When to use Linear Stability Analysis?

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  Let me try to share some insight that I got from taking George Haller's course on Non-Linear Dynamics. 

Most of the times when we try to analyze the stability of a dynamical system we start by finding the fixed points,

The fixed points are anchor points in the phase space and determine the dynamics across it.After we find the fixed points, we need to determine the characteristic of the fixed point, is it a stable/unstable/saddle fixed point. 

If it is a 1-D plot we were told to draw the F(x) vs x plot and draw arrows along +/- x-axis depending on the value of F(x). If in the neighborhood of a fixed point the arrows are pointing to it, then it's a stable point. If it's pointing away from it in both sides, it is an unstable fixed point. If it's pointing towards it on one side and pointing away from it on another side then its a saddle fixed point.

The funny business starts once you go to higher dimensions. How do we determine the stability of a fixed point in higher dimensions? The relevance of this question is huggeeeee. Examples include, parking spots for satellites(Lagrange Points) and stability analysis of a falcon X rocket re-entering earth upright (Cool Video) (Thesis on vertical landing). These sort of systems have large-dimensional phase space and the cost of getting it wrong is in Billions of USD and human lives.

We are taught a very naive trick in undergrad of doing a linear expansion about the fixed point to gauge the stability profile of the fixed point. What we are not taught most of the times, is that there is a criterion for using this linear analogue for the non-linear system.

We define the Jacobian of F(x) and the criteria you need to satisfy to use the linearization to characterize the fixed point is that the fixed point should be hyperbolic. This means that Real part of non of the Eigen values of the Jacobian should be zero. This is called as the Hartman-Grobman Theorem. Please do keep in mind when you do stability analysis next time.

PS: I have assumed necessary smoothness conditions in certain parts (C^k)


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