The twist-bend nematic phase, $N_{\mathrm{TB}}$, may be viewed as a heliconical molecular arrangement in which the director $\mathbf{n}$ precesses uniformly about an extra director field, $\mathbf{t}$. It corresponds to a nematic ground state exhibiting nanoscale periodic modulation. To demonstrate the stability of this phase from the elastic point of view, a natural extension of the Frank elastic energy density is proposed. The elastic energy density is built in terms of the elements of symmetry of the new phase in which intervene the components of these director fields together with the usual Cartesian tensors. It is shown that the ground state corresponds to a deformed state for which $K_{22}>K_{33}$. In the framework of the model, the phase transition between the usual and the twist-bend nematic phase is of second order with a finite wave vector. The model does not require a negative $K_{33}$ in agreement with recent experimental data that yield $K_{33}>0$. A threshold is predicted for the molecular twist power below which no transition to a twist-bend nematic may occur.

We study the relationship between classical and quantum structures for a map on the sphere whose behavior can be chaotic in the classical limit. On the classical side we implement an efficient method to locate periodic points on symmetry lines. On the quantum side we show how matrix elements of the propagator in the coherent-state representation are connected to classical structures. Diagonal and off-diagonal matrix elements are related to periodic points, symmetry lines, and other invariant structures in phase space, both in the time and in the energy domains. The scarring phenomena related to the short periodic orbits and their homoclinic neighborhoods are discussed.