Generative eBooks

Julia Sets

Generative eBooks — 2016-06-08T13:09:18Z

We end by complex dynamics, a fascinating field of mathematics, widely known because of the famous Mandelbrot set. We consider the simplest example. Consider the quadratic map $f_c(z)=z^2+c$. Then define
$$ z_n+1=f_c(z_n)=z_n^2+c $$
where $z_n$ and $c$ are complex numbers. Given an initial condition $z_0$, we can compute its image $z_1=f_c(z_0)$, next the image of its image, namely $z_2=f_c(z_1)=f_c^2(z_0)$, and so forth and so on. Here $f_c^2$ means $f_c$ composed with itself, i.e.,

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Turing's patterns

Generative eBooks — 2016-06-08T13:08:35Z

It is obvious that many natural phenomena are not captured by differential equations because there are changes both in time and space. Mathematically, this means that we have to consider partial differential equations. In the so-called reaction-diffusion equations, there are local reactions in which the substances are transformed into each other and can be degraded, and diffusion which causes the substances to spread out in space. Such equations display a wide range of behaviors, such as (…)

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A discrete-time prey-predator model

Generative eBooks — 2016-06-08T13:07:41Z

In the population models we have seen, differential equation models imply a continuous overlap of generations. But many species have no overlap whatsoever between successive generations and so population growth is in discrete steps. We get difference equations defined by a mapping. As we have seen, one needs at least three variables (i.e., a three-dimensional phase space) to get deterministic chaos with differential equations. Remarkably, in discrete time, chaos can show up with a single (…)

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