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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 $δ$?

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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:

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

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  • $\begingroup$ Thank you for your reply and providing the link - both are useful. However, there are certain terms that are unclear. Could you please elaborate? (1) I am representing the trajectory in binary so the initial condition is represented in binary (symbolic dynamics). What is meant by diameter of the attractor? $\endgroup$
    – SKM
    May 25, 2016 at 17:56
  • $\begingroup$ (2) In the link you say that $x_i$ is the $i-th$ bit. Isn't number of bits = number of iterations of the map, $n$? Say starting from an initial condition $x_0 = 0.1$ if the map is iterated $n=10$ times, then I should get 10 digits after the real valued numbers are binarized using the critical value of the map (generating partition). $\endgroup$
    – SKM
    May 25, 2016 at 17:57
  • $\begingroup$ (3) Which equation can be used to determine the bounds? $\endgroup$
    – SKM
    May 25, 2016 at 17:58
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    $\begingroup$ @SKM: 1) The attractor of the (symmetric) tent map is $0 ∪ {ℝ ∩ [0,1]}$. The diameter of this set is $1$ (also see the edit to my answer). 2) Defining $x_i$ as the $i$th bit is just the notation I use in that answer. The index does not indicate iterations of the tent map. 3) See my edit. $\endgroup$
    – Wrzlprmft
    May 25, 2016 at 18:18

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