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In Von Neumann's stability analysis of Finite Difference Equations, the Euler's formula is used to describe a perturbation. Now the finite difference equation is stable if this perturbation does not grow in time.

Now, my question really is: how do I plot the perturbation (Equation 1) three-dimensions(?)/Im vs Real dimensions?

$$\epsilon(x,t) = \sum_{m=1}^{M} e^{at} e^{ik_m x} \ldots \ldots \text{(1)}$$

I realize that the the Euler's formula $e^{ik_m x}$ describes a circle. Clearly the multiplicative amplitude factor $e^{at}$ either "grows" this circle or decreases it whether or not $e^{at}$ is greater than 1 or not.

I would like to describe that as a Manipulate or a ListAnimate in Mathematica but I don't understand which function to use.

So far I have understood that PolarPlot and ParametricPlot draw this circle in 2D.

Parametric Plot example

ParametricPlot[G {Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, 2 \[Pi]}, 
 PlotRange -> {{-1, 1}, {-1, 1}}]

enter image description here

Polar Plot example

PolarPlot[{Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, Pi}]

enter image description here

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Your question asks, in part, "Clearly the multiplicative amplitude factor $e^{at}$ either "grows" this circle or decreases it whether or not $e^{at}$ is greater than 1 or not. I would like to describe that as a Manipulate or a ListAnimate in Mathematica." Here's one way to visualize the growing and/or shrinking.

Manipulate[
 ParametricPlot[a {Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, 2 \[Pi]}, 
   PlotRange -> {{-1, 1}, {-1, 1}}], {a, 0.1, 2}]

This draws the circle and let's you change the radius by moving the slider to specify different a values.

enter image description here

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