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Simulations using time series

Time series were generated using the following classical examples:
  1. The Hénon map, governed by the equations [Grassberger et al., 1988,Hénon, 1976]
    $\displaystyle x_{n+1}$ $\textstyle =$ $\displaystyle 1 - ax_{n}^{2} + by_{n}$ (15)
    $\displaystyle y_{n+1}$ $\textstyle =$ $\displaystyle x_{n}$  

    with $a = 1.4$ and $b = 0.3$. $x_{0}$ and $y_{0}$ were chosen uniformly in $[-1,1]$ and $[-0.1,0.1]$ respectively. Initial conditions within this area cause the iterates to approach the attractor [Hénon, 1976]. Literature values are: $\nu = 1.22$ and $K_2 = 0.325$ [Grassberger and Procaccia, 1983a].
  2. The logistic map, governed by the equation [Grassberger et al., 1988]
    \begin{displaymath}
x_{n+1} = 1 - ax_{n}^{2}
\end{displaymath} (16)

    For $a = 2$, analytical results [Grassberger et al., 1988] are $\nu = 1$ and $K_{2} = \ln 2$. $x_{0}$ was chosen uniformly in $[-1,1]$, so that any $x_{0}$ is near the attractor.
  3. The sine wave
    \begin{displaymath}
x_{n} = \sin (\omega n + \omega_{0})
\end{displaymath} (17)

    with $\omega = \frac{2}{\pi}$, resulting in approximately ten points per cycle and $\omega_{0}$ chosen uniformly in $[0,2\pi]$. The values for $\nu$ and $K_{2}$ are 1 and 0 respectively.
The $x$-variable was used to generate a time series of length $L$ ($\Delta t = 1$); the first $L_{s}$ iterates were discarded to avoid transient effects.



Subsections
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