init_nsm_nsm.m

计算动力学系统的分岔图

function [x0,v0]= init_NSm_NSm(mapfile, x, p, ap,n)

%

% Initializes a Neimark_Sacker bifurcation continuation from a NS point

% 
nsmds=[];
global cds nsmds
% check input
if size(ap,2)~=2
    errordlg('Two active parameter are needed for a NS-curve continuation');
end
v0=[];
% initialize nsmds
nsmds.mapfile = mapfile;
func_handles = feval(nsmds.mapfile);
nsmds.func = func_handles{2};
nsmds.Jacobian  = func_handles{3};
nsmds.JacobianP = func_handles{4};
nsmds.Hessians  = func_handles{5};
nsmds.HessiansP = func_handles{6};
nsmds.Der3      = func_handles{7};
nsmds.Der4      = func_handles{8};
nsmds.Der5      = func_handles{9};
nsmds.Niterations      = n;
siz = size(func_handles,2);
if siz > 9
    j=1;
    for i=10:siz
        nsmds.user{j}= func_handles{i};
        j=j+1;
    end
end

nsmds.nphase = size(x,1);
nsmds.ActiveParams = ap;
nsmds.P0 = p;
cds.curve = @neimarksackermap;
cds.ndim = length(x)+3;
x0=[x;nsmds.P0(ap)];
[x1,p] = rearr(x0); p = n2c(p);
curvehandles = feval(cds.curve);
cds.curve_func = curvehandles{1};
cds.curve_options = curvehandles{3};
cds.curve_jacobian =curvehandles{4};
cds.curve_hessians = curvehandles{5};
cds.options = feval(cds.curve_options); 
cds.options = contset(cds.options,'Increment',1e-5);
jac = nsmjac(x1,p,n);
nphase = size(x1,1);
nap = length(nsmds.ActiveParams);
% calculate eigenvalues and eigenvectors
[V,D] = eig(jac);
d = diag(D);
% find a 'neutral pair'
idx1=0;idx2=0;
for s=1:nphase
  for j=s+1:nphase
      if (abs(1-d(s)*d(j))<0.00001)
         idx1=s; idx2=j;
      end
  end
end
dx1=idx1;dx2=idx2;
if idx1==0
  debug('No neutral pair found!');
  result='Could not find appropriate eigenvalues',return;
end
%check whether complex or real.
if abs((imag(d(idx1)))<0.00001)
   %check whether R1 or R2
    [V1,D1] = eig(jac-eye(nphase));
    [s,idx2] = min(abs(diag(D1)));
    if(s < 1e-5) %R1
      vext=real(V1(:,idx2));   
      [V1,D1]=eig(jac'-eye(nphase));
      [s,idx2]=min(abs(diag(D1)));
      wext=real(V1(:,idx2));
      Bord = [ jac-eye(nphase)  wext;vext' 0];
      genvext =  Bord\[vext; 0];genvext=genvext(1:nphase);
      genwext = Bord'\[wext; 0];genwext=genwext(1:nphase);
      [Q,R,E] = qr([vext genvext]);      
      nsmds.borders.v = Q(:,1:2);     
      [Q,R,E] = qr([wext genwext]); 
      nsmds.borders.w = Q(:,1:2);
      k=1;
      else
        
      [V2,D2] = eig(jac+eye(nphase));   
      [s,idx2] = min(abs(diag(D2)));
      if(s < 1e-5)  %R2
          vext=real(V2(:,idx2));
          [V2,D2]=eig(jac'+eye(nphase));
          [s,idx2]=min(abs(diag(D2)));
          wext=real(V2(:,idx2));
          Bord = [ jac+eye(nphase)  wext;vext' 0];
          genvext =  Bord\[vext; 0];genvext=genvext(1:nphase);
          genwext = Bord'\[wext; 0];genwext=genwext(1:nphase);
          [Q,R,E] = qr([vext genvext]);     
          nsmds.borders.v = Q(:,1:2);
          [Q,R,E] = qr([wext genwext]);     
          nsmds.borders.w = Q(:,1:2);
          k=-1;
          end
    end
   elseif (imag(d(dx1))==0) & (imag(d(dx2))==0) %Neutral saddle
        debug('Neutral saddle\n');
        [Q,R,E]=qr([V(:,dx1) V(:,dx2)]); 
        nsmds.borders.v = Q(:,1:2);         
        [V4,D4] = eig(jac'-d(dx1)*eye(nphase));
        [s4,d4]=min(abs(diag(D4)));
        [V5,D5] = eig(jac'-d(dx2)*eye(nphase));
        [s5,d5]=min(abs(diag(D5)));             
        [Q,R,E]=qr([V4(:,d4) V5(:,d5)]);
        nsmds.borders.w = Q(:,1:2); 
        k=0.5*(d(dx1)+d(dx2));
    
elseif (d(dx1)== conj(d(dx2)))
        
     %  V=V(:,idx1);
        [Q,R,E] = qr([real(V(:,dx1)) imag(V(:,dx1))]); %reza
        nsmds.borders.v = Q(:,1:2);
        % find pair of complex eigenvalues for transposed
        [V6,D6] = eig(jac'-d(dx1)*eye(nphase));
        [s6,d6]=min(abs(diag(D6))); 
        [Q,R,E] = qr([real(V6(:,d6)) imag(V6(:,d6))]);
        nsmds.borders.w = Q(:,1:2); 
        temp=idx1;  
        k  = real(d(idx1));
       
else
      debug('There is no such pair of eigenvalues\n');return,     
end 

x0 = [x0;k];
% calculate eigenvalues

% ERROR OR WARNING
RED  = jac*jac-2*k*jac+eye(nsmds.nphase);
jacp = nsmjacp(x1,p,n);
A = [jac-eye(nsmds.nphase)  jacp zeros(nsmds.nphase,1)];  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[Q,R] = qr(A');
Bord  = [RED nsmds.borders.w;nsmds.borders.v' zeros(2)];
bunit = [zeros(nsmds.nphase,2);eye(2)];
vext  = Bord\bunit;
wext  = Bord'\bunit;

wext1=jac'*wext(1:nsmds.nphase,1);
vext1=vext(1:nsmds.nphase,1);
nphase=size(x1,1); 
xx1=x1;
AA=zeros(nphase,nphase,n);
AA(:,:,1)=nsmjac(xx1,p,1);
xit=zeros(nphase,n);
for m=2:n    
 xx1=feval(nsmds.func,0,xx1,p{:});
 xit(:,1)=xx1;
 AA(:,:,m)=nsmjac(xx1,p,1);
end  
gx1=nsvecthessvect(xit,p,vext1,wext1',AA,n);
wext2=wext(1:nsmds.nphase,1)';
vext2=jac*vext(1:nsmds.nphase,1);
gx2=nsvecthessvect(xit,p,vext2,wext2,AA,n);
wext3=-2.0*k*wext(1:nsmds.nphase,1)';
vext3=vext(1:nsmds.nphase,1);
gx3=nsvecthessvect(xit,p,vext3,wext3,AA,n);
gxx1=gx1+gx2+gx3;
wext12=wext(1:nsmds.nphase,1)'*jac;
vext12=vext(1:nsmds.nphase,2);
gx12=nsvecthessvect(xit,p,vext12,wext12,AA,n);

wext22=wext(1:nsmds.nphase,1)';
vext22=jac*vext(1:nsmds.nphase,2);
gx22=nsvecthessvect(xit,p,vext22,wext22,AA,n);

wext32=-2.0*k*wext(1:nsmds.nphase,1)';
vext32=vext(1:nsmds.nphase,2);
gx32=nsvecthessvect(xit,p,vext32,wext32,AA,n);
gxx2=gx12+gx22+gx32;

wext31=wext(1:nsmds.nphase,2)'*jac;
vext31=vext(1:nsmds.nphase,1);
gx31=nsvecthessvect(xit,p,vext31,wext31,AA,n);

wext32=wext(1:nsmds.nphase,2)';
vext32=jac*vext(1:nsmds.nphase,1);
gx32=nsvecthessvect(xit,p,vext32,wext32,AA,n);

wext33=-2.0*k*wext(1:nsmds.nphase,2)';
vext33=vext(1:nsmds.nphase,1);
gx33=nsvecthessvect(xit,p,vext33,wext33,AA,n);
gxx3=gx31+gx32+gx33;

wext41=wext(1:nsmds.nphase,2)'*jac;
vext41=vext(1:nsmds.nphase,2);
gx41=nsvecthessvect(xit,p,vext41,wext41,AA,n);

wext42=wext(1:nsmds.nphase,2)';
vext42=jac*vext(1:nsmds.nphase,2);
gx42=nsvecthessvect(xit,p,vext42,wext42,AA,n);

wext43=-2.0*k*wext(1:nsmds.nphase,2)';
vext43=vext(1:nsmds.nphase,2);
gx43=nsvecthessvect(xit,p,vext43,wext43,AA,n);
gxx4=gx41+gx42+gx43;
for i = 1:nsmds.nphase
    gx(1,i)=gxx1(:,i);
    gx(2,i)=gxx2(:,i);
    gx(3,i)=gxx3(:,i);
    gx(4,i)=gxx4(:,i);
end
gk(1,1) =2*wext(1:nsmds.nphase,1)'*jac*vext(1:nsmds.nphase,1);
gk(2,1) =2*wext(1:nsmds.nphase,1)'*jac*vext(1:nsmds.nphase,2);
gk(3,1) =2*wext(1:nsmds.nphase,2)'*jac*vext(1:nsmds.nphase,1);
gk(4,1) =2*wext(1:nsmds.nphase,2)'*jac*vext(1:nsmds.nphase,2);
wext1=wext(1:nsmds.nphase,1)'*jac;
vext1=vext(1:nsmds.nphase,1);
gx1=nsvecthesspvect(xit,p,vext1,wext1,AA,n);

wext2=wext(1:nsmds.nphase,1)';
vext2=jac*vext(1:nsmds.nphase,1);
gx2=nsvecthesspvect(xit,p,vext2,wext2,AA,n);

wext3=-2.0*k*wext(1:nsmds.nphase,1)';
vext3=vext(1:nsmds.nphase,1);
gx3=nsvecthesspvect(xit,p,vext3,wext3,AA,n);
gp1=gx1+gx2+gx3;

wext12=wext(1:nsmds.nphase,1)'*jac;
vext12=vext(1:nsmds.nphase,2);
gx12=nsvecthesspvect(xit,p,vext12,wext12,AA,n);

wext22=wext(1:nsmds.nphase,1)';
vext22=jac*vext(1:nsmds.nphase,2);
gx22=nsvecthesspvect(xit,p,vext22,wext22,AA,n);

wext32=-2.0*k*wext(1:nsmds.nphase,1)';
vext32=vext(1:nsmds.nphase,2);
gx32=nsvecthesspvect(xit,p,vext32,wext32,AA,n);
gp2=gx12+gx22+gx32;

wext31=wext(1:nsmds.nphase,2)'*jac;
vext31=vext(1:nsmds.nphase,1);
gx31=nsvecthesspvect(xit,p,vext31,wext31,AA,n);

wext32=wext(1:nsmds.nphase,2)';
vext32=jac*vext(1:nsmds.nphase,1);
gx32=nsvecthesspvect(xit,p,vext32,wext32,AA,n);

wext33=-2.0*k*wext(1:nsmds.nphase,2)';
vext33=vext(1:nsmds.nphase,1);
gx33=nsvecthesspvect(xit,p,vext33,wext33,AA,n);
gp3=gx31+gx32+gx33;

wext41=wext(1:nsmds.nphase,2)'*jac;
vext41=vext(1:nsmds.nphase,2);
gx41=nsvecthesspvect(xit,p,vext41,wext41,AA,n);

wext42=wext(1:nsmds.nphase,2)';
vext42=jac*vext(1:nsmds.nphase,2);
gx42=nsvecthesspvect(xit,p,vext42,wext42,AA,n);

wext43=-2.0*k*wext(1:nsmds.nphase,2)';
vext43=vext(1:nsmds.nphase,2);
gx43=nsvecthesspvect(xit,p,vext43,wext43,AA,n);
gp4=gx41+gx42+gx43;
for i = 1:nap
    gp(1,i)=gp1(:,i);
    gp(2,i)=gp2(:,i);
    gp(3,i)=gp3(:,i);
    gp(4,i)=gp4(:,i);
end

A = [A;gx gp gk]*Q;
Jres = A(1+nsmds.nphase:end,1+nsmds.nphase:end)';
[Q,R,E] = qr(Jres');
index = [1 1;1 2;2 1;2 2];
[I,J] = find(E(:,1:2));
nsmds.index1 = index(I(1),:);%index(I(1),:),pause
nsmds.index2 = index(I(2),:);%index(I(2),:),pause
rmfield(cds,'options'); 


% ---------------------------------------------------------------

function [x,p] = rearr(x0)
% [x,p] = rearr(x0)
% Rearranges x0 into coordinates (x) and parameters (p)

global cds nsmds
nap = length(nsmds.ActiveParams);
p = nsmds.P0;
p(nsmds.ActiveParams) = x0((nsmds.nphase+1):end);
x = x0(1:nsmds.nphase);