“Linear Stability”的概念、定义、习题解题方法及翻译-数学辞典

单词

Linear Stability

释义

Linear Stability

Consider the general system of two first-order Ordinary Differential Equations

			(1)
			(2)

Let

and

denote Fixed Points with

, so

			(3)
			(4)

Then expand about

To first-order, this gives

(7)

where the

Matrix is called the Stability Matrix.

In general, given an -D Map , let be a Fixed Point, so that

(8)

Expand about the fixed point,


			(9)

(10)

The map can be transformed into the principal axis frame by finding the Eigenvectors and Eigenvalues of the Matrix A

(11)

so the Determinant

(12)

The mapping is

(13)

When iterated a large number of times,

(14)

only if

for

, ...,

but

if any

. Analysis of theEigenvalues (and Eigenvectors) of A therefore characterizes the typeof Fixed Point. The condition for stability is

for

, ...,

See also Fixed Point, Stability Matrix

References

Tabor, M. ``Linear Stability Analysis.'' §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20-31, 1989.

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