Dynamical systems

These images were made by mucking around with equations producing (to quote from the Pickover chapter that inspired them) “time-discrete phase planes associated with the cyclic systems:

(1)   \begin{equation*} \left\{   \begin{array} {lr}    \dot{x}(t)=-f(y(t))\qquad \longleftarrow\quad \dot{x}(t)\text{ means the rate of change of }x\text{ at time }t,\\    \dot{y}(t)=f(x(t))\qquad\qquad\qquad \text{like the speed of a car at a certain moment }t.\\   \end{array}   \right.  \end{equation*}

In phase space each dimension can represent one of the variables in the differential equation. …Here, these functions are generally of the form often found in frequency modulation synthesis applications:

    \[f(x)=\sin[x+\sin(\rho x)]\]

By modulating a sine wave (often called the “carrier”) by another with a different frequency (controlled by \rho), a large variety of complex waveforms can be generated … Using this technique, a graphically “rich” function can be produced with only a small number of input parameters.
The discretization of Equation (1) for implementation on a computer takes the following simple form (known as the forward Euler approximation):

    \[ \left\{   \begin{array}{lr}   x_{t+1}-x_t=-hf(y_t)\\   y_{t+1}-y_t=hf(x_t)   \end{array}   \right. \]

where h>0 is a constant known as the step size of the numerical solution. In this section, h is kept small (h\sim0.1). … Despite the phase portraits’ complexity, they possess universal features shared by entire classes of nonlinear processes. For example, a range of real physical systems can be described by similar systems of differential equations.”{}^1
I’ve not used the trajectories of randomly-chosen points, as he does, but instead traced the trajectory of every point on the screen a certain small number of steps, and colouring them according to the distance travelled from the starting point, with colours similar to those used in coloured relief maps – blue(ocean) \rightarrow dark green \rightarrow light green \rightarrow yellow \rightarrow brown \rightarrow white(snow).
Some of the images have the 2 main equation lines pasted on them, from the program that drew them – e.g. the 11th picture below, the first one with a formula on it, says in C++ :
realtemp = real-mult*sin(27*imag+sin(7*imag));
imag += mult*sin(30*real+sin(28*real));
I made those numbers up pretty much randomly, to see what would happen. This translates into the above style as:

    \[\left\{\begin{array}{lr}   x_{t+1}=x_t-h\sin(27y_t+\sin(7y_t))\\   y_{t+1}=y_t+h\sin(30x_t+\sin(28x_t))   \end{array} \right. \]

{}^1Computers, Pattern, Chaos and Beauty: Graphics from an Unseen World – Clifford Pickover, 1990


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