Embedding dimension $m>4$ for high-dimensional chaos.

Question

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.

High-dimensional chaos is usually used to refer to dynamics with more than one positive Lyapunov exponent. There is a Scholarpedia article on this topic. Here, it is written: "the minimal dimension for a (continuous) hyperchaotic system is 4". This means that for a high-dimensional chaotic system, $m$ is at least equal to $4$?

Could I have a good reference (or an answer) to the topic of my queston?

Thanks in advance.

Answer 1

The easiest way to see this is in your case is probably that the central feature of attractor reconstruction is to preserve dynamically relevant features. In case of high-dimensional chaos, there are at least four dynamically relevant Lyapunov exponents (see, e.g., the source you linked). A three-dimensional attractor reconstruction, however, only has three Lyapunov exponents (one for each dimension of space).

More generally, if your chaotic attractor has a dimension $d$, the embedding dimension $m$ must at least be $d+1$ – at least if you want to obtain a properly embedded reconstruction without spurious overlaps (see this answer of mine for more on this issue). The additional dimension is coming from the requirement of topological mixing, i.e., that the dynamical flow returns to visit any vicinity again and you have an attractor in the first place. (Note that you may need a much higher embedding dimension, see yet another answer of mine.) The relation between the attractor dimension $d$ and the Lyapunov exponents is in turn addressed in the Kaplan–Yorke conjecture. Most importantly, $d$ is at least as high as the number of non-negative Lyapunov exponents. So in your case $m ≥ d+1 = 3+1 =4$.

Answer 2

I've encountered the terms in the following contexts:

1. 'hyperchaos' as defined on Scholarpedia: at least two largest Lyapunov exponents are positive;
2. 'high-dimensional chaos' e.g. in the research of Predrag Cvitanović - i.e., systems in $\sim 10^4$ dimensions (that's really high-dimensional!)
3. there was a search of 'low-dimensional chaos' in spin-down rates of pulsars in 2013, which was focused on $m=3$; however, there are also researches on the (possible) fractal dimensions of blazar time series. In this case, an unsuccessful search was performed up to $m=20$.

So, hyperchaos is unambiguously defined, while low- and high-dimensional chaos is subjective in what I encountered.