Fixed Points and Stability

the solid black dot is a stable fixed point (the local flow is toward it) and the open dot is an unstable fixed point (the flow is away from it). 

In terms of the original differential equation, fixed points represent equilibrium solutions (sometimes called steady, constant, or rest solutions, since if X = X initially, then x(z) = x for all time). An equilibrium is defined to be stable if all sufficiently small disturbances away from it damp out in time. Thus stable equilibria are represented geometrically by stable fixed points. Conversely, unstable equilibria, in which disturbances grow in time, are represented by unstable fixed points. 

Find all fixed points for x = x -1, and classify their stability. 

Solution This type of circuit problem is probably familiar to you. It is governed by linear equations and can be solved analytically, but we prefer to illustrate the geometric approach. 

First we write the circuit equations. As we go around the circuit, the total voltage 

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Fixed points and stability matrix eigenvalues in the first order of the e, <5 - expansion [72]. 

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