold_branch.parameter.min_bound=[2 Qmin; 4 Emin];
fold_branch.parameter.max_bound=[2 Qmax; 4 Emax];
fold_branch.parameter.max_step=[2 0.02; 4 0.02];

figure;
[fold_branch,s,f,r]=br_contn(funcs,fold_branch,300);
fold_branch = br_rvers(fold_branch);
[fold_branch,s,f,r]=br_contn(funcs,fold_branch,300);

fold_branch.method.bifurcation.plot_testfunctions=1;

fold_branch = br_stabl(funcs,fold_branch,0,0);
fold_branch = br_bifdet(funcs,fold_branch);

BR_CONTN warning: boundary hit.
BR_CONTN warning: boundary hit.
BR_BIFDET: Bogdanov-Takens detected near par(2) = 0.0000000000, par(4) = 0.4913668450.
BR_BIFDET: Failed to correct Bogdanov-Takens point.

BR_BIFDET: Bogdanov-Takens detected near par(2) = 0.0199989457, par(4) = 0.4835320970.
BR_BIFDET: Failed to correct Bogdanov-Takens point.

BR_BIFDET: Bogdanov-Takens detected near par(2) = 0.0399981183, par(4) = 0.4757164567.
BR_BIFDET: Failed to correct Bogdanov-Takens point.

BR_BIFDET: Cusp detected near par(2) = 1.3179630819, par(4) = 0.0462438690.
BR_BIFDET: Failed to correct cusp.

BR_BIFDET: Bogdanov-Takens detected near par(2) = 1.3179630819, par(4) = 0.0462438690.
BR_BIFDET: a = -0.1903821241, b = -0.9519106203, par(2) = 1.3000000000, par(4) = 0.0505079212.

BR_BIFDET: Cusp detected near par(2) = 1.5990696079, par(4) = -0.0000094571.
BR_BIFDET: Cusp located at  par(2) = 1.6000000000, par(4) = -0.0000000000.
BR_BIFDET: Normal form coefficients: b = -0.0000002484, c = 0.5555557861.

BR_BIFDET: Bogdanov-Takens detected near par(2) = 1.5811766051, par(4) = -0.0008560336.
BR_BIFDET: Failed to correct Bogdanov-Takens point.

BR_BIFDET: Cusp detected near par(2) = 1.2852191807, par(4) = -0.0540913384.
BR_BIFDET: Failed to correct cusp.

BR_BIFDET: Bogdanov-Takens detected near par(2) = 1.2852191807, par(4) = -0.0540913384.
BR_BIFDET: a = 0.1903821241, b = 0.9519106203, par(2) = 1.3000000000, par(4) = -0.0505079212.


Published with MATLAB® R2014b