Linear stability analysis

Linear stability analysis exploits the Hartman-Grobman theorem to gain insight into the stability of a fixed point via the vector field's Jacobian at that point.

Differential system

Given a differential system $˙ 𝐱 = 𝑓 (𝐱)$ with a fixed point $𝐱 *$ , the local behaviour at $𝐱 *$ resembles the linear dynamics of the Linear differential system $˙ 𝐲 = 𝐷 𝑓 (𝐱 *) 𝐲$ . Thus if $𝐯 𝑖$ are each eigenvectors of $𝐷 𝑓 (𝐱 *)$ with eigenvalues $𝜆 𝑖$ , the local behaviour follows

𝐲 (𝑡) = \sum 𝑐 𝑖 𝐯 𝑖 𝑒 𝜆 𝑖 𝑡

where

$𝜆 𝑖 > 0$ indicates instability along eigendirection $𝐯 𝑖$
$𝜆 𝑖 < 0$ indicates stability along eigendirection $𝐯 𝑖$
$𝜆 𝑖, 𝑗 = 𝛼 \pm 𝛽 𝑖$ with $𝛼 \neq 0$ indicates a spiral in the plane of $𝐯 𝑖$ and $𝐯 𝑗$ . $𝛼$ indicates stability
$𝜆 𝑖, 𝑗 = \pm 𝛽 𝑖$ indicates the fixed point is a linear centre but gives no guarantee of a nonlinear centre¹

If any eigenvalues have a real part of zero, the Hartman-Grobman theorem does not apply and linear stability analysis fails, since the stability at $𝐱 *$ depends on higher order terms of the Taylor expansion.²

Iterated map

Given an Iterated map $𝐱 𝑛 + 1 = 𝑓 (𝐱 𝑛)$ with a fixed point $𝐱 *$ , the local behaviour at $𝐱 *$ resembles that of $𝐱 𝑛 + 1 = 𝐷 𝑓 (𝐱 *) 𝐱 𝑛$ . Thus if $𝐯 𝑖$ are each eigenvectors of $𝐷 𝑓 (𝐱 *)$ with eigenvalues $𝜆 𝑖$ , the local behaviour follows

𝐲 𝑛 = \sum 𝑐 𝑖 𝐯 𝑖 (𝜆 𝑖) 𝑛

where

$| 𝜆 𝑖 | > 1$ indicates instability along eigendirection $𝐯 𝑖$
$| 𝜆 𝑖 | < 1$ indicates stability along eigendirection $𝐯 𝑖$

If any eigenvalues are one, linear stability is insufficent.³ In the 1D case, this can easily be identified by the nature of the intersection between $𝑓$ and $i d$ (which is the defining feature of fixed points).

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2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, §6.3, p. 170 ↩
2022. Nonlinear Dynamics: A Concise Introduction Interlaced with Code, §1.4, pp. 9–12 ↩
2022. Nonlinear Dynamics: A Concise Introduction Interlaced with Code, §1.4, p. 12 ↩

Linear stability analysis

Differential system

Iterated map

Footnotes