The equation of the system is
dx(t)/dt=A(t)x(t) where A is 2*2 matrix
In floquet-lyaponov theory it is said that using two independent initial conditions we can calculate PHI(t) matrix which is state transition matrix.
let A=[0 exp(t) ;
0 1]
By taking independent initial conditions [1;0] and [0;1] how we can calculate phi matrix?
Solve the ODE for closed form solutions {x_{11}(t), x_{21}(t)} and {x_{12}(t), x_{22}(t)} for the two initial conditions. Then \Pi(t) = \mx{cc}{ x_{11}(t) & x_{12}(t) \\ x_{21}(t) & x_{22}(t) } is the fundamental matrix. The state transition matrix \Phi(t,t_0) = \Pi(t) (\Pi(t_0))^{-1}.
hi Zigang Pan.
i am not getting your notations can you demonstrate the answer for given example.
I am using bellow document tell me how they are getting (STM) in example one
This idea works equally well in the n-dim case: if [s_1,...,s_n] would be linearly independent vectors in R^n, then solve dx/dt=Ax with each initial condition x(0)=X_j to get a solution x_j(t). Let S be the matrix be the matrix with the columns [s_1..] and let X(t) be the matrix with the columns [x_1(t),...,x_n(t)]. Then Phi(t)=X(t)S^{-1}.
use pseudo-inverse Penrose and you have to see there are singularities or not then use svd and also regularization factor
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