asymptotic stability

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asymptotic stability

[ā‚sim′täd·ik stə′bil·əd·ē]
(mathematics)
The property of a vector differential equation which satisfies the conditions that (1) whenever the magnitude of the initial condition is sufficiently small, small perturbations in the initial condition produce small perturbations in the solution; and (2) there is a domain of attraction such that whenever the initial condition belongs to this domain the solution approaches zero at large times.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Observe that if all roots of the equation det (z[alpha](z)I - A) = 0 are inside the unit circle, then system (5) (or equivalently, (6)) is asymptotically stable, see for instance .
If [R.sub.0] [less than or equal to] 1, then the infection-free equilibrium [E.sub.0] is globally asymptotically stable and it becomes unstable if [R.sub.0] > 1.
The equilibrium points [x.sup.*] are locally asymptotically stable if all the eigenvalues [[lambda].sub.j] (j = 1,2, ..., n) of the Jacobian matrix A = [partial derivative]f/[partial derivative]x evaluated at the equilibrium points [x.sup.*] satisfy the following condition:
Hence, the low-criminality equilibrium [P.sub.l] = ([S.sub.0], [D.sub.0], [C.sub.1], [S.sub.1], [D.sub.1], 0,0,0) is locally asymptotically stable if [R.sup.*.sub.1] < 1, where [R.sup.*.sub.1], defined as Criminality Reproduction Number (CRN), is given by
Further, (i) if [R.sub.0] < 1, the disease-free equilibrium [P.sub.02]([K.sub.p], 0) is locally asymptotically stable and (ii) if [R.sub.0] > 1, the disease-free equilibrium [P.sub.02]([K.sub.p], 0) is unstable and the endemic equilibrium [P.sup.*]([S.sup.*.sub.p], [I.sup.*.sub.p]) exists and is locally asymptotically stable.
Then system (1) is asymptotically stable. Combined with the above two cases, as long as the linear matrix inequality (11) is satisfied, system (1) is asymptotically stable.
The system [mathematical expression not reproducible] is asymptotically stable if [absolute value of arg(eig(A))] > [alpha][pi]/2 where 0 < [alpha] < 2, and eig(A) are the eigenvalues of matrix A.
Based on the Routh-Hurwitz criterion and the discussion in , it follows that the positive equilibrium [E.sub.*] is locally asymptotically stable if the following condition holds: ([H.sub.1]):
The positive equilibrium point [E.sup.*]([x.sup.*], [y.sup.*]) is locally asymptotically stable, if (I) holds.
Thus, according to the Routh-Hurwithz theorem, we know that if conditions ([H.sub.11]) [a.sub.10]- > 0, [a.sub.13] > 0, and [a.sub.1]3 [a.sub.12] > [a.sub.11] hold, then viral equilibrium [E.sub.*]([S.sub.*])[L.sub.*],[A.sub.*],[R.sub.*]) of system (2) without delay is locally asymptotically stable.
They showed using LaSalle's invariance principle that the disease-free equilibrium, [P.sub.0], is globally asymptotically stable in T if [R.sub.0] [less than or equal to] 1 and unstable if [R.sub.0] > 1.

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