The Hartman-Grobman Theorem guarantees the existence of a homeomorphism that conjugates the nonlinear equation to its linearization It happens to be that this is not always a diffeomorphism. In this homework problem, we see how resonances can conspire to prevent the smoothness of the conjugacy map. Consider
x' = -x
y' = -2y + x²
a) Write down the general solutions for this system of equations. b) Find a conjugacy to the linearized system.
u' = -u
z' = -2z
Note: it is clear that w= x since the equations are identical. Show that
y = z + αw2 (log)w
is a conjugacy for the y component. Note that log refers to natural logarithm. Find the α that makes this a conjugacy. c) Vorify that the conjugacy is C¹ but not C².
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