Asymptotic Stability of Dynamical System by Lyapunov's Direct Method

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.


Contributed by: Housam Binous and Ahmed Bellagi (April 2015)
Open content licensed under CC BY-NC-SA




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

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.