Lagrangian descriptors for the Bunimovich stadium billiard

Using Lagrangian descriptors to calculate the Maslov index of periodic orbits

The Maslov index of a periodic orbit is an important piece in the semiclassical quantization of nonintegrable systems, while almost all existing techniques that lead to a rigorous calculation of this index are elaborate and mathematically demanding. In this paper, we describe a straightforward technique, for systems with two degrees of freedom, based on the Lagrangian descriptors. Our method is illustrated by applying it to the two-dimensional coupled quartic oscillator.

Lagrangian descriptors: The shearless curve and the shearless attractor

Hamiltonian systems with a nonmonotonic frequency profile are called nontwist. One of the key properties of such systems, depending on adjustable parameters, is the presence of a robust transport barrier in the phase space called the shearless curve, which becomes the equally robust shearless attractor when dissipation is introduced. We consider the standard nontwist map with and without dissipation. We derive analytical expressions for the Lagrangian descriptor (LD) for the unperturbed map and show how they are related to the rotation number profile. We show how the LDs can reconstruct finite segments of the invariant manifolds for the perturbed map. In the conservative case, we demonstrate how the LDs distinguish the chaotic seas from regular structures. The LDs also provide a remarkable tool to identify when the shearless curve is destroyed: we present a fractal boundary, in the parameter space, for the existence or not of the shearless torus. In the dissipative case, we show how the LDs can be used to localize point attractors and the shearless attractor and distinguish their basins of attraction.