Gdddk

I would like to solve a $2times 2$ system of the form
$$fracddthetaT=TA,quad T(0)=Id$$
where $theta$ is real and $A$ is of the form
$$A=beginpmatrix 0 & frace^-i thetalambda\ frac136e^-ithetaleft(9lambda + 2(lambda-1)^2 (6costheta + cos2theta + 6)right) & 0endpmatrix,$$
with $lambda$ a free parameter in the unit circle.

In particular I'm interested in obtaining numeric solutions at $theta=2pi$ depending on the extra parameter $lambda$. I'm fairly new using Mathematica, and this is what I have tried so far:

T[θ_] = T11[θ], T12[θ], T21[θ], T22[θ];
A[θ_] = 
 0, E^(-I θ)/λ,
 1/36 E^(-I θ) (9 λ + 2 (-1 + λ)^2 (6 + 6 Cos[θ] + Cos[2 θ])), 0
;
sys = T'[θ] == T[θ].A[θ];

The previous code sets the system that I want to solve and now I try to solve numerically. First I've tried

NSol = NDSolve[
 sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1,
 T11[θ], T12[θ], T21[θ], T22[θ],
 θ,
 θ, 0, 2 Pi
];

which gives me the output

NDSolve::dupv: "Duplicate variable θ found in NDSolve[<<1>>]."

I have also tried

Nsol2 = ParametricNDSolve[
 sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1,
 T11, T12, T21, T22,
 θ, 0, 2 Pi,
 λ
];

which gives me as output $T_11,dots,T_22$ as ParametricFunctions depending on each other and on $lambda$.

I don't know if this is the right approach and, if so, how to extract a numeric expression depending on $lambda$ from the last output - all that I've seen in the documentation are examples that are plotted for specific values of the parameter. Any help is much appreciated.

EDIT

Following the comments in one of the answers I'd like to explain further: the output that I would like to obtain is some sort of function depending of the parameter $lambda$ that I can manipulate afterwards. Say for instance, computing the series expansion of powers of $lambda$ of my solution. I don't know how to treat the ParametricFunction that I obtain to do such computations.

Here is how to obtain the series expansion of the matrix components. First, using a tweaked version (solving for t[2π] instead of t) of JM's formulation:

A[θ_] = 
 0, E^(-I θ)/λ,
 1/36 E^(-I θ) (9 λ + 2 (-1 + λ)^2 (6 + 6 Cos[θ] + Cos[2 θ])), 0
;

pf = ParametricNDSolveValue[
 
 t'[θ] == t[θ].A[θ], t[0] == IdentityMatrix[2]
 ,
 t[2π],
 θ, 0, 2π,
 λ
];

Then, pf will return the matrix value at $theta = 2 pi$. For example:

pf[1]
pf[Exp[I Pi/3]]

1. + 3.62818*10^-9 I,
6.82646*10^-8 - 5.73536*10^-9 I, 1.70661*10^-8 - 1.43384*10^-9 I,
1. + 3.62818*10^-9 I

0.985595 + 1.17074 I, -0.425572 + 0.737112 I, -0.788363 - 1.36549 I,
0.985595 - 1.17074 I

Finding the series expansion is simple:

DecimalForm[Series[pf[λ], λ, 1, 5], 4,4] //TeXForm

$left(
beginarraycc
1.0000 & 6.2830 \
1.5710 & 1.0000 \
endarray
right)+left(
beginarraycc
0.0000+0.0000 i & -6.2830+0.0000 i \
1.5710+0.0000 i & 0.0000+0.0000 i \
endarray
right) (lambda -1)+left(
beginarraycc
0.0000-0.7869 i & 0.6691-0.0000 i \
0.8799+0.0000 i & -0.0000+0.7869 i \
endarray
right) (lambda -1)^2+left(
beginarraycc
0.0000+0.7869 i & -1.3380+0.0000 i \
0.0000+0.0000 i & 0.0000-0.7869 i \
endarray
right) (lambda -1)^3+left(
beginarraycc
-0.0152-0.4524 i & 1.8410-0.0000 i \
-0.5708-0.0000 i & -0.0152+0.4524 i \
endarray
right) (lambda -1)^4+left(
beginarraycc
0.0305+0.1179 i & -2.1780+0.0000 i \
0.5708-0.0000 i & 0.0305-0.1179 i \
endarray
right) (lambda -1)^5+Oleft((lambda -1)^6right)$

As far as I can tell, you did everything right in your second approach. Classically, Mathematica returns lists of rules as results of solving functions, but in this case, I find this tradition rather confusing and prefer to use ParametricNDSolveValue; it returns a ParametricFunction object that, when applied to a numerical parameter, returns a list of 4 InterpolatingFunction for your 4 functions.

T[θ_] = T11[θ], T12[θ], T21[θ], T22[θ];
A[θ_] = 0, E^(-I θ)/λ, 1/36 E^(-I θ) (9 λ + 2 (-1 + λ)^2 (6 + 6 Cos[θ] + Cos[2 θ])), 0;
sys = T'[θ] == T[θ].A[θ];

Tsol = ParametricNDSolveValue[sys, T11[0] == 1, T12[0] == 0, 
 T21[0] == 0, T22[0] == 1,
 T11, T12, T21, T22,
 θ, 0, 2 Pi,
 λ
 ];

In order to obtain the numerical values for all the solutions at θ = 2 Pi for a given parameter, say λ = 0.1, you may use Through:

Through[Tsol[0.1][2. Pi]]

-0.545795 + 1.00532 I, -1.43497 - 7.95125*10^-7 I, -0.215035 -
2.80298*10^-8 I, -0.545795 - 1.00532 I

In order to make that into a function, you may use

f = λ [Function] Through[Tsol[λ][2. Pi]]

For this case, you don't even need to write out the components of your matrix function:

pf = ParametricNDSolveValue[t'[θ] == t[θ].0, Exp[-I θ]/λ,
 Exp[-I θ] (9 λ + 2 (λ - 1)^2
 (6 Cos[θ] + Cos[2 θ] + 6))/36, 0,
 t[0] == IdentityMatrix[2], t, θ, 0, 2 π, λ,
 Method -> "StiffnessSwitching"];

sol = pf[(3 + 4 I)/5];

ParametricPlot[ReIm[Tr[sol[θ]]], θ, 0, 2 π]

ParametricPlot[ReIm[Det[sol[θ]]], θ, 0, 2 π]

You can even make a plot where the parameter is varying:

Plot[Re[Tr[pf[Exp[I ϕ]][2 π]]], ϕ, 0, 2 π]