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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.

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to my knowledge, the logistic equation modelizes population growth (among other things) and chaos is not observed in such problems – glougloubarbaki Feb 11 '13 at 13:09
The logistic map $x\mapsto \lambda x(1-x)$ was popularised by a biologist, Robert May, in a 1976 paper in Nature, 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 logistic equation $\dot{x} = \lambda x(1-x)$ also models population growth, but without chaos.) – Vaughn Climenhaga Feb 11 '13 at 15:00
@vaughn : yes, but my question is precisely : is such chaos observed in his experiment ? equivalently, is the model relevant ? – glougloubarbaki Feb 11 '13 at 16:22
As a mathematician and not a biologist, I'm not the most qualified to speak to the biological relevance of the model. The fact that it was a biologist and not a mathematician who wrote the paper popularising the model suggests to me that it is of more than just mathematical interest. I think his paper itself may have a better discussion of this issue. – Vaughn Climenhaga Feb 11 '13 at 16:31
up vote 3 down vote accepted

The most natural way in which discrete time dynamics appears in physical systems is stroboscopically. The stroboscopic description of a periodically driven Hamiltonian system leads to the standard map, which exhibits a chaotic phase space. These systems have been realized with microwaves and with atomic matter waves. The discrete time, stroboscopic description of a periodically driven system is essential here, since these are all one-dimensional systems which would not exhibit chaos if the Hamiltonian would be time independent.

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As another example, iterates of the Arnol'd-Avez cat map correspond to unit-frequency stroboscopic projections onto the standard 2-torus of trajectories on the plane generated by the Hamiltonian $H(\xi; p) := K(p^2 -\xi^2 + \xi p)$, where $K = \sinh^{-1}(\sqrt{5}/2)/\sqrt{5}$. – Steve Huntsman Feb 11 '13 at 16:04
The geodesic flow on a surface of negative curvature is an archetype of chaos, and is very conveniently described (stroboscopically) as well. I sketch this and my preceding comment as background in – Steve Huntsman Feb 11 '13 at 16:06
thank you, very nice answer ! – glougloubarbaki Feb 13 '13 at 13:56

One of the popular modern topics in discrete-time dynamics is Iterated Function Systems (IFS). They have a lot of applications both in mathematics and outside, including the image processing. The literature is enormous. Look under the keywords IFS, Fractals.

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A key technique in studying continuous-time systems is provided by Poincaré Sections. You place a hyperplane in your phase space in a suitable manner; instead of studying the full continuous-time system, you can then study the dynamics of the first-return map to this section.

In fact, the famous Hénon map was introduced to capture the behavior of a Poincaré section of the Lorenz attractor. Letting one of the parameters in the Hénon family degenerate actually leads to the real quadratic family, aka the "logistic family" as mentioned above. This is described very nicely by Lyubich in his 2000 AMS Notices article

There are probably more direct applications of discrete dynamical systems (look for processes that naturally work in discrete steps). However, in my opinion the above is among the strongest "justifications" for studying discrete-time dynamical systems (if such justification is necessary beyond the beauty, and fundamental nature, of the theory itself).

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Many systems are best probed stroboscopically. For e.g. in the design of space mission trajectories, it is customary to use the restricted-three body problem as the model for dynamics of the spacecraft. This system is a 4 (or 6 in 3D case) dimensional Hamiltonian system with chaotic trajectories. To analyze the dynamics, it is often useful to take a Poincare section at 'special' sections in the trajectory, such as the periapse (closest point to the central body) or apoapse (farthest point) of the trajectory.

Taking the Poincare section reduces the dimension of the system, results in a discrete time system and simplifies the analysis.

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Sometimes discrete models are easier to work with, and it is important to have well-thought out discrete maps that are expected to exhibit generic phenomena that any dynamical system -- or at least a particular class -- of discrete or continuous dynamical systems have.

One example that comes to mind is the Standard Map ( It is much easier to do numerical simulations for this map than solving a planar time-periodic ODE, yet the dynamical phenomena observed in this map is very universal. For example, the same behavior exists in the periodically forced pendulum. It is also used in plasma physics, statistical mechanics, etc. as a toy model to work with.

Aslo, whenever one simulates an ODE or PDE, one is essentially applying a discrete map. To elaborate, consider the ODE

$\dot{x} = v(x, t)$

Many numerical schemes to solve the ODE is of the form

$ x_{n + 1} = x_n + f(x_n, h)$

Where $f$ is a map that takes one from the current time step at position $x_n$ to $x_{n + 1}$ and $h$ is a time-step. Thus, $f$ is really a discrete map. The same goes when discretizing a PDE in space and/or time. When analyzing a numerical scheme it is important to make sure the dynamics of the numerical scheme match the dynamics of the system.

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Here You find I nice pyhsical system to observe chaos at home:

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