Math Insight

Applet: From function iteration to continuous evolution

The evolution of the difference equation $x(t+\Delta t) - x(t) = \Delta t \, g(x(t))$ at discrete time steps of length $\Delta t$ is illustrated by a plot of $x(t)$ versus $t$ as well as in a list on the right. The format is nearly identical to the function iteration applet with $f(x) = x+\Delta t \, g(x)$. Clicking the “iterate” button calculates a value of $x(t+\Delta t)$ for the next time step $t+\Delta t$. The “refine” button cuts the time step $\Delta t$ in half and doubles the number of time points so that the same final value $x(t)$ is calculated with smaller time steps. You can also adjust $\Delta t$ independently by typing a value in its box. The function $g(x(t))$ is specified by entering a value in the box; however, for the applet to work, one must type the variables as $x$ rather than $x(t)$. When the “details” box is unchecked, the points and lines indicating the values of $x(t)$ at the discrete points disappear, revealing just a plot of the function $x(t)$. When $\Delta t$ is small, $x(t)$ approximates the solution to the differential equation $\diff{x}{t} = g(x)$. You can zoom the horizontal axis with the +h and -h buttons and the vertical axis with the +v and -v buttons. You can pan up and down with the buttons labeled by arrows.

Applet file: from_function_iteration_to_continuous_evolution.ggb

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