Very different starting conditions into similar outcomes within Chaos Theory.

Question

Chaos Theory demonstrates how systems with very similar starting conditions can end up in very different states, but can it demonstrate whether systems with very different starting conditions can end up in very similar states?

Answer 1

The exponential divergence of initial conditions only holds if they are very close to each other (and only in average). The trajectories emerging from two distant initial conditions may very well converge to each other (only to eventually separate again due to the positive Lyapunov exponent of the system). To see that this must happen, consider the evolution of phase-space volumes¹:

• If we consider a small bounded volume of initial conditions (e.g., a cube), the volume covered by the states arising from the evolution of these points blows up over time, until the entire attractor is covered.

• On the other hand, the attractor itself is an invariant set (under the dynamics), i.e., it is unchanged by temporal evolution and so is its volume.

So, if we cover the entire neighbourhood of the attractor in small cubes, the future of any of these phase-space volumes is again a (smaller) neighbourhood of the same attractor. Therefore some initial conditions from different (distant) cubes must be evolved to be close to each other.

This property is called topological mixing and one of the defining features of chaos.

A simple example that illustrates this is the bit-shift map $T$: If we represent the state of the system in binary, applying the map corresponds to removing the first digit after the radix point. Suppose we choose two arbitrary irrational numbers $x$ and $y$ as initial conditions, and denote their $i$th digits with $x_i$ and $y_i$ respectively. The distance after $n$ iterations ($δ:=|T^n(x)-T^n(y)|$) depends primarily on how many of their digits following the $n$th one are identical, i.e., what is the largest $p$ such that $x_{n+i}=y_{n+i} ∀ i ∈ {0,…,p}$. Now, unless you carefully craft your initial conditions to avoid this, you will eventually find a block of $p$ matching digits for any $p$, and thus at some time in the future, $δ$ will become arbitrarily small.

^{¹ assuming for simplicity that there is only one attractor, whose basin of attraction is the entire phase space}