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Observations and simulations of barred spiral galaxies have shown that, ingeneral, the spiral arms rotate at a different pattern speed to that of thebar. The main conclusion from the bibliography is that the bar rotates fasterthan the spiral arms with a double or even a triple value of angular velocity.The theory that prevails in explaining the formation of the spiral arms in thecase of a barred spiral galaxy with two pattern speeds is the manifold theory,where the orbits that support the spiral density wave are chaotic, and arerelated to the manifolds emanating from the Lagrangian points L_1 and L_2 atthe end of the bar. In the present study, we consider an alternative scenarioin the case where the bar rotates fast enough in comparison with the spiralarms and the bar potential can be considered as a perturbation of the spiralpotential. In this case, the stable elliptical orbits that support the spiraldensity wave (in the case of grand design galaxies) are transformed intoquasiperiodic orbits (or 2D tori) with a certain thickness. The superpositionof these perturbed preccesing ellipses for all the energy levels of theHamiltonian creates a slightly perturbed symmetrical spiral density wave.

Perturbed precessing ellipses as the building blocks of spiral arms in a barred galaxy with two pattern speeds