3

This is more of a conceptual question but closely related with how non-linear dynamics simulations results should be interpreted. I am confused about the relationship of "period" in the context of period-doubling bifurcation, quasi-periodicity and dimension of the torus attractor. I find the wikipedia article on quasi-periodicity very concise so I decided to ask here.

For example,

enter image description here

This figure shows period-doubling bifurcation of a 3-D system. Although the period went from 2 in a), i.e. limit cycle, to 4 in b), 8 in c) and 16 in d), there are only two types of attractors here, which is the 1D limit cycle in a), and different (?) 2D tori (as this is a 3D system so there can't be non-2D tori) in the rest, so it seems the dimension of the tori is not related with the "period" in the context of period-doubling bifurcation? This should also mean that b)-d) have the same "quasi-periodicity" when you characterize it by the dimension of the attractor, but I think the Lyapunov exponents are different in each cases?

What is an example of a quasi-periodic motion given by a 3D torus? In time series, does it look to be just irregular?

Axel Wang
  • 197
  • 7

1 Answers1

2

I did some more thinking/digging on this and will try to answer myself.

The cases b)-d) indeed all have a 2D torus as an attractor. The reason the trajectories do not trace out the entire torus (aka. "fill" the torus) is because the two frequencies here are commensurate (their ratio can be represented as a ratio of two integers), resulting in periodic rather than quasi-periodic motion. In other words, just because the attractor is a torus doesn't mean the dynamics is quasi-periodic.

On a 3D torus, there still is a possibility of having strictly periodic motion, if the 3 frequencies are commensurate. However, even for a real quasi-periodic motion on a 2D torus, I think just by eye it is almost indistinguishable from chaos. An example of this is Fig. 6.1 of this book.

Axel Wang
  • 197
  • 7