The Lyapunov function method

We will begin the description of the Lyapunov method pf examination of the stability of a stationary point with the analysis of a straightforward example. Consider the following system of equations  

The function W x, y) is thus seen to decrease, except for the point (0, 0), for increasing t. 

0) is the point which is approached by the phase trajectories of the system (A 17). The point (0, 0) satisfies the equations x = y = 0 and thus is a stationary point of the system (A 17). 

The analysis of properties of the function W(x, y) has led us to a conclusion that (0, 0) is an (asymptotically) stable stationary point. Note that to arrive at this conclusion it was not necessary to solve the system (A17). 

Considerations of this type are characteristic of the Lyapunov method of examination of the stability of a stationary point the function W defined by equation (A18) is an example of the so-called Lyapunov function. Note also that to draw the conclusion about an asymptotic stability of the stationary point (0, 0), in addition to inequality (A 19) deriving from properties of the Lyapunov function and properties of the investigated system, inequality (A20) was also required. 

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The stability of stationary points in cases (al), (a2), (d) is proven in Appendix 45 using the Lyapunov function method. 

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