What is the probability of an arbitrary nonlinear dynamical system to be chaotic?

Question

Particularly, how to characterize a set of chaotic nonlinear dynamical systems as a subset of nonlinear dynamical systems with respect to the set cardinality?

To explain the question more, a simple similar question would be "if I point to an arbitrary number on a real number line, what is the probability of this number to be irrational". Here, this would be 1 which is explained by the cardinalities of rational and of the irrational subsets of a real set.

Though it's interesting in general, the fact that linear dynamical systems can be chaotic as well is dismissed in the question since that, if I'm right, requires infinite dimensionality. I'm doing theoretical neuroscience research and here nonlinear dynamics are largely used to model motor control and perceptual phenomena.

I can't find the answer within research nor textbook so far, and I'm also new to dynamical systems.

Answer 1

This is an important and interesting question; unfortunately it's very hard to say anything in much generality.

First one needs to establish within which class of dynamical systems the question is being asked, and then to establish a notion of "largeness" within that class. Cardinality is not the most useful because whatever precise definition of "system" and "chaotic" we settle on, both classes are likely to be uncountable. So we may try to impose a measure on the space of systems, and ask if a positive measure set (or full measure set) of systems are chaotic. (Again, deferring for the moment the question of just what chaotic means.) Or we may ask whether the set of chaotic systems is residual (countable intersection of open dense sets), so that a topologically generic system is chaotic.

One immediate difficulty is that the space of systems is infinite-dimensional. More precisely, if "dynamical system" means "diffeomorphism of a given manifold", then it's not clear what measure to use on the space of systems. There are notions of prevalence that address this issue, but I won't go into them here.

We still haven't said what chaotic means. Let's agree that a dynamical system is a smooth map $f$ from some manifold $M$ to itself, and that chaotic means "$f$ has a physical measure with non-zero Lyapunov exponents". This means that there is an $f$-invariant probability measure $\mu$ on $M$ such that

1. all the Lyapunov exponents of $\mu$ are non-zero ($\mu$ is a hyperbolic measure), and at least one is positive; moreover,
2. there is a set of points $G_\mu\subset M$ such that $G_\mu$ has positive volume (with respect to the Lebesgue measure induced by a Riemannian structure compatible with the smooth structure of $M$) and every $x\in G_\mu$ is generic for $\mu$, meaning that it satisfies the conclusion of the Birkhoff ergodic theorem for every continuous observable $\phi$.

Now one can ask: within the space of smooth maps, how large is the set of maps for which the above conditions hold? This at least gives a more precise formulation of the question. Here's what I know.

1. The answer is very different depending on whether one considers the space of $C^1$ maps or the space of $C^2$ maps. I won't go into the reasons for this. Let's consider $C^2$ maps, which is where I know more - other people are more familiar with the $C^1$ case.
2. There is a series of conjectures due to Jacob Palis which address the question in this general formulation (see "A global view of dynamics and a conjecture on the denseness of finitude of attractors", in Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque, 2000, pp. xiii-xiv, 335-347). These conjectures are very hard and not too much is known in full generality.
3. Rather more can be said in specific classes of examples. In particular, one-dimensional maps are reasonably well-understood. If one considers the case where $M=[0,1]$ is an interval and $\{f_\lambda \mid \lambda\in [a,b]\}$ is a one-parameter family of smooth maps satisfying certain technical conditions -- in particular, the family of logistic maps $f_\lambda(x) = \lambda x(1-x)$ satisfies these with $\lambda\in [0,4]$ -- then the following is known.
   • Write $R\subset [a,b]$ for the set of parameters $\lambda$ where $f_\lambda$ has regular behaviour, meaning that every orbit eventually is attracted to a stable periodic orbit (and the system is non-chaotic). Then $R$ is open and dense.
   • Write $S\subset [a,b]$ for the set of parameters $\lambda$ where $f_\lambda$ has stochastic behaviour, meaning that there is a hyperbolic physical measure as discussed above. Then $S$ has positive Lebesgue measure.
   • $R\cup S$ has full Lebesgue measure. A very nice account of all this is given by Lyubich in the October 2000 Notices of the AMS, entitled "The Quadratic Family as a Qualitatively Solvable Model of Chaos".

Numerical experiments suggest that the behaviour observed for one-dimensional maps -- namely, existence of chaotic behaviour for a positive measure set of parameters together with open sets of parameters with regular behaviour -- occurs more broadly than just in this setting. But rigorous results are quite hard to come by.

One final remark: You don't need nonlinearity to get chaotic behaviour, but for a linear system you do need non-trivial topology. The doubling map $x\mapsto 2x \pmod 1$ has all the chaotic behaviour you could want, but relies on the non-trivial topology of the circle; similarly, the Arnold cat map $x\mapsto \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}x \pmod {\mathbb{Z}^2}$ as a map on the two-torus is both linear and chaotic.