The extent of chaos

In chaotic systems the typical situation is that at a low level trajectories of points are wild, but overall there is a nice statistical description of the system.

For example, consider the trajectories of a system of differential equations. Suppose there is some sort of chaos like in the Lorenz system of differential equations. Each trajectory is very wild and sensitive to the initial condition. However, on the global scale there is a strange attractor that attracts all initial conditions.

It seems that this is the typical situation, that there are finitely many attractors that attract all initial conditions, and on each one of them there is a measure which describes the frequency of visits of each trajectory to some region of the attractor. That is, one can say on average how many times a point visits a certain region near the attractor. Moreover, this structure is stable.

I would like to understand why chaotic, irregular behavior of trajectories lead to a sort of regular behavior when one considers all trajectories together and takes a global point of view?

In order to understand the situation a bit better, I like to understand what is in the complement. That is, are there chaotic systems where there is no nice, global picture as described above? How wild should the trajectories of a system (say in $\mathbb{R}^3$) be so that the global picture is also chaotic''? Is this even possible?

What does it even mean for the global picture to be chaotic? Infinitely many strange attractors? Or the other extreme: lack of a strange attractor and any specific pattern?

What is the most chaotic system one can have?

There's a lot here, so I'll try to address "why chaotic, irregular behavior of trajectories leads to a sort of regular behavior".

The most classical version of this theory is that of $C^2$ diffeomorphisms $f$ with compact hyperbolic attractors ($C^2$ is needed so that the derivative $df$ is bounded in operator norm). In this situation, the diffeomorphism may behave however 'orderly' or 'chaotically' you can imagine away from the attractor, but in a neighborhood of the attractor, you can apply a very nice theory (that of SRB measures) which indicates that the statistics of orbits (of a Lebesgue-full set of initial conditions) in that neighborhood are governed by a nice measure, which is supported on the attractor itself. Precisely, there is a probability measure $\mu$ such that, given a continuous observable $f$ and a generic point $x \in U$ where $U \supset \Lambda$ is attracted to the attractor $\Lambda$, we have $$\frac{1}{n} \sum_{k = 0}^{n-1} f^k x \rightarrow \int_{\Lambda} f d\mu$$ Moreover, the measure $\mu$ is in the best possible sense "compatible" with the Lebesgue measure.

The reason "why" is interesting as well: it turns out that in the above situation, there is a way to symbolically encode the dynamics on $\Lambda$ as a subshift of finite type (aka a topological markov shift). This encoding gives rise to a special measure $\mu$ on $\Lambda$ with all the nice properties I've mentioned.

I should mention that hyperbolicity is important: it means more-or-less that the asymptotic dynamics is governed by the derivative term at a point (technically, it means that the eigenvalues of $df$ are bounded away from the unit circle).

This stuff is the province of differentiable ergodic theory. For reference, you should google the authors Ruelle, Bowen, and Lanford.

• Thank you for your answer. I am aware of the existence of SRB measures for $C^2$ hyperbolic diffeomorphisms. In fact, in the first part of my question I was trying to describe the picture you described but in a non-mathematical way. What I'm most interested is that which does not fall into the picture you described. In the last two paragraphs of my question I say that I am interested in the complement, i.e. situations that don't follow the picture described. The idea is to understand how far one can deviate from that picture. Jan 22 '13 at 13:27
• You would definitely benefit reading some theory on dispersing billiards, for instance. A lot of work has gone into obtaining the dividends of the thermodynamic formalism for billiard systems over the past twenty or so years. Complications with billiards: the dynamics are only piecewise continuous, and where they are continuous, the derivative has singularities (from so-called grazing reflections). The story of billiards begins with the work Sinai did in the late seventies, proving ergodicity and mixing properties (you can see Chernov & Markarian for a reference). Jan 23 '13 at 6:11
• Another direction is towards "partially" or "nonuniformly" hyperbolic dynamics- my advisor, Prof. Young at Courant in NYU, is credited with obtaining SRB measures for Henon attractors, which are merely partially hyperbolic. Jan 23 '13 at 6:13
• There's an entire industry built around deriving sufficient conditions for the existence of SRB measures. Criteria are often like: the tangent bundle of the attractor decomposes into a hyperbolic subbundle and a one-dimensional center subbundle. That's just one I happened to read earlier this week on arxiv. Jan 23 '13 at 6:19

Each trajectory is very wild and sensitive to the initial condition. However, on the global scale there is a strange attractor that attracts all initial conditions.

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I would like to understand why chaotic, irregular behavior of trajectories lead to a sort of regular behavior when one considers all trajectories together and takes a global point of view?

I do not quite share this impression: The trajectories are as wild or as regular as the the attractor. If you ignore transients, the trajectories are even just a part of the attractor. Moreover, any trajectory becomes the attractor if its length goes towards infinity, if you so wish (again, ignoring transients).

Taking another point of view, the sensitivity on the initial conditions is not totally wild either but features a lot of regularity:

• If two initial conditions are very close to each other, the respective trajectories won’t just instantly start running of in different directions but are very likely to stay together for a while and only then separate. Moreover, the closer initial conditions are, the longer it takes on average until they separate.

• In case of one attractor, the qualitative dynamical behaviour of all trajectories is the same. They also all become the same attractor if we let the trajectory length go towards infinity. For multiple attractors, comparable statements apply.