The Degenerate Node

So for tomorrow, we will finish classifying the orbit structure of linear maps of the plane, basically concluding that orbits of planar linear maps follow similar patterns as that found in the trajectories of linear, homogeneous, autonomous ODEs in 2-dimensions. But there is a twist, in that it is the sign of the eigenvalues that determines the type and stability of the origin, as an isolated equilibrium of a nondegenerate linear system.  But it is the magnitude of the eigenvalues of a linear map from the plane to itself, that determines the type and stability of the isolated fixed point at the origin.  The secret as to why lies in calculating the time-1 map of a linear system of ODEs.  THAT is a linear map from the plane to the plane.  But that linear map does NOT have the same matrix as the linear system.  It is actually the matrix exponential of the matrix that determines the linear system, and that is sometimes a bit difficult to calculate.  We will spend a short time on this. 

For today, we will discuss linear maps of the plane, leading to a classification that is quite similar to that of linear systems of ODEs, but with some important differences.  Once classified, we will use a particular example to highlight what divergence may mean for an orbit of a linear dynamical system in real two space and just how an orbit can go to infinity.  From this, we get a way to create a functional form for the elements of a Fibonacci (or similar) sequence.