On the equivalence of three-dimensional differential systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

In the present section, we shall briefly introduce the related concepts which will be used throughout the rest of this article.
Consider differential system which has a continuously differentiable right-hand side and a general solution ( ) φ t t x ; , o 0 . For such a system, the reflective function is defined as ( ) = (− ) F t x φ t t x , ; , [1]. A continuous differentiable vector function ( ) F t x , on × R R n is the reflective function of system (1.1) if and only if the following basic relation holds.
To check whether the above systems compatible, we can use the Frobenius theorem [2]. Doing this in practice, however, is a very hard task.
If we can neither solve the system (1.1) nor the problem (1.2), then it is good enough to construct any system (1.3), which is equivalent to system (1.1). To do this, sometimes we can use: be solutions of the equation and ( ) ( = … ) α t k m 1, 2, , k are arbitrary continuous odd functions. Then all the perturbed systems of the form are equivalent to each other and to system (1.1) (here m is any natural number or even = ∞ m ).
Thus, if we find some solutions of equation (1.4), we can construct systems (1.5), which are equivalent to system (1.1). It can be seen that the solution of equation (1.4), that is, the reflective integral [3] of system (1.1), is particularly important to determine the equivalence of two differential systems. In addition, if these equivalent systems are ω 2 -periodic with respect to t, then their Poincare mappings coincide, the initial conditions at = − t ω of their ω 2 -periodic solutions and their stability characters are the same. Thus to study the equivalence of two differential systems is very important and interesting.

Main results
Now, we focus on the equivalence conditions for the general three-dimensional differential system x p t x p t x P t x y q t y q t y Q t y z r t x r t y r t z r t x r t xy r t y r t xz r t z r t yz R t x y z , , , (2.2) as well as the characteristics of ( ) 1, 2; 1, 2, , 9 are continuously differentiable functions. Let us outline the investigation method. In accordance with Lemma 1.1, we seek systems equivalent to system (2.2) in the set of systems (1.5). Usually, however, we cannot find out all solutions of equation (1.4) in the case under consideration. Therefore, we seek only polynomial solutions ( ) t x y z Δ , , , of equation (1.4), namely, solutions of the form In the following, we denote k k , 0 1 and k 2 are arbitrary constants.
Proof. Using relation (2.4), we can get i.e., Equating the coefficients of the same powers of x y , and z, we have  (2.14) ω ω , 0 1 and ω 2 are arbitrary constants.
Proof. Using relation (2.4), we can get Summarizing the above, the proof is finished.   γ γ x γ y γ z γ x γ y γ z γ xy γ xz γ yz γ γ x γ y γ z p x p x γ γ x γ y γ z q y q y γ γ x γ y γ z r x r y r z r x r xy r y r xz r z r yz r r x r y r z μ μ x μ x r r x r y r z s s y s y r r x r y r z γ γ x γ y γ z γ x γ y γ z γ xy γ xz γ yz   i.e.   is also the reflective function of system (2.55), i.e. systems (2.54) and (2.55) are equivalent. Obviously, system (2.54) has no periodic solutions. Considering equivalence between systems (2.54) and (2.55), system (2.54) has no periodic solutions too.