Asymptotic Stability of Dynamical System by Lyapunov's Direct Method

Consider the two-dimensional autonomous dynamical system:
,
.
There are two steady states, and , shown by the green and red points.
Restricting the parameters to and , we compute and plot the region of asymptotic stability, shown in orange. All trajectories originating at an initial condition (IC) chosen in this orange region end at the . Trajectories diverge otherwise. The blue region is also a basin of attraction for . This region is obtained using Lyapunov's direct method and the Lyapunov function . The blue region cannot contain because it is an unstable steady state. defines a closed curve containing the origin if is a positive constant. The largest such region can be obtained by having at its frontier, or . You can drag the locator to change the IC.

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Reference
[1] A. Varma and M. Morbidelli, Mathematical Methods in Chemical Engineering, New York: Oxford University Press, 1997.
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