Minimal demo - Normal forms of Hopf bifurcations

This part creates the computations of normal form coefficients along Hopf bifurcations, requiring theextension ddebiftool_extra_nmfm. This demo requires minimal_demo_stst_psol.html to have run beforehand.

(c) DDE-BIFTOOL v. 3.1.1(74), 31/12/2014

Contents

%#ok<*SAGROW>
hopfcurves={hopf,hopf1};
ind_hoho=0;
for i=1:length(hopfcurves)

Compute Lyapunov coefficient L1 along Hopf curves

    [L1{i},L1low]=HopfLyapunovCoefficients(funcs,hopfcurves{i});
    fprintf('maximal error of L1 along hopf branch=%g\n',norm(L1{i}-L1low,'inf'));

maximal error of L1 along hopf branch=2.0088e-09

maximal error of L1 along hopf branch=3.44257e-09

Hopf bifurcation changes criticality

Detect generalized Hopf bifurcation

    [genh{i},genhlow,hopfref{i},ind_genh(i)]=GeneralizedHopfNormalform(funcs,hopfcurves{i},...
        find(diff(sign(L1{i}))~=0,1)+(0:1));
    fprintf(['Generalized Hopf point at (b,tau)=(%g,%g)\n',...
        'with L1=%g, L2=%g,\n',...
        'L1 error est=%g, L2 error est=%g.\n'],...
        genh{i}.parameter(indb),genh{i}.parameter(indtau),...
        genh{i}.nmfm.L1,genh{i}.nmfm.L2,...
        abs(genh{i}.nmfm.L1-genhlow.nmfm.L1),abs(genh{i}.nmfm.L2-genhlow.nmfm.L2));

Bisection residual: res(1)=0.0193873, res(2)=-0.00850967
Bisection: new residual=0.00556438
Bisection: new residual=-0.00143977
Bisection: new residual=0.00207033
Bisection: new residual=0.000317312
Bisection: new residual=-0.000560717
Bisection: new residual=-0.000121575
Bisection: new residual=9.79013e-05
Bisection: new residual=-1.18288e-05
Generalized Hopf point at (b,tau)=(0.0369058,14.8349)
with L1=-1.18288e-05, L2=-1.49612,
L1 error est=3.48683e-10, L2 error est=0.000216606.

Bisection residual: res(1)=0.00488743, res(2)=-0.018094
Bisection: new residual=-0.00649218
Bisection: new residual=-0.00077775
Bisection: new residual=0.00206062
Bisection: new residual=0.000642925
Bisection: new residual=-6.70357e-05
Bisection: new residual=0.000288038
Bisection: new residual=0.000110526
Bisection: new residual=2.17533e-05
Generalized Hopf point at (b,tau)=(0.0378536,6.08046)
with L1=2.17533e-05, L2=-0.587593,
L1 error est=1.35718e-09, L2 error est=0.00074193.

Compute stability along Hopf curve

This shows that there are several HopfHopf interactions: detect them and compute their normal form.

    [nunsth{i},dum,dum,hopfref{i}.point]=GetStability(hopfref{i},'funcs',funcs,...
        'exclude_trivial',true,'locate_trivial',@(p)[-1i*p.omega,1i*p.omega]); %#ok<ASGLU>
    ind_hh=find(abs(diff(nunsth{i}))==2);
    for k=1:length(ind_hh)
        ind_hoho=ind_hoho+1;
        [hoho{ind_hoho},hoho_low]=HopfHopfNormalform(funcs,hopfref{i},ind_hh(k)+(0:1));
        fprintf(['Normal form coefficients of Hopf-Hopf point\n',...
            'at (b,tau)=(%g,%g) with omega1=%g, omega2=%g:\n'],...
            hoho{ind_hoho}.parameter(indb),hoho{ind_hoho}.parameter(indtau),...
            hoho{ind_hoho}.omega1,hoho{ind_hoho}.omega2);
        disp(hoho{ind_hoho}.nmfm);
        fprintf('Error of normal form coefficients: %g\n',...
        norm(struct2array(hoho{ind_hoho}.nmfm)-struct2array(hoho_low.nmfm),'inf'));
    end

Bisection residual: res(1)=0.000694535, res(2)=-0.00146838
Bisection: new residual=-0.000386969
Bisection: new residual=0.000153767
Bisection: new residual=-0.000116604
Bisection: new residual=1.85803e-05
Bisection: new residual=-4.90122e-05
Bisection: new residual=-1.5216e-05
Bisection: new residual=1.68216e-06
Normal form coefficients of Hopf-Hopf point
at (b,tau)=(0.354452,8.72869) with omega1=1.09756, omega2=0.708364:
    g2100: 0.1385 + 0.1232i
    g1011: 0.3121 + 0.2776i
    g1110: -0.3656 + 0.2017i
    g0021: -0.2059 + 0.1136i
    theta: -1.5155
    delta: -2.6393
Error of normal form coefficients: 2.15019e-08
Bisection residual: res(1)=-0.00435055, res(2)=0.000946688
Bisection: new residual=-0.00171931
Bisection: new residual=-0.000390537
...

Bisection residual: res(1)=-0.00159524, res(2)=0.00260849
Bisection: new residual=0.00052094
Bisection: new residual=-0.000533554
Bisection: new residual=-5.41024e-06
Bisection: new residual=0.000257989
Normal form coefficients of Hopf-Hopf point
at (b,tau)=(0.355063,8.72902) with omega1=0.708358, omega2=1.09771:
    g2100: -0.2055 + 0.1132i
    g1011: -0.3653 + 0.2012i
    g1110: 0.3106 + 0.2773i
    g0021: 0.1380 + 0.1233i
    theta: -2.6465
    delta: -1.5114
Error of normal form coefficients: 2.75342e-09

end

Save and continue

For continuation of folds and torus bifurcations of periodic orbits, see minimal_demo_extra_psol.html. Final results in minimal_demo_plot_2dbif.html.

save('minimal_demo_extra_nmfm_results.mat')

Published with MATLAB® 7.14