Upper and lower bound for the separation of two trajectories of a dynamical system

Question

Consider trajectories $x(n)$ and $y(n)$ of the tent map, starting from initial conditions $x(0)$ and $y(0)$. Then the distance $δ$ between the trajectiories is:

$δ = |x(n) - y(n)| = \exp (λ n)|x(0) - y(0)| $

where $λ$ is the largest Lyapunov exponent, which for the tent map is $\ln (2) = 0.6590$.

I want to know if there is an upper and lower bound for $δ$ that can be deduced from the above equation and the maximum Lyapunov exponent. I do not know what $|x(0)−y(0)|$ would be and how to find the upper and lower bounds of $δ$?

Answer 1

I want to know if there is an upper and lower bound for $δ$ that can be deduced from the above equation […]

There are such bounds, but they cannot be deduced from the equation you gave. The latter only holds for small separations $|x(0)-y(0)|$ and small $n$ and completely brakes down, once the distance between the trajectories has reached the order of magnitude of the diameter of the attractor. It is for this reason that the separation is frequently rescaled when this equation is used to empirically measure the Lyapunov exponent.

Now for other ways to determine your bounds:

• The diameter of the attractor, i.e., the longest distance between any two states of the system, is an upper bound of $δ$ if your initial conditions lie on the attractor. Otherwise it may be higher, but not much, as long as your initial conditions are near the attractor. For the tent map, this upper boundary would be $1$.

• Quite obviously, $0$ is a lower bound for $δ$.

• That these boundaries are optimal follows from the property of chaotic dynamical systems called topological mixing: The future of any small volume of initial conditions will eventually cover the entire attractor, and hence you can observe $δ$ to become arbitrarily close to the diameter of the attractor. On the other hand, to allow for small phase-space volumes to expand to the entire attractor on temporal evolution, they must mix with the future of other initial conditions, some of which become arbitrarily close. For a more detailed explanation and examples for the tent map, see this answer of mine.

So, to summarise:

$$ 0 ≤ δ ≤ \max \left\{ \left|x(i)-x(j)\right| ~∀ i,j ∈ ℕ \right\}$$