For the values of and , sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.
The trajectories for all nine initial values is
Consider the Eigen values and .
As , the future and past of the system toward outside.
Draw the graph of and as follows:
If the Eigen values of the system are and such that then the phase portraits of the system is
By the trajectory definition, draw the graph of the trajectory of the system as follows:
Hence the trajectories for all nine initial values and traces of the future and past of the system are sketched.
Consider the IVP with where A is an upper triangular matrix with m distinct diagonal entries . See the examples in Exercise 45 and 46.
(a) Show that this problem has a unique solution whose components are of the form
for some polynomials . Hint: Find first , then , and so on.
(b) Show that the zero state is a stable equilibrium solution of this system if (and only if) the real part of all the is negative.
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