# Strange attractors: what is the difference between a map and differential equation system?

As far as I have been able to understand, there are two main ways of generating (or finding) a strange attractor:

1. Using a map. E.g. the Hénon map (for a given $a,b$):

$$x_{n+1} = 1 - a x_n^2 + y_n \ \ \ , \ \ \ \ y_{n+1} = b x_n$$

1. Using differential equations. E.g. the Lorentz system (for a given $\delta,\rho,\beta$):

$$\dot x = \delta (y-x) \ \ \ , \ \ \ \ \dot y = x( \rho -z)-y \ \ \ , \ \ \ \ \dot z = x y - \beta z$$

What I do not understand very well is what differences are between a map and differential equations in this respect.

This is what I guess:

1. A map is always a discrete-time dynamical system, so no differential equations are required to generate the strange attractor.

2. In the other hand, a differential equation system is per se a continuous-time dynamical system (due to the fact that it is based indeed on differential equations).

Are the above assumptions correct, or are the differences between a map and a differential-equations-based dynamical system more than that? Can a differential equations system be converted into a map (probably adding some restrictions), or likewise, a map into a differential equations system and be able to reproduce the same strange attractor (or a restricted version of the same)?

• As a side note. There is a notion of Poincaré map which in some cases reduces the studying of dynamics of the ODE to the dynamics of this map. In some cases people say that every diffeomorphism (a differentiable bijective map whose inverse is also differentiable) can be realized as a Poincaré map of some system of ODEs (see suspension flow). Oct 18 '17 at 7:36
• @Evgeny true! As far as I understand (as per the Wikipedia) is a kind of cut of a more than two dimensions dynamical system into a two dimensional plane. Thank you for the reference to suspension flow. Oct 18 '17 at 8:38
• @Evgeny by the way, your question regarding dynamical systems is very useful: math.stackexchange.com/questions/510291/… Oct 18 '17 at 8:41
• Oh, thank you :) I still don't know any way to get the list of such books and articles on these topics. I'm glad that this list is useful for someone else besides me. Oct 18 '17 at 9:31

are the differences between a map and a differential-equations-based dynamical system more than that?

Well, first of all, there is the practical difference that maps are usually easier to analyse while differential equations are closer to reality.

Of course, some theoretical statements need to be translated. For example you need three dimensions in a differential equation to obtain chaos, while one dimension suffices for maps.

Besides there are some phenomena (like weak ergodicity breaking) that have only been observed in carefully constructed maps, as far as I know.

Can a differential equations system be converted into a map […]?

Sure, whenever we solve a differential equation numerically, we are essentially turning it into a (complicated) map. IIRC, some prominent chaotic maps have been obtained this way, though I cannot name one.

A more sophisticated approach is making use of Poincaré sections, i.e., you consider intersections of the trajectory with some plane or manifold in phase space and the mapping of one intersection to the next. In fact, this is how the Hénon map was obtained from the Lorenz system.

Can […] a map [be converted] into a differential equations system […]?

I can see how for many maps, you could carefully construct differential equations that have the map as a Poincaré section, but what would you gain?

be able to reproduce the same strange attractor (or a restricted version of the same)?

Either way, you would gain or lose trivial properties, e.g., the connectedness of the attractor. Note for example that if you connect temporally adjacent points of the Hénon attractor with lines, they would be all over the place, while this does not apply to the Lorenz attractor.