# Chaos and dynamical systems

At the end of 1960's and during the 1970's, chaos theory was a bit of a hype, and many books appeared on the fun-side of chaos as well as serious mathematical books developing the theory of dynamical systems that could lead to chaotic behaviour. Feldman describes his book as an introduction to the subject that is somewhere in between those two approaches. Chaos, he says, is a phenomenon, not a theory. He of course has to explain concepts and technical terms, but he does it without being too abstract. His approach is more on an intuitive basis, letting the reader be surprised by the results produced, and then asking (and answering) the questions that the reader naturally would ask: Why is it like that? How is this possible? Feldman has a lot of teaching experience and is able to explain everything requiring only a minimum of mathematics (the derivative is necessary because he uses differential equations, but no integrals are used, so that high school mathematics suffice).

He considers two types of dynamical systems: iterated function systems and differential equations. So explaining what these are (in one dimension) is the subject of the first two chapters (together with definitions of stable/unstable equilibrium, orbit, etc.). It is followed by an almost philosophical chapter about what a mathematical model is supposed to be if it should reflect a physical reality from a world that is ruled by an unrealistic Laplace's demon. In many cases a simplified model is much better to highlight exactly that aspect that one is interested in, rather than confusing it with a minute imitation of every detail that is not relevant. There are many kinds of models that can be used.

The main purpose of the book starts in chapter 4 where the (discrete) logistic equation is introduced. It describes the evolution of a population. The sensitivity to initial conditions is defined as the butterfly effect (including the Lyapunov exponent) and, depending on the parameters, the system has different limiting behaviour. This moves on smoothly to the next chapter which illustrates that, although the system is deterministic, the sequence that is generated is random (to be distinguished from sequences produced by stochastic systems). The logistic differential equation (which also describes population dynamics, but now with harvesting) is used to illustrate bifurcation with a reference to catastrophe theory, tipping points, and hysteresis. Next chapter fully explores the associated bifurcation diagram with its period-doubling route to chaos and a definition of the Feigenbaum constant. Most surprisingly, this constant turn up again and again in different systems. This is called universality.

Here the reader may be wondering how this is possible and therefore Feldman makes an excursion to universality in physics and what is called there renormalization. That gives an intuitive idea about a stretching and folding process, which is the key to the self-similarity of the bifurcation diagram. This is used as an explanation for a more abstract setting in which it is stated (and graphically checked) that all the iteration functions mapped by this kind of self similarity mapping, is converging to a universal function which is the attractor. This explains universality, i.e., why many different iteration functions all move to the same asymptotic behaviour. Some remarks are given on phase transition, critical phenomena and power laws. The link with fractals could have been made here, but Feldman avoids this byroad.

With the Lotka-Volterra differential equations and the Rössler system, Feldman introduces multi-dimensional systems and the phase plane. The bifurcation diagrams of the first part is replaced now by strange attractors. A completely new concept because bifurcation spreads out more and more, the attractor pulls in every trajectory. There is however randomness in the attractor as there was in bifurcation and the stretch and fold operation is also here applicable. The Lorentz attractor is a famous example that is at the origin of the term "butterfly effect". It is less obvious that some form of reverse engineering is possible: just selecting the maxima from the chaotic behaviour, allows to produce the Lorentz map, which reveals the iteration function used. Using the time series with delays of one coordinate, a Poincaré map can be constructed that gives an idea of the strange attractor in phase space, provided the correct delays are chosen.

At several places Feldman refers to complex systems, which is a field in which many components interact with each other like on the Internet, global climate, etc. This is "dynamical systems brought to the next level" and complex systems are still a subject of intense research. In this book references are made to agent based systems, nonlinearity, emergence, and the limits of universality, concepts that return when studying complex systems. So it should be no surprise that in the concluding chapter, Feldman, besides giving a brief summary of what has been discussed, also points to some links and differences with complex systems.

With a minimum of mathematics, Feldman succeeds in introducing the reader to the world of dynamical systems and the, almost mythical, chaos that they can produce. He explains that there is nothing magic about sudden bifurcation phenomena and that there is nothing strange about a strange attractor, but that there are simple mathematical explanations for these phenomena. The reader can just be satisfied with his explanations, but Feldman is obviously hoping that some of his readers are genuinely interested in the mathematics, and for them he provided extensive sections where he gives advise about what literature to consult for further details. I could spot a few typos not detectable by a spell checker, but none serious. Some in the style of the footnote on page 151: "...is not a always group".

**Submitted by Adhemar Bultheel |

**14 / Aug / 2019