10 - Epilogue: outlook

Summary

In this concluding chapter, we present a brief overview of some phenomena and concepts, the detailed investigation of which is beyond the scope of this introductory book, but whose inclusion may provide (along with the bibliography) further understanding.

First and foremost, we emphasise that chaotic behaviour can be observed in laboratory experiments. The validity of the physical laws determining the motion of macroscopic systems is beyond doubt; consequently, the phenomena found in numerical simulations are also present in the real world. The chaotic feature of many of our examples (magnetic pendulum, ball bouncing on a double slope or on a vibrating plate, or the mixing of dyes) can be demonstrated by relatively simple equipment. In the cases of the periodically driven pendulum, the spring pendulum, the driven bistable system or chaotic advection, the chaos characteristics have been determined by precise laboratory measurements, and the transitions towards chaos have also been investigated. In other branches of science, numerous processes are also known whose chaoticity is supported by observational or experimental evidence (see Box 9.3).

In this book we have presented the simplest forms of chaos and interpreted them as the consequence of hyperbolic periodic orbits. In general, however, non-hyperbolic effects also play a role due to the existence of orbits whose local Lyapunov exponents are zero. One example of this is the algebraic (non-exponential) decay of the lifetime distribution in chaotic scattering due to the existence of KAM tori (see equation (8.10)).