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Team:SJTU-BioX-Shanghai/judge a system
From 2009.igem.org
Project introduction. Inspired by the natural regulator of circadian bioclock exhibited in most eukaryotic organisms, our team has designed an E.coli-based genetic network with the toxin-antitoxin system so that the bacterium oscillates between two states of dormancy and activity (more...)
How to judge a system that cannot oscillate?
After we have tried so many groups of parameters, the question approaches that why some of the systems are asymptotic stable. This asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem.
In our system, suppose 3-order differential equations are denoted as following form:
Let J(p) be the 3×3 Jacobian matrix at the point p. If all eigenvalues of J(p) have strictly negative real part then the solution is asymptotically stable. This condition can be tested using the Routh–Hurwitz criterion.
Example: If a = 26,b = 1,c = 6,d = 21,e = 1,f = 46,g = 21,h = 1,we obtain the fixed point p = (4.1967, 0.7953, 0.0000); Then we evaluated the eigenvalues for the jacobian :
-32.0086 -0.5849 + 5.3151i -0.5849 - 5.3151i
The solution is asymptotically stable, since their real parts are negative. This result corresponds with the following figure.
Reference:
- Hurwitz, A. "‘On the conditions under which an equation has only roots with negative real parts". Selected Papers on Mathematical Trends in Control Theory. 1964
- Routh, E. J.A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion. 1987
- Stability theory, Wikipedia, the free encyclopedia
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