Using spectral methods, we analyse the orbital structure of prolate/triaxial dark matter (DM) halos in N-body simulations to understand the processes that drive the evolution of shapes of DM halos and elliptical galaxies in which central masses are grown. A longstanding issue is whether the change in the shapes of DM halos is the result of chaotic scattering of box orbits, or whether they change shape adiabatically in response to the evolving galactic potential. We use orbital frequencies to classify orbits, to quantify orbital shapes, and to identify resonant orbits and chaotic orbits. The frequency-based method overcomes the limitations of Lyapunov exponents which are sensitive to numerical discreteness effects. Regardless of the distribution of the baryonic component, the shape of a DM halo changes primarily due to changes in the shapes of individual orbits within a given family. Orbits with small pericentric radii are more likely to change both their orbital type and shape than orbits with large pericentric radii. Whether the evolution is regular (and reversible) or chaotic (and irreversible), depends primarily on the radial distribution of the baryonic component. The growth of an extended baryonic component of any shape results in a regular rather than chaotic change in orbital populations. In contrast the growth of a massive and compact central component results in chaotic scattering of a significant fraction of both box and long-axis tube orbits. The growth of a disk causes a significant fraction of halo particles to become trapped by major global orbital resonances. Despite the fact that shape of a DM halo is always quite oblate following the growth of a central baryonic component, a significant fraction of its orbit population has characteristics of its triaxial or prolate progenitor (ABRIDGED).
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