I've been reading a fair amount about the relationship between fractality and chaos. While approximating the dominant Lyapunov exponent seems to be the preferred method for diagnosing chaos, it seems that many sources claim that fractal dimension is enough to diagnose chaos in simulated data (or sufficiently noise-free real-world data).
However, it is also claimed that some types of noise have finite correlation dimension such as power law noise. My question is then, can fractal dimension alone (for instance the correlation dimension) really be used to detect chaos. Follow-up: Is any non-zero fractal dimensional estimate enough to diagnose chaos in these cases or do nonchaotic systems exist with periodic behaviour that is multidimensional?
Just so we are talking about the same thing: The general notion you are referring to is:
As you already suspected, this notion is not entirely correct.
No. As you already mention, it was shown for certain types of noise that the correlation dimension can be finite and non-integer. See, e.g., Osborne, Provencale – Finite correlation dimension for stochastic systems with power-law spectra.
The only thing that you can safely deduce from a finite non-integer dimension is that the dynamics is not a deterministic regular one.
The attractor of a proper periodic dynamics always has a dimension of 1. This can be easily seen as you need exactly one real number to describe its state, namely the phase with respect to the periodic behaviour.
Regular dynamics with higher integer dimensions are quasiperiodic and their attractors are tori or hypertori. For example, a system consisting of two driven and damped, non-interacting harmonic oscillators with incommensurable frequencies is two-dimensional: You need both oscillators’ phases to describe the system’s state.