What is the difference between high dimensional and low dimensional chaos? Often I read of high and low dimensional chaos. But, I don't know what is their difference. I have thought the following answer.
Let us consider a time series $\{x_i\}_{i\in\mathbb N}$. According to Packard et al. and Takens, the reconstructed attractor of the original system is given by the vector sequence
$$\textbf{p}(i)=\left(x_i,x_{i+\tau},x_{i+2\tau},\dots,x_{i+(m-1)\tau}\right)$$
where $\tau$ and $m$ are the embedding delay and the embedding dimension, respectively. Maybe, the difference between high and low dimensional chaos can be expressed in terms of embedding dimension: for example, for $m\leq3$ we have low dimensional chaos and for $m>3$ we have high dimensional chaos.
Could I have a good reference for the topic of my queston?
Thanks in advance.
 A: High-dimensional chaos is usually used to refer to dynamics with more than one positive Lyapunov exponent. If you so wish, there are two growing (and hence chaotic) dimensions in the local phase-space deformation through temporal evolution. I quote a footnote from Harrison et al. – Route to high-dimensional chaos:

While there has been no formal definitions of low-dimensional
  versus high-dimensional chaos, here we take the notion that
  low-dimensional chaos is characterized by one positive
  Lyapunov exponent, and high-dimensional chaos by more than
  one.

Thus high-dimensional chaos is the same as hyperchaos, which the same authors state in another paper:

High-dimensional chaos also has been known as hyperchaos.

Now, hyperchaos is a much more popular term, which should allow you to find further resources. For starters, there is a Scholarpedia article on this topic.

High-dimensional chaos in this sense is not linked to the dimension of the dynamical system or the embedding dimension (which is only relevant if you want to reconstruct the attractor from time series). I have seen 20000-dimensional dynamics that still only have one positive Lyapunov exponent. The only constraint is that a continuous-time dynamical system must at least have four Lyapunov exponents (and hence be a four-dimensional dynamical system) to be able to exhibit high-dimensional chaos: one Lyapunov exponent must be negative; one must be zero; and at least two shall be positive.
