Examples of transient chaos in the quasi-periodic route to chaos?

Question

I have read that transient chaos has been observed experimentally where a quasiperiodic steady state is involved (e.g. in Ytrium Crystals). I have also read about examples of equations which show transient chaos for the period doubling route to chaos. However, I am looking for examples of ODEs which admit quasiperiodic solutions and which exhbit transient chaos. However, I can't seem to find any and would be grateful for any pointers.

FOR A LITTLE CONTEXT:

I am simulating a system that has been shown to be chaotic for certain parameter values. Other value look periodic but upon estimating the correlation dimension I found that it was $\ge 2$ with a non-positive dominant Lyapunov exponent. This led me to conclude that the system must be quasiperiodic.

However, the "quasiperiodic" steady state is preceded by an erratic looking transient that resembles the system at chaotic parameter values.

Answer 1

You can always “cheat”:

1. Let $\dot{y} = f(y)$ with $f:ℝ^n→ℝ^n$ describe some system that exhibits transient chaos with a periodic attractor.
2. For some vector $z ∈ℝ^{2n}$, let $\hat{z}∈ℝ^n$ be the vector containing the first $n$ components of $z$ and $\check{z}∈ℝ^n$ be the vector containing the second $n$ components of $z$, i.e., $z_{n+1}, …, z_{2n}$.
3. Define $g$ such that $g_i(z) := f_i(\hat{z})$ for $i∈{1,…,n}$ and $g_i(z) := π·f_i(\check{z})$ for $i∈{n+1,…,2n}$.
4. Consider the dynamics described by $\dot{z}=g(z)$.

This dynamics is composed of two independent dynamics as described by $f$ running at different, incommensurable speeds. Hence this dynamics would have a quasiperiodic attractor and exhibit transient chaos, as long as one of the individual dynamics has not settled on its attractor.

If this is too much cheating for you, add a small coupling term between the two subsystems that is designed such that it vanishes on their respective attractors. If the term is sufficiently small, it should not affect the properties you are interested in. (Also it may help to keep in mind that every quasiperiodic motion can somehow be separated into independent periodic ones.)