this is for holding javascript data
Benedict Irwin added main.tex
over 9 years ago
Commit id: e759b75cea611ce89c32aee3c60a5c364c9c61c6
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We may write a generalised iterative scheme where \begin{equation}
[I_\alpha(\beta)]_0=\int_0^{1} \alpha - \beta x \; \mathrm{dx} \\
[I_\alpha(\beta)]_i=\int_0^{[I_\alpha(\beta)]_{i-1}} \alpha - \beta x \; \mathrm{dx} \\
\end{equation}
Then we may request the limit \begin{equation}
\lim_{i\to\infty}[I_\alpha(\beta)]_i=I_\alpha(\beta)=\int_0^{I_\alpha(\beta)} \alpha - \beta x \; \mathrm{dx} \\
\end{equation}
Provided the first iteration does not equal zero, the scheme can be coded nicely.
Then one may vary $\alpha$ and $\beta$. Due to initial reasons $\beta$ was set to $0.5$, and $\alpha$ varied from $1$ till $4$, at low values of $\alpha$ the integral converges nicely, at values less than $3$ there is a point where the scheme hops between two converant values. Luckily, it was noticed at $3.5$ four convergent values appeared, and from prior knowledge (a guess at Lyapunov/Feigenbaum behaviour) I took a sweep for suspected bifurcations. Amazingly, the solution to $I_\alpha(\beta)$ for any $\alpha$ is the logistic map! The tradiditional bifurcation diagram when taking the iteration \begin{equation}
x_{n+1}=rx_n(1+x_n)
\end{equation}
