Overview

In this section, we consider some problems of chaotic dynamics of a vibrating system that possesses two minima and one maximum of the potential energy. Further on, the system will be referred to as the two-well system . Its mathematical model was originally derived in 1979 as a single-mode equation of a buckled beam [5]. First results of a related physical experiment were presented by F.Moon in 1980 [8]. In the following years, the model has found applications in several branches of physics. It soon appeared that the system was so rich in various nonlinear phenomena that it became an archetypal model being explored in many textbooks, also in books written by applied mathematicians [31].

Chapter 3 that presented a case of a pendulum familiarized the Reader with such concepts of nonlinear dynamics as: saddle-node bifurcation, period-doubling bifurcation, Poincar map, bifurcation diagram, basins of attraction, strange attractor, fractals and others. In the present chapter, we try to explain and illustrate new phenomena and concepts such as:

• boundary crisis of the chaotic attractor;

• unpredictability of the system behavior following the destruction of chaotic attractor;

• intermittent transition to chaos;

• Melnikov criterion;

• Lyapunov exponents.