Introduction

We study dynamical systems, described by ordinary differential equations or iterated maps, and depending on a parameter. Assuming that the bifurcation diagram of this system is known, we would like to understand the behaviour of the new equation obtained by varying the parameter slowly in time. These adiabatic systems often display memory effects, such as bifurcation delay and hysteresis.

We are particularly interested in the following questions:

• If the parameter moves periodically, does the system admit periodic or chaotic orbits?
• Does the system display hysteresis?
• How do the properties of hysteresis cycles, such as their area, scale with the adiabatic parameter?

This research is discussed in detail in my Ph.D. thesis. See this abstract for a summary.