ABSTRACT
In this chapter, we describe the major notions and techniques used in the study of chaotic dynamical systems. In Sec. 1.2, we describe the Poincaré map technique in order to study bifurcations and chaos in dynamical systems. In Sec. 1.3, we discuss the Smale horseshoe as a fundamental tool for studying chaos in dynamical systems. The symbolic dynamics representation of a solution is presented in Sec. 1.4. The notion of chaos and strange attractors in dynamical systems is presented and discussed in Sec. 1.5 with many numerical examples. Since chaotic dynamical systems can have more than one attractor depending on the choice of initial conditions, the notion that basin of attraction comes from this property is discussed in Sec. 1.6. In Sec. 1.7, we discuss the possible relations between robustness and hyperbolicity discussed in Sec. 8.10. The statistical properties of chaotic attractors are presented in Sec.1.8. Different types of entropies are defined for dynamical systems, including the physical (or Sinai-Ruelle-Bowen) measure, Hausdorff dimension, topological entropy, Lebesgue (volume) measure and Lyapunov exponents.
Now, proving chaos in 1-D dynamics needs the so called Period 3 implies chaos theorem presented in Sec. 1.9. The generalization of this theorem to high dimensions is the subject of Sec. 1.10, where the Snap-back repeller is defined and used instead of period 3 orbit. The Shilnikov criterion for the existence of chaos in autonomous systems is presented in Sec. 1.11. These results use the notion of homoclinic and heteroclinic orbits. The resulting chaos is called horseshoe-type or Shilnikov chaos.
