- Open Access
Integrability and linearizability for Lotka-Volterra systems with the resonant saddle point
© Wang and Huang; licensee Springer. 2014
Received: 25 June 2013
Accepted: 23 December 2013
Published: 16 January 2014
Integrability and linearizability of a Lotka-Volterra system in a neighborhood of the singular point with eigenvalues 3 and any negative integer −q are studied completely. By computing the singular point quantities and generalized period constants, we obtain, respectively, the integrable and linearizable necessary conditions for this class of systems. Then we apply some effective ways to prove the sufficiency. Here the algorithms of finding necessary conditions are all linear and readily done using computer algebra system such as Mathematica or Maple, and these play an important role in solving completely the integrability and linearizability for the resonant case.
where . For the above systems, most of the known work was focused on some special resonant, yet for the general case there are few results. For the resonant saddle point of quadratic systems, by computing the saddle numbers to get the necessary conditions, the authors completely solved the integrability problem in [1, 2], and furthermore in , mainly by annihilating the coefficients of the normal form and finding the Darboux factor, the authors obtained the necessary and sufficient conditions (15 cases) for the linearizable systems. For the resonant cases, integrability and linearizability of Lotka-Volterra systems were solved, respectively, in  and . As for the general case , the authors of  fully discussed the two problems by generalizing and expanding two methods already known: the power expansion of the first integral and the transformation of the saddle into a node. For the general case of the resonance, the authors of  studied systematically the mechanisms which lead to the origin being linearizable, integrable or normalizable. Recently the authors in  gave the center variety for families of resonant polynomial vector fields, and moreover derived an algorithm for computing the focus quantities. Especially there followed, for cubic Lotka-Volterra systems, some good results, obtained in [7–9]. However, one is far from finishing the investigation of all Lotka-Volterra systems.
where , , , and , are homogeneous polynomials in , i.e., , .
where , , , and are polynomials of , with rational coefficients, and , can be determined uniquely, for .
We write , , , .
Definition 1.1 For any positive integer k, is called the k th singular point quantity of the origin of system (4), and the origin of system (4) is called the generalized center, i.e., system (4) is integrable at the origin if , i.e., , .
Definition 1.2 For any positive integer k, are called the k th generalized period constants of the origin of system (4). And system (4) can be called linearizable at the origin if , i.e., , .
System (4) is linearizable at the origin if it is integrable at the origin and there exists an analytic change of variables (9), such that one of the two equations and holds.
In the next section, we deduce a recursive formula to compute the singular point quantities of system (3) and the integrable necessary conditions for system (3) are obtained. In Section 3, we deduce also a recursive formula to compute generalized period constants of the systems, and at the same time, all linearizable necessary conditions of the origin for system (3) are obtained. In Section 4, the proofs of sufficient conditions are given completely.
2 Generalized center at the origin
First we discuss computational method of singular point quantities for system (3).
The relations between and () are as follows.
The converse case holds as well.
Remark 1 Similar to the proof procedure in [11, 15], we can obtain the above relation, thus and are called algebraic equivalent, i.e., [14, 15]. Further, if and the coefficients of system (4) are all real, namely it is just system (1), for the resonant focus number in , then , . So we can apply directly the above method to find the necessary conditions of integrability for solving the problem of the generalized center.
Now we consider the real system (3).
in the above expressions, if or , let , and is the mth singular point quantity of the origin of system (3).
Corollary 2.4 For , the origin of system (3) is a generalized center if and only if the following conditions are satisfied:
Proof By applying the recursive formulas, one can obtain the necessary conditions (18) and (19). It is easily verified that the above conditions are identical to the corresponding results in [1, 4]. The proof of these sufficient conditions will not be given any longer. □
Remark 2 For system (3), according to Theorem 2.3, we can get the recursive formulas to compute the singular point quantities for investigating the generalized center for any positive integer q. The integrability and linearizability for Lotka-Volterra systems with or , , resonances have been studied in  completely, so we only need to consider the case of ().
For system (3), we can compute the singular point quantities and obtain the following result.
Theorem 2.5 The first 20 singular point quantities for the origin of system (3) are as follows:
where , and are different for different , , similarly and for . In the above expression of , we have already let , .
Then we have the following.
- (B)When , , the origin of system (3) is a generalized center if and only if one of the following conditions is satisfied:(27)(28)(29)(30)(31)
Proof We have to find necessary conditions of the generalized center from the vanishing of all first singular point quantities. We can let the first 20 singular point quantities in equation (20) or (21) be zero. It is easy to obtain the above five necessary conditions.
Next, the sufficient conditions need to be proved. On the one hand, for the conditions (22), (23), (24) and (27), (28), (29) will be proved sufficiency, respectively, in Section 4. And furthermore, from Lemma 2.7, the condition (25) or (30) is sufficient.
where , .
where , .
However, the conditions () and () will be proved sufficient, respectively, in Section 4. □
Lemma 2.7 (, Theorem C)
System (3) is integrable if .
3 Linearizability at the origin
Now, we discuss generalized isochronicity of the origin for system (3), namely we figure out all linearizable conditions of the system. First we introduce the algorithm of computing the generalized period constants , , which has been given in .
In the expressions (36), (37), (38), and (39), we have let , , and if or , we let .
The relations between , and , () are as follows.
The converse also holds true.
The algorithm of Theorem 3.1 and Theorem 3.2 gives a method to determine linearizable systems and find the necessary conditions for system (3). By applying the recursive formulas, the authors of  verify the linearizable conditions for Lotka-Volterra systems with the and resonance, which have been studied in [1, 4], respectively.
Now we consider the linearizability by investigating the two former cases of generalized center conditions, respectively, for , .
at the same time, for the case of , if equation (27) or (28) or (29) or (33) holds, there exist similar results. Therefore the above ones are judged as linearizable conditions at the origin.
Case (b): Consider the generalized center conditions in Case (iv) and (IV). When (iv) or (IV) holds, obviously we may consider only the case of , from , we can also have two constants r and s such that and hold. And furthermore from equation (25) or (30), we can get the only condition, , and then putting it into the recursive formulas in Theorem 3.1, from Theorem 3.2 we have the first 20 pairs of coefficients of the normal form for the saddle point, which are as follows:
However, some conditions are included in condition (22), (23) or (27), (28), excluding the conditions, so we can get the only following conditions:
when (), it is linearizable at the origin if and only if one of the conditions (22) (23), (24), (32), and (45) is satisfied;
when (), it is linearizable at the origin if and only if one of the conditions (27) (28), (29), (33), and (46) is satisfied.
The necessary conditions of Theorem 3.3 are obvious, the proof of sufficient conditions will be given in Section 4.
4 The proof of sufficient conditions
In the process of proving the sufficient conditions in Theorems 2.6 and 3.3, we will apply several well-known results.
Lemma 4.1 ()
for λ is always linearizable if .
From in condition (i) or (I), we can apply Lemma 4.1, so system (3) is linearizable and also integrable.
Lemma 4.2 (, Theorem D)
For the real system (3) with and , if and , hold, for arbitrary parameter λ, it is linearizable at the origin.
If the condition (ii) or (II), namely equation (23) or (28) holds, we can apply Lemma 4.2, thus it is linearizable at the origin of system (3).
Lemma 4.3 (, Theorem E)
If the condition (iii) or (III), namely equation (24) or (29) holds, we can apply (E2) and (E3) in Lemma 4.3 except for the case of , however, for the case of , we can also apply Lemma 4.1. Therefore, under the condition (iii) or (III), it is linearizable at the origin of system (3).
Furthermore, we have the following.
thus system (3) is linearizable.
Theorem 4.5 For system (3), if , , , , , , hold, it is linearizable at the origin.
such that and ; system (48) is therefore linearizable at the origin. □
Theorem 4.6 For system (3), if , , , , , , hold, it is linearizable at the origin.
which ensures that the node of system (52) is linearizable, and similar to the proof procedure in Theorem 4.5, one can see that system (51) is also linearizable at the origin. □
From Theorems 4.4, 4.5, 4.6 and (E4) in Lemma 4.3, one can see that if condition (45) or (46) holds, it is linearizable at the origin of system (3).
Furthermore, we have the following.
Theorem 4.7 For system (3), if , , , , , hold, it is linearizable at the origin.
which ensures that the node of system (54) is linearizable, and similar to the proof procedure in Theorem 4.5, one can see that system (53) is also linearizable at the origin. □
Theorem 4.8 For system (3), if , , , hold, it is linearizable at the origin.
which ensures that the node of system (56) is linearizable, and similar to the proof procedure in Theorem 4.5, one can see that system (55) is also linearizable at the origin. □
and moreover system (3) is linearizable.
Obviously, applying Lemma 1.2, we can prove the above theorem easily.
Theorem 4.10 For system (3), if , , , , , hold, it is linearizable at the origin.
such that , and we can apply Lemma 1.2, so the system is linearizable. □
From Theorems 4.7, 4.8, 4.9, and 4.10, one can see that if condition () or (), namely equation (32) or (33) holds, it is linearizable at the origin of system (3).
Thus, the proofs of Theorems 2.6 and 3.3 have been completed.
Remark 3 For system (3), in fact, if , namely the case of , the conclusions of Theorems 2.6 and 3.3 still hold. Thus we solve completely the integrability and linearizability of system (3).
This work was supported by the doctor’s scientific research foundation of Hezhou University (No. HZUBS201302), Guangxi Key Laboratory of Trusted Software (No. kx201336) and the National Natural Science Foundation of China (Nos. 11261013, 11371373).
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