Dynamical behavior of a parametrized family of one-dimensional maps

Abstract

The connection of these maps to homoclinic loops acts like an amplifier of the map behavior, and makes it interesting also in the case where all map orbits approach zero (but in many possible ways). We introduce so-called 'flat' intervals containing exactly one maximum or minimum, and so-called `steep' intervals containing exactly one zero point of $f_{\mu,\omega}$ and no zero of $f'_{\mu,\omega}$. For specific parameters $\mu$ and $\omega$, we construct an open set of points with orbits staying entirely in the 'flat' intervals in section three. In section four, we describe orbits staying in the 'steep' intervals (for open parameter sets), and in section five (for specific parameters) orbits regularly changing between 'steep' and 'flat' intervals. Both orbit types are described by symbol sequences, and it is shown that their Lebesgue measure is zero.

Keywords

• homoclinic behavior
• one-dimensional maps
• symbolic dynamics
• measure