void verifyExistenceOfPeriodicOrbit(IPoincareMap& pm, IVector X, int period)
{
IVector center = midVector(X);
interval returnTime;
// Center is 2-dimensional. Embed it into Poincare section, i.e. add first coordinate x=0.
// Define a tripleton representation of the center of X.
C0HOTripletonSet s1({interval(0.),center[0],center[1]});
// Compute iteration of Poincare map at the center (3-dim object is returned).
IVector y = pm(s1,returnTime,period);
// Project it onto 2-dim Poincare section, first coordinate is ignored.
IVector imCenter(2,y.begin()+1);
// Derivative of PM on the set X.
// Define doubleton representation of the first order variational equations.
C1HORect2Set s2({interval(0.),X[0],X[1]});
// The matrix monodromyMatrix will store derivatives of the FLOW not Poincare map.
IMatrix monodromyMatrix(3,3);
y = pm(s2,monodromyMatrix,returnTime,period);
// This member function recomputes derivatives of the flow into derivatives of Poincare map
IMatrix DP = pm.computeDP(y,monodromyMatrix);
// We extract a 2x2 slice from 3x3 DP matrix and subtract identity.
IMatrix DP_Minus_Id(2,2);
DP_Minus_Id[0][0] = DP[1][1] - 1.;
DP_Minus_Id[0][1] = DP[1][2];
DP_Minus_Id[1][0] = DP[2][1];
DP_Minus_Id[1][1] = DP[2][2] - 1.;
// Compute interval Newton operator.
IVector N = center - capd::matrixAlgorithms::gauss(DP_Minus_Id,imCenter-center);
// Verification if N is a subset of X
cout << "\n---------------------------------------------------\n\nN = " << N << endl;
cout << "X = " << X << endl;
cout << "Return time: " << returnTime << endl;
if(subsetInterior(N,X))
cout << "the existence of period " << period << " orbit verified\n";
else
{
cout << "N is not a subset of X\n\n";
cout << "diam(N)=" << diam(N) << endl;
cout << "diam(X)=" << diam(X) << endl;
cout << "N-X" << N-X << endl;
}
}
static auto prepare_it(IIterator ifirst, IIterator ilast, EIterator efirst, EIterator elast, IVector& ivec, EVector& evec) {
std::copy(ifirst, ilast, std::back_inserter(ivec));
auto input_first = ivec.begin();
auto input_last = ivec.end();
if (Denoising) {
std::copy(efirst, elast, std::back_inserter(evec));
auto expected_first = evec.begin();
auto expected_last = evec.end();
return std::make_tuple(input_first, input_last, expected_first, expected_last);
} else {
return std::make_tuple(input_first, input_last, input_first, input_last);
}
}
