📂Dynamics

Sensitivity to Initial Conditions

Definition ¹

Let space $X = \left( \mathbb{R}^{n} , \left\| \cdot \right\| \right)$ and a smooth function $f,g : X \to X$ be given, expressing a vector field and map as follows. $$\dot{x} = f(x) \\ x \mapsto g(x)$$

$\phi (t, \cdot)$ is the flow of the vector field $\dot{x} = f(x)$, $g^{n}$ is the map applied $n$ times to the map $g$, and let $\Lambda \subset X$ be an invariant compact set under $\phi (t, \cdot)$ or $g(\cdot)$.

$\phi (t,x)$ or $g(x)$ is said to have sensitive dependence on initial conditions in $\Lambda$ if for every $x \in \Lambda$ there exists $\varepsilon > 0$ satisfying the following, and for every neighborhood $U$ of $x$ there exist $y \in U$ and $t > 0$ meeting the following conditions. $$\begin{align*} \left\| \phi (t,x) - \phi (t,y) \right\| > \varepsilon \text{ or } \left\| g^{n} (x) - g^{n} (y) \right\| > \varepsilon \end{align*}$$

Explanation

The above equation precisely describes how, depending on changes in initial conditions, the difference can be as large as desired after a sufficiently short time. This adequately illustrates the concept of ‘sensitive dependence on initial conditions’.

See Also

• Chaos in Invariant Sets
• Chaos in Multidimensional Maps
• Chaos in Systems Expressed by Differential Equations