The authors in Ref.  [1] pointed out that an ordinary differential equation can be written in a “ linear-gradient system” when its one or more integrations or Lyapunov function is given. The skew-gradient system is a kind of linear-gradient systems which has important significance, and is different from the gradient system given by Ref.  [2]. The former’ s matrix is antisymmetric; while the latter’ s matrix is symmetric, and is an identity matrix. Quispel et al.^{[3, 4]} proved that an autonomous ordinary differential equation has a first integral if and only if it can be written as a skew-gradient system, and presented a general method to rewrite such a system as a skew-gradient system, then from which constructed several new integral-preserving discrete gradient methods. Hong et al.^[5] discussed the skew-gradient representation of autonomous stochastic differential equations with a conserved quantity, then from which constructed direct/indirect discrete gradient approaches. The Birkhoffian system is a natural generalization of the Hamiltonian system. Great progress on its dynamical research has been achieved.^{[6– 14]} The generalized Birkhoffian system refers to the Birkhoffian system with added terms.^[14] We have studied the gradient representation for the generalized Birkhoffian system.^[15] In this paper, we will give a condition under which a generalized Birkhoffian system can be rewritten in a skew-gradient system, and discuss the dynamic behaviors, especially integration and stability of the generalized Birkhoffian system by using the property of the skew-gradient system.

The differential equations of the skew-gradient system have the form^[1]

where

where V is called “ energy” in Ref.  [1]. The gradient system given in Ref.  [2] has the form

It is clear that equation  (1) is different from Eq.  (3).

The skew-gradient system has the following important properties. (i) An autonomous ordinary differential equation has a first integral if and only if it can be written as a skew-gradient system. (ii) The energy function V is an integral of the system  (1). (iii) If V can be a Lyapunov function, then the solution x_i = x_i0 (i = 1, 2, … , m) of the system  (1) is stable.

The skew-gradient form  (1) has many uses. For example, if the system has a first integral V, then its motion stays on the level set Σ _C : = {X : V(X) = C}. Preserving this property leads to good nonlinear stability, especially if Σ _C is compact.^[1] Another example, systems with a Lyapunov function V can be “ weak” or “ strong” . Preserving these properties is important because they are equivalent to the existence of a stable, respectively asymptotically stable, attracting set.^[1]

The skew-gradient representation can be used not only to discuss the integration and stability, but also to construct the integral-preserving discrete gradient approaches for the system. In addition, Hamiltonian systems, generalized Hamiltonian systems, and autonomous Birkhoffian systems are skew-gradient systems. Naturally, the results and methods on the skew-gradient system can be applied to these systems.

The generalized Birkhoffian system refers to the Birkhoffian system with added terms. Great progress has been made on its dynamical research.^[16]

In general, a generalized Birkhoffian system cannot be rewritten in a skew-gradient system. In the following, we give a condition under which a generalized Birkhoffian system can become a skew-gradient system. In such a case, one can easily discuss the integration and stability or construct integral-preserving discrete gradient methods for the system by using the results of the skew-gradient system.

The nonsingular generalized Birkhoffian equations have the form^[14]

where and in the following, the same indices in a term indicate to take the sum with it. In Eqs.  (4), B = B(t, a) is the Birkhoffian, R_μ = R_μ (t, a) are Birkhoff’ s functions, and Λ _μ = Λ _μ (t, a) are the added terms, and

is called Birkhoff’ s tensor, satisfying

For Eqs.  (4), if

then we have

In general, equations  (8) still cannot become a skew-gradient system. If there is a function V = V(a), satisfying

then equations  (8) become

Obviously, it is a skew-gradient system. So, under conditions  (7) and (9), the generalized Birkhoffian system  (1) becomes the skew-gradient system  (10).

Taking the derivative of V with respect to time according to Eqs.  (10), we have

So, V is an integral of system  (10). If V can further become a Lyapunov function, then one can discuss the stability of generalized Birkhoffian system  (1) by using the property of the skew-gradient system.

For a Birkhoffian system, there are Λ _μ = 0 (μ = 1, 2, … , 2n), then an autonomous Birkhoffian system is a skew-gradient system, whose Birkhoffian is an integral. Similarly, if B can further become a Lyapunov function, then one can discuss the stability of the Birkhoffian system.

Example 1 A generalized Birkhoffian system of order 4 is

Try to rewrite it in a skew-gradient system, and discuss the stability of zero solution.

Clearly, condition  (7) is satisfied. Condition  (9) gives

From which one can take

Then equations  (10) give

Take function V as a Lyapunov function, which is positive-definite in the neighborhood of a¹ = a² = a³ = a⁴ = 0. Taking V̇ according to the above equation obtains

According to the Lyapunov theorem, zero solution a¹ = a² = a³ = a⁴ = 0 is stable.

Example 2 The Hojman– Urrutia equation has a Birkhoffian representation^[6]

Try to rewrite it in a skew-gradient system.

Equations  (10) give

Clearly, it is a skew-gradient system, and function B is an integral of the system. But B cannot become a Lyapunov function.

Generally speaking, a generalized Birkhoffian system is not a skew-gradient system. In this paper, we obtained a condition under which the generalized Birkhoffian system can become a skew-gradient system, namely expressions  (7) and (9). When a generalized Birkhoffian system is rewritten in a skew-gradient system, its dynamic behaviors can be discussed by using the properties of the skew-gradient system. Especially, if function V can become a Lyapunov function, then one can discuss the stability of the system.