  Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model
without an external field on a closed chain of spins of width 2 in the
direction of interaction of only nearest neighbors and length L in the
direction of interaction of nearest neighbors and next nearest neighbors, we
found exact values of the partition function in a finite strip of length L ,
free energy, internal energy per node, heat capacity in a finite strip of
length L and in the thermodynamic limit. The entire spectrum of the 16x16
transfer matrix and the structure of all transfer matrix eigenvectors are
found. The problem of finding the spectrum of the transfer matrix is reduced to
finding the spectrum of two matrixes of size 8x8. The characteristic polynomial
of the first matrix is decomposed into two linear polynomials of multiplicity
two and two square trinomials. The characteristic polynomial of the second
matrix is decomposed into two polynomials of the fourth degree, the roots of
which are found by the Ferrari method. The free energy in the thermodynamic
limit has a very simple form and is expressed in terms of the logarithm of the
root of the quadratic equation. Two theorems on the physical characteristics of
the ANNNI model in a finite closed strip of length L and in the thermodynamic
limit are formulated and proved. An example of calculation is given for some
values of the model parameters in the thermodynamic limit, heat capacity graphs
are plotted. A comparative analysis of the heat capacity for the
double-stranded ANNNI model and the ANNNI model on the entire flat lattice is
given. The closest connection and similarity of the physical characteristics of
the two-chain model and the model on the entire flat lattice is shown.
