We present an observational and theoretical study of the spiral structure in galaxy NGC 1566 .
A digitized image of NGC 1566 in I -band , obtained with the Australian National University 30in telescope , was used for measurements of the radial dependence of amplitude variations in the spiral arms .
The azimuthal variations of the surface brightness in the I -band are about \pm 7 % at 50 ^ { \prime \prime } radius and increase up to 57 % at 100 ^ { \prime \prime } .

We use the known velocity dispersion in the disk of NGC 1566 , together with its rotation curve , to construct linear and 2D nonlinear simulations which are then compared with observations .
The linear stability analysis of the disk of NGC 1566 , made under the assumption that the ratio of the vertical and the radial velocity dispersions is equal to 0.6 , shows that the disk is stable with respect to spiral perturbations .
We confirm this conclusion with 2D hydrodynamical simulations .
The disk of NGC 1566 is unstable if the ratio of the vertical and the radial velocity dispersions c _ { z } / c _ { r } is of order 0.8 - 1.0 .
Under this assumption , a two-armed spiral constitutes the most unstable global mode in the disk of NGC 1566 .

The 2D hydrodynamic simulations of the unstable disk seeded with random perturbations show the exponential growth of two unstable modes , m = 2 , and m = 3 .
The growth rates of the global modes seen in the nonlinear simulations are in good agreement with the results of the linear modal analysis .
The m = 3 global mode is less important , however , in comparison with the main m = 2 global mode , and the overall evolution of the perturbations is determined by the two-armed spiral mode , which saturates at a level \log _ { 10 } A _ { 2 } \approx - 0.5 .
The theoretical surface amplitude and the velocity residual variations across the spiral arms calculated during the nonlinear phase of instability are in qualitative agreement with the observations .

The spiral arms found in the linear and nonlinear simulations are considerably shorter than those observed in the disk of NGC 1566 .
The nonlinear phase of instability is characterized by the transport of angular momentum towards the disk center , making the surface density distribution quite steep in comparison to the observations .
We argue therefore , that the surface density distribution in the disk of the galaxy NGC 1566 was different in the past , when spiral structure in NGC 1566 was linearly growing .