Linearly stable / non-linearly unstable map example

[Mattbringssoda]

Homework Statement

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Give examples of fixed points of vector fields and maps that are stable in the linear approximation but are nonlinearly unstable

Homework Equations

The Attempt at a Solution

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I was able to find an example in a vector field that, when the Jacobian is found and the fixed point at the origin is analyzed, gives eigenvectors of +/- i, indicating Lyapunov stability but not asymptotic stability. When this equation is converted to polar coordinates, it is show clearly that the r' = r^3 and so the graph actually spirals away from the origin.

However, I have been unsuccessful in finding a map version of this. I am taking a few stabs in the dark and they don't seem to be getting me anywhere, and I'm not sure how to go about constructing the answer...

Thanks for any and all help!

 

[mighty2000]

One example of a map that is stable in the linear approximation but is nonlinearly unstable is the logistic map. This map is commonly used in population dynamics and is given by the equation x_n+1 = rx_n(1-x_n), where r is a constant. When analyzing the fixed points of this map, it can be shown that the fixed point at x=0 is stable in the linear approximation (since the derivative at this point is less than 1), but is actually nonlinearly unstable. This can be seen by plotting the map for different values of r, where it is observed that for r>3, the fixed point at x=0 becomes unstable and the trajectory of the map diverges away from it. This is an example of a nonlinear instability, as the linear approximation does not accurately predict the behavior of the map.

Another example of a map that exhibits this behavior is the Henon map, given by the equations x_n+1 = 1-ax_n^2 + y_n and y_n+1 = bx_n, where a and b are constants. This map has a fixed point at (0,0) which is stable in the linear approximation, but is nonlinearly unstable for certain values of a and b. This can be seen by plotting the map for different values of a and b, where it is observed that for certain values, the trajectory of the map diverges away from the fixed point at (0,0). This again shows that the linear approximation does not accurately predict the behavior of the map.