0 Introduction

In Ref.[1], the author pointed out that it is especially suitable to study the gradient systems by using of Lyapunov functions. More general linear-gradient system has already been put forward in Ref.[2]. In addition to usual gradient system, there are three other kinds of gradient systems, i.e., the skew-gradient system, the symmetric negative definite gradient system, and the semi-definite gradient system. Thus, there are four kinds of gradient systems, which have important values for us to study integral and stability of solution of the system. For the applications of gradient system on mechanical systems, there have been some results already, such as Refs.[3-8]. Based on those researches, the paper mainly tries to transform an autonomous Birkhoffian system^[9-10] into one of the four kinds of gradient systems with some conditions, and then use the properties of four kinds of gradient system to study the problems of integration and stability of the Birkhoffian systems.

1 Four kinds of gradient systems and Birkhoff's equations

The differential equations of motion of the general gradient systems are

$ \begin{equation} \label{eq1} \dot {x}_i =-\frac{\partial V}{\partial x_i }, \quad i=1, 2, \cdots, m, \end{equation} $

(1)

where V(x) is called potential function and x=(x₁, x₂, ……, x_m). The general gradient systems consist of the following importance properties:

1) The function V is a Lyapunov function of the system (1) and $\dot V$=0 if and only if x=(x₁, x₂, ……, x_m) is an equilibrium point.

2) For the linear system of the gradient system (1), there are only real characteristic roots at any equilibrium points.

Similarly, the skew-gradient systems can be formula as

$ \begin{equation} \label{eq2} \dot {x}_i =b_{ij} \frac{\partial V}{\partial x_j }, \quad i, j=1, 2, \cdots, m, \end{equation} $

(2)

where b_ij (x)=-b_ji (x) and the unified subscript denotes summation convention as well as in the following text. In addition, V=V(x) is called the energy function. The skew-gradient systems have the following importance properties:

1) The function V=V(x) is an integration of the system (2).

2) If V is a Lyapunov function, the solution of the system (2) is stable.

The differential equations of the gradient systems with symmetric negative definite matrices are^[2]

$ \begin{equation} \label{eq3} \dot {x}_i =S_{ij} \frac{\partial V}{\partial x_j }, \quad i, j=1, 2, \cdots, m, \end{equation} $

(3)

where S_ij=S_ij (x) is symmetric negative definite matrix. This system has the property as follows: If V is a Lyapunov function, the solution of the system (3) is stable.

The differential equations of the gradient systems with negative semi-definite matrices are^[2]

$ \begin{equation} \label{eq4} \dot {x}_i =a_{ij} \frac{\partial V}{\partial x_j }, \quad i, j=1, 2, \cdots, m, \end{equation} $

(4)

where a_ij=a_ij (x) is symmetric negative semi-definite matrix. The property of system (4) as follows: if V is a Lyapunov function, the solution of the system (4) is stable.

In order to convenience our statement, the above four kinds of gradient systems are separately called as the first to the fourth class gradient system.

2 The autonomous Birkhoff's equations

The gradient representations of autonomous Birkhoffian systems have the form

$ \begin{equation} \label{eq5} \dot {a}^\mu =\Omega ^{\mu \nu }\frac{\partial B}{\partial a^\nu }, \quad \mu, \nu =1, 2, \cdots, 2n, \end{equation} $

(5)

where B=B(a), a=(a¹, a², ……, a²ⁿ) and we have

$ \begin{array}{*{20}{l}} {{\Omega ^{\mu \nu }}{\Omega _{\nu \rho }} = \delta _\rho ^\mu, } \\ {{\Omega _{\nu \rho }} = \frac{{\partial {R_\rho }}}{{\partial {a^\nu }}} - \frac{{\partial {R_\nu }}}{{\partial {a^\rho }}}, \quad \mu, \nu, \rho = 1, 2, \cdots, 2n.} \\ {{R_\nu } = {R_\nu }\left( a \right), } \end{array} $

(6)

For the system (5), if there exist matrices (b_μν(a)), (S_μν(a)), (S_μν(a)), (a_μν(a)) and a function V=V(a) satisfying the following conditions

$ \begin{equation} \label{eq7} \Omega ^{\mu \nu }\frac{\partial B}{\partial a^\nu }=-\frac{\partial V}{\partial a^\mu }, \end{equation} $

(7)

$ \begin{equation} \label{eq8} \Omega ^{\mu \nu }\frac{\partial B}{\partial a^\nu }=b_{\mu \nu } \frac{\partial V}{\partial a^\nu }, \end{equation} $

(8)

$ \begin{equation} \label{eq9} \Omega ^{\mu \nu }\frac{\partial B}{\partial a^\nu }=S_{\mu \nu } \frac{\partial V}{\partial a^\nu }, \end{equation} $

(9)

$ \begin{equation} \label{eq10} \Omega ^{\mu \nu }\frac{\partial B}{\partial a^\nu }=a_{\mu \nu } \frac{\partial V}{\partial a^\nu }, \end{equation} $

(10)

it can be transformed into one of the four kinds of gradient systems which be defined at the end of section 2. Clearly, Eq. (8) can be satisfied easily, but not the case of Eqs. (7), (9), and (10).

3 Illustrative examples

Consider a Birkhoffian system with^[9]

$ \begin{array}{*{20}{l}} {{R_1} = {a^2} + {a^3}, \quad {R_2} = 0, \quad {R_3} = {a^4}, \quad {R_4} = 0, } \\ {B = \frac{1}{2}\{ {{({a^3})}^2} + 2{a^2}{a^3} - {{({a^4})}^2}\} .} \end{array} $

(11)

Try to transform it into a gradient system and study the stability of solution.

From the Birkhoffian equation (5), we know that

$ {\dot a^1} = {a^3}, \quad {\dot a^2} = {a^4}, \quad {\dot a^3} = - {a^4}, \quad {\dot a^4} = - {a^2}. $

Let V=B, we have

$ \left( {\begin{array}{*{20}{c}} {{{\dot a}^1}} \\ {{{\dot a}^2}} \\ {{{\dot a}^3}} \\ {{{\dot a}^4}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1&0&0 \\ { - 1}&0&0&{ - 1} \\ 0&0&0&1 \\ 0&1&{ - 1}&0 \end{array}} \right)\left( \begin{gathered} \frac{{\partial V}}{{\partial {a^1}}} \hfill \\ \frac{{\partial V}}{{\partial {a^2}}} \hfill \\ \frac{{\partial V}}{{\partial {a^3}}} \hfill \\ \frac{{\partial V}}{{\partial {a^4}}} \hfill \\ \end{gathered} \right), $

which is the second kind of gradient system. V is an integral of the system but cannot be a Lyapunov function. Consider the second and the fourth identities of the above equation and they can be rewritten as

$ \left( {\begin{array}{*{20}{c}} {{{\dot a}^2}} \\ {{{\dot a}^4}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1 \\ { - 1}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{{\partial V}}{{\partial {a^2}}}} \\ {\frac{{\partial V}}{{\partial {a^4}}}} \end{array}} \right), $

where

$ V = \frac{1}{2}{({a^2})^2} + \frac{1}{2}{({a^4})^2}, $

which is not only an integral but also a Lyapunov function, therefore the solutions a²=a⁴=0 are stable.

Birkhoffian system is

$ \begin{equation} R_1 =a^2, \quad R_2 =0, \quad B=(a^1)^2+(a^2)^2+a^1a^2-\frac{1}{3}(a^1)^3+\frac{1}{3}(a^2)^3. \end{equation} $

(12)

Try to transform it into a gradient system and study stability of solution.

From the Birkhoffian equation (5), we know that

$ \begin{gathered} {{\dot a}^1} = {a^1} + 2{a^2} + {({a^2})^2}, \hfill \\ {{\dot a}^2} = - 2{a^1} + {({a^1})^2} - {a^2}. \hfill \\ \end{gathered} $

It can be rewritten as

$ \left( {\begin{array}{*{20}{c}} {{{\dot a}^1}} \\ {{{\dot a}^2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1 \\ { - 1}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{{\partial V}}{{\partial {a^1}}}} \\ {\frac{{\partial V}}{{\partial {a^2}}}} \end{array}} \right), $

where the matrix is anti-symmetric and V satisfy

$ V = B, $

which is the second kind of gradient system. V is an integral of the system and a Lyapunov function. Therefore, the solutions a¹=a²=0 are stable.

Birkhoffian system is

$ \begin{align} R_1 =a^2, \quad R_2 =0, \quad B=\frac{1}{2}(a^1)^2-\frac{1}{4}(a^2)^2. \end{align} $

(13)

Try to transform it into a gradient system and study stability of solution.

The Birkhoffian equation (5) gives

$ {\dot a^1} = - \frac{1}{2}{a^2}, \quad {\dot a^2} = - {a^1}, $

or

$ \left( {\begin{array}{*{20}{c}} {{{\dot a}^1}} \\ {{{\dot a}^2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 1{\text{ }}}&{\;\;1} \\ {\;1}&{ - 2} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{{\partial V}}{{\partial {a^1}}}{\text{ }}} \\ {\frac{{\partial V}}{{\partial {a^2}}}{\text{ }}} \end{array}} \right), $

where the matrix is symmetric negative definite and V is

$ V = \frac{1}{2}{({a^1})^2} + \frac{1}{4}{({a^2})^2} + {a^1}{a^2}, $

which is the third kind of gradient system. V is not a Lyapunov function. The characteristic equation has positive real roots. Therefore, the solutions a¹=a²=0 are not stable.

4 Conclusions

We know that the autonomous Birkhoffian systems consist of the following properties, i.e., Birkhoffian function B is integral of the system, and if B can also be a Lyapunov function, the solutions of the system are stable, which is obtained in view of gradient system in this paper. It is easy to see that the autonomous Birkhoffian system is the second kind system naturally, but it is difficult to become three other kinds of gradient systems. Even through an autonomous Birkhoffian system can be transformed into three other kinds of gradient systems, V is also very difficult to become the Lyapunov function. In this case, the first-order approximation theory can be used if we want to study the stability of solution.