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We propose an approach for constructing Lyapunov function in quadratic form of a differential system. First, positive polynomial system is obtained via the local property of the Lyapunov function as well as its derivative. Then, the positive polynomial system is converted into an equation system by adding some variables. Finally, numerical technique is applied to solve the equation system. Some experiments show the efficiency of our new algorithm.

Analysis of the stability of dynamical systems plays a very important role in control system analysis and design. For linear systems, it is easy to verify the stability of equilibria. For nonlinear dynamical systems, proving stability of equilibria of nonlinear systems is more complicated than linear systems. One can use the Lyapunov function at the

For an autonomous polynomial system of differential equations, how to compute the Lyapunov function at

In this paper, we suppose Lyapunov function has quadratic form and some coefficients of Lyapunov function are unknown numbers. Some positive polynomials are obtained using the technique mentioned in [

The rest of this paper is organized as follows: Definitions and preliminaries about the Lyapunov function and the asymptotic stability analysis of differential system are given in Section

In this section, some preliminaries on the stability analysis of differential equations are presented.

In this paper, we consider the following differential equations:

In general, there exists two techniques to analyze the stability of an

Let

For a small system, it is easy to obtain the eigenvalues of the matrix

Another method to determine asymptotic stability is to check if there exists a Lyapunov function at the point

Given a differential system and a neighborhood

:

:

Solving polynomial system has been one of the central topics in computer algebra. It is required and used in many scientific and engineering applications. Indeed, we only care about the real roots of a polynomial system arising from many practical problems. For zero dimensional system, homotopy continuation method [

Due to the importance of this problem, many approaches have been proposed. The most popular algorithm which solves this problem is CAD; another is the so-called critical point methods, such as Seidenberg’s approach of computing critical points of the distance function [

Recently, Wu and Reid [

Let

There may exist some components which have no intersection with these random hyperplanes. Some points on these components must be the solutions of the Lagrange optimization problem:

In this section, we will present an algorithm for constructing the Lyapunov function. Our idea is to compute positive polynomial system which satisfies the definition of Lyapunov function first. Then we solve the polynomial system deduced from the positive polynomial system using homotopy algorithm; at this step, we use the famous package hom4ps2 [

Given a quadratic polynomial

Let

By the theory of linear algebra, one knows that the symmetric matrix

Let

Suppose all the roots of a real polynomial

Combine Theorems

Suppose we have obtained the positive polynomial system as in (

Note that the number of variable is more than the number of equation in system (

Recall the algorithm mentioned in Section

In the following we propose an algorithm to determine if there exists a Lyapunov function at the

Input: a differential system as defined in (

Output: a Lyapunov function or UNKNOW.

Construct the positive polynomial.

Convert the positive polynomial system into positive dimensional system defined in system (

We choose

Let

for

if the norm of imaginary part of

End for.

Construct polynomial system

Solve

return UNKNOW.

In the following, we present a simple example to illustrate our algorithm.

This is an example from [

Let Lyapunov function

We obtain the positive polynomial using Theorems

Convert system (

Construct two hyperplanes

Compute the roots of the augmented system

We obtain the first approximate real root of the system

Thus,

If the random hyperplanes

Solving the system

Thus,

In this section, some examples are given to illustrate the efficiency of our algorithm.

This is an example from [

We assume that

This is an example from a classic ODE’s textbook:

Assume that

This is another example from an ODE's textbook:

Assume that

For a differential system, based on the technique of computing real root of positive dimensional polynomial system, we present a numerical method to compute the Lyapunov function at

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was partially supported by the National Natural Science Foundation of China (11171053) and the National Natural Science Foundation of China Youth Fund Project (11001040) and cstc2012ggB40004.