Hello,

I am a graduate student in the field of discrete-time dynamics. I am wondering about applications of this field outside of mathematics. More precisely, I would like to know if there are "real life" situations where dynamical notions provide a significant insight, or even better, a power of prediction.

For example, is there a situation which is naturally modelized by discrete-time dynamics in which chaos is observed (I know about Lorenz attractor and meteorology in the continuous-time case) ? Or a situation in which estimations of the radius of an attractor is helpful (let's say outside of algorithms to find numerical roots), or structural stability, Lyapunov exponents, entropy, etc. play a concrete role ?

Sorry if this question is a bit too general.

map$x\mapsto \lambda x(1-x)$ was popularised by a biologist, Robert May, in a 1976 paper inNature, where it is indeed motivated by considering the dynamics of a population with non-overlapping generations. Presence or absence of "chaotic" behaviour depends in a quite subtle manner on the parameter $\lambda$, but for a positive measure set of parameter values, there is an absolutely continuous invariant measure and positive Lyapunov exponent, which is interpreted as chaos. (The logisticequation$\dot{x} = \lambda x(1-x)$ also models population growth, but without chaos.) – Vaughn Climenhaga Feb 11 '13 at 15:00