Summary

Due to some limitations, the work I’ve done this some was still fairly basic. So far, the research involves some computer programing and mathematical proofs. The coding part includes two Mathematica programs to numerically solve stable regions and Python program to run Monte Carlo trials. With the help of these programs, I looked into the stable region of system A-Pt where A and P are 3 by 3 real matrices when t progresses from negative infinity to positive infinity.  I found the possible number of times that the stability of the system could change and found substantial examples for each situation. Regarding the situations when the system’s stability changes for an odd number of times, some observations are also made and proved.  I also digressed a little bit to look into the probability that a random 3 by 3 real matrix is stable. I got an approximation with numerical experiment, but I haven’t started any theoretical proofs.

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Some Other Observations and a Short Digression

We can categorize matrix P by its Jordan normal form to deal with the problem.

p_jordan

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Stability Switches: Examples

When A and P are arbitrary 3 by 3 matrices, the system A – Pt where t ∈R can have 0,1,2,3,4,6 stability switches. I have found the following systems as examples.

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On the Number of Stability Switches

As previously mentioned, by Routh-Hurwitz Stability Criterion,

rh

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Getting the Programs Ready

This research focuses on the dynamical system A-Pt where A and P are both matrices.

When A and P are 2 by 2 matrices:

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