📂Dynamics

Chaos in One-Dimensional Maps

Definition

Chaotic Orbit¹

An orbit of map $f : \mathbb{R} \to \mathbb{R}$ is said to be chaotic if it satisfies the following conditions:

• (i) It is not asymptotically periodic.
• (ii): $h (x_{1} ) > 0$

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• A bounded orbit is an $M \in \mathbb{R}$ for which there exists a $|x_{n} | < M$ that satisfies all $n \in \mathbb{N}$.
• $h(x_{1} )$ refers to the Lyapunov exponent.

Chaotic Map

A map $f$ is said to be chaotic if there exists a periodic-$n$ orbit for all $n \in \mathbb{N}$.

Explanation

• Based on English pronunciation, Chaos is closer to [kay-oss] rather than [ka-oss], and Chaotic is closer to [kay-ott-ik] instead of [ka-ot-tik], hence the transliteration to chaotic.

In mathematics, when conditions and equations are given, one can find the desired answers. However, in a chaotic orbit, since the Lyapunov exponent is positive, no matter how much the map is iterated, it does not synchronize or attract, and it is even impossible to find a similar periodic orbit. This defines mathematically the fact that the distant future cannot be predicted, no matter how well the current conditions are known.

One point to clarify is that the existence of a chaotic orbit in a system created by map $f$ does not mean that the system itself is chaotic.

Generalization

• Chaos in Multidimensional Maps