StabilityLinearization - 1.4 Stability and Linearization...

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§ 1.4 Stability and Linearization 39 1.4 Stability and Linearization Suppose we are studying a physical system whose states x are described by points in R n . Assume that the dynamics of the system is governed by a given evolution equation dx dt = f ( x ) . Let x 0 be a stationary point of the dynamics, i.e., f ( x 0 ) = 0. Imagine that we perform an experiment on the system at time t = 0 and determine that the initial state is indeed x 0 . Are we justiFed in predicting that the system will remain at x 0 for all future time? The mathematical answer to this question is yes, but unfortunately it is probably not the question we really wished to ask. Experiments in real life seldom yield exact answers to our idealized models, so in most cases we will have to ask whether the system will remain near x 0 if it started near x 0 . The answer to the revised question is not always yes, but even so, by examining the evolution equation at hand more carefully, one can sometimes make predictions about the future behavior of a system starting near x 0 . A simple example will illustrate some of the problems involved. Consider the following two di±erential equations on the real line: x ± ( t )= - x ( t ) (1.4.1) and x ± ( t x ( t ) . (1.4.2) The solutions are, respectively, x ( x 0 ,t x 0 e - t (1.4.3) and x ( x 0 x 0 e + t . (1.4.4) Note that 0 is a stationary point of both equations. In the Frst case, for all x 0 R , we have lim t →∞ x ( x 0 ) = 0. The whole real line moves toward the origin, and the prediction that, if x 0 is near 0 then x ( x 0 ) is near 0, is justiFed. On the other hand, suppose we are observing a system whose state x is governed by equation (0.1.2). An experiment telling us that at time t = 0, x (0) is approximately zero will certainly not permit us to conclude that x ( t ) stays near the origin for all time, since all points except 0 move rapidly away from 0. ²urthermore, our experiment is unlikely to allow us to make an accurate prediction about x ( t ) because if x (0) < 0, x ( t ) moves rapidly away from the origin toward -∞ , but if x (0) > 0, x ( t
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.

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StabilityLinearization - 1.4 Stability and Linearization...

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