Intereting Posts

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I am studying dynamical systems as part of a research project.

I have been using Newton’s Method and studying the dynamic properties.

Does anyone know where I could find a relatively simple example of bifurcation using Newton’s Method?

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Here is an illustration of the comment that I made 5 hours ago. The idea is to examine the dynamics of $f_c(x)=x^2+c$ for various choices of $c$. When $c<0$, there are two real roots. When $c>0$, there are no real roots. Thus, we expect a distinct change in behavior as $c$ passes through zero.

In the animation below, $f_c(x)=x^2+c$ is the light parabola and the corresponding Newton’s method iteration function

$$n_c(x)=\frac{x^2-c}{2x}$$

is shown in blue. The diagonal black line is the graph of $y=x$ and the intersection of this line with the graph of $n_c(x)$ are the fixed points of $n_c$. The extreme bifurcation at zero happens because the fixed points of $n_c$ correspond to the roots of $f_c$, which disappear.

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