Difference Equation versus Differential Equation

Let be a generic point in the plane. The figure illustrates the relation between the difference equation and the differential equation for the particular case . For decreasing values of the step size parameter and for a chosen initial value , you can see how the discrete process (in white) tends to follow the trajectory of the differential equation that goes through (in black). The background is colored using the norm of the expected motion, rescaled to be in the interval .

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