Fixed point

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 with a fixed point , the local behaviour at resembles the linear dynamics of the Linear differential system . Thus if are each eigenvectors of with eigenvalues , the local behaviour follows

where

• indicates instability along eigendirection
• indicates stability along eigendirection
• with indicates a spiral in the plane of and . indicates stability
• 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 depends on higher order terms of the Taylor expansion.²

Iterated map

Given an Iterated map with a fixed point , the local behaviour at resembles that of . Thus if are each eigenvectors of with eigenvalues , the local behaviour follows

where

• indicates instability along eigendirection
• indicates stability along eigendirection

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 and (which is the defining feature of fixed points).

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