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The generalized polynomial Moon–Rand system
Nonlinear Analysis: Real World Applications 39 (2018) 411–417
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
The generalized polynomial Moon–Rand system Jaume Ginéa, *, Claudia Vallsb a
Departament de Matemàtica, Inspires Research Centre, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain b Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal
article
abstract
info
Article history: Received 8 March 2016 Received in revised form 14 July 2017 Accepted 15 July 2017 Available online 10 August 2017
The Moon–Rand systems, developed to model control flexible space structures, are systems of differential equations in R3 of the form u˙ = v,
v˙ = −u − uw,
w˙ = −λw + f (u, v).
We give a partially positive answer to a recently conjecture for a special class of such systems, called the generalized polynomial Moon–Rand systems in the case when λ ∈ (0, ∞) and f is either a homogeneous cubic, quartic, quintic or sextic polynomial. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Moon–Rand system Center–focus problem Center manifolds First integrals
1. Introduction and statement of the main results The Moon–Rand systems are systems of differential equations on R3 with polynomial or rational right hand sides having an isolated singularity at the origin at which the linear part has one negative and one pair of purely imaginary eigenvalues for all choices of the parameters. These systems were developed to model control of flexible space structures. More specifically, Moon and Rand [1] (see also Exercise 5 of Section 5.5 of [2]) introduced the following system of differential equations u˙ = v,
v˙ = −u − uw,
w˙ = −λw + f (u, v),
(1)
which we shall call the Moon–Rand system and where f (u, v) = c20 u2 + c11 uv + c02 v 2 or f (u, v) = c11 uv/(1 + ηu2 ). Here λ, η, c20 , c11 and c02 are real numbers and λ > 0 and η > 0. The authors showed that in the polynomial case the origin is asymptotically stable for the flow restricted to the center manifold if 2c20 − 2c02 − λc11 < 0. Recently in [3] the authors have given a complete analysis of the flow restricted to a neighborhood of the origin in any center manifold for arbitrary values of η and strictly positive values of λ so that the singularity at the origin becomes isolated.
*
Corresponding author. E-mail addresses: [email protected] (J. Gin´ e), [email protected] (C. Valls).
http://dx.doi.org/10.1016/j.nonrwa.2017.07.006 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
412
J. Giné, C. Valls / Nonlinear Analysis: Real World Applications 39 (2018) 411–417
In [3] is given a natural generalization of system (1) in the polynomial case. More precisely, the authors in [3] consider that ∑ f (u, v) = cjk uj v k , j+k=n
and the resulting systems with such f are called the generalized polynomial Moon–Rand systems. In this paper, we study a conjecture posed in [3] related to these generalized polynomial Moon–Rand systems. More concretely, in [3] the authors conjectured that in the case of f being a homogeneous polynomial of degree n, called fn , then system (1) has a center on the local center manifold at the origin if and only if n fn (u, v) = c un + c un−1 v. λ They proved this for the case λ ∈ (0, ∞) and f = f2 , i.e., f being a homogeneous polynomial of degree two and also in the case λ = 1 and f = f3 , that is, f being a homogeneous polynomial of degree three. In [4] the author proves the conjecture for the case λ = 2 and f = f3 a homogeneous polynomial of degree three. Due to the enormous computation restrictions the authors in [3,4] state that they do not consider the general case of λ ∈ (0, ∞) and f = f3 . We recall that the case f = f3 is computationally much harder than the case f = f2 due to the presence of an extra parameter. Moreover, the parameter λ appears in the denominator of the focus quantities and growing λ brings computational difficulties. Finally, we want to mention that going from a fixed value of λ to considering it as a parameter provides again enormous computation difficulties. This is the main issue so that the authors either in [3] or in [4] could not consider the general case. In this paper using another smart approach we are able to partially solve the conjecture for the case λ ∈ (0, ∞) and f = f3 , f = f4 , f = f5 or f = f6 . This gives the intuition that the conjecture must be right. More concretely, we prove the following results. We recall that a monodromic system with a nondegenerate singular point has a center if and only if there exists a locally analytic first integral defined in a neighborhood of it. We define Property A: If system (1) with λ ∈ (0, ∞) and f a homogeneous polynomial of degree n has a center on the center manifold at the origin, then either fn = c1 un + nλ c1 un−1 v, or it has a local analytic first integral in the variables x and y and continuous in the parameters of the system in a neighborhood of zero in the parameter space formed by the free parameters once we have a fixed center. Theorem 1. System (1) with λ ∈ (0, ∞) and f a homogeneous polynomial of degree three satisfies Property A if and only if f = c1 u3 + λ3 c1 u2 v. Theorem 1 is proved in Section 3. Theorem 1 is also true in the case of a homogeneous polynomial of degrees four, five or six. The proof in these cases is analogous to the proof when we have a homogeneous polynomial of degree three and we will not do it to save the reader of repetitive developments. Thus, we state the results without proof. The approach used in this work is also used in [5,6]. Theorem 2. System (1) with λ ∈ (0, ∞) and f a homogeneous polynomial of degree n for n = 4, 5 and 6 satisfies Property A if and only if f = c1 un + nλ c1 un−1 v. Note that Theorems 1 and 2 prove the conjecture stated in [3] for the case of f being a cubic, quartic, quintic or sextic homogeneous polynomial and λ ∈ (0, ∞) with the restriction that the system satisfies Property A. We believe that the same method of proof works for the case λ ∈ (0, ∞) and f = fn a homogeneous polynomial of degree n ≥ 7 but it is computationally too hard and we cannot treat it now with our computational facilities. In the next section we provide general techniques and definitions necessary for the proof of our main theorem. We recall that in [3] the authors provide the Mathematica code for automatic computation of the coefficients of the lowest order terms in the expansion of the center manifold in a neighborhood of the origin and of the first focus quantities in an abbreviated form due to its length.
J. Giné, C. Valls / Nonlinear Analysis: Real World Applications 39 (2018) 411–417
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2. Preliminary results First, we briefly discuss the procedure to study the center problem on a center manifold for the threedimensional system (1). It is well-known that system (1) admits a local center manifold at the origin. We will denote by X the corresponding vector field X =v
∂ ∂ ∂ + (−u − uw) + (−λw + f (u, v)) . ∂u ∂v ∂w
A local analytic first integral of system (1) is a nonconstant differentiable function I that is analytic and constant on the trajectories of (1), that is, X I ≡ 0. A formal first integral for system (1) is a nonconstant formal power series I in the variables u, v, w and satisfying X I = 0. In [7] is given the following theorem for knowing whether a system has a center on a center manifold. Theorem 3. The following statements are equivalent for system (1). (i) The origin is a center for the vector field X restricted to the center manifold. (ii) Admits a local analytic first integral Φ. It can be always chosen of the form Φ = u2 + v 2 + · · · in a neighborhood of R3 (the dots indicate higher order terms) (iii) Admits a formal first integral. Hence the computation of center conditions in [3] is constructing a formal first integral for system (1). Here the computation of the center conditions is done in the following way: We begin by introducing an approximation of the center manifold w = h(u, v) up to certain order. Next we substitute this approximation in system (1) obtaining the two-dimensional differential system u˙ = v,
v˙ = −u − uh(u, v).
(2)
For system (2) we apply the classical method of passing to polar coordinates u = r cos θ, v = r sin θ, and propose a formal first integral of the form H = r2 /2 + · · · to find the obstructions to its existence, see for instance [8]. The following lemma is established in [9]. Here is adapted in the case we use real coordinates. Lemma 4. Let X and Y be two smooth differential systems with a singular point at the origin. If one of the equations has a center at the origin and X ∧ Y = µ(x2 + y 2 )j + O(2j + 1),
µ ∈ R,
(3)
then a necessary condition for the other system to have a center at the origin is µ = 0. Proof . Notice that the wedge product gives the scalar product between the vector field X and the orthogonal of the vector field Y. Therefore, if µ ̸= 0 the above equality implies that, in a neighborhood of the origin, the level curves of the solutions of the system having a center are without contact for the flow associated to the other equation, giving the impossibility of having a center for the second system. Hence µ = 0 is a necessary condition to have a center for the other differential system. □ In fact, in expression (3) for the term multiplied by µ it is only necessary to have a definite term of lower degree with respect to the rest of the terms.
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3. Proof of Theorem 1 We consider system (1) with f = f3 a homogeneous polynomial of degree 3 that we write as f = f3 = c30 u3 + c21 u2 v + c12 uv 2 + c03 v 3 . Hence we consider the system u˙ = v, v˙ = −u − uw, w˙ = −λw + c30 u3 + c21 u2 v + c12 uv 2 + c03 v 3 .
(4)
We will separate the proof of Theorem 1 into different lemmas. Lemma 5. System (4) with c12 = c03 = 0 has a center on the center manifold if and only if c21 = 3c30 /λ. Proof . First we construct a polynomial approximation of any center manifold at the origin up to certain degree. Thus we express the center manifold as w = h(u, v) = h20 u2 + h11 uv + h02 v 2 + · · · , where hij are parameters to be determined. The coefficients hij are found by equating coefficients in the expression that determines the center manifold, ∂h ∂h u˙ + v˙ = −λh + c30 u3 + c21 u2 v. ∂u ∂v Next we consider the two-dimensional differential system (2) where w is replaced by the computed approximation up to certain degree, see for instance [10]. We also consider the polynomial differential system u˙ = v,
v˙ = −u −
c u4 , 3
(5)
where c is an arbitrary parameter. System (5) has a center at the origin because it is Hamiltonian. Now we consider X and Y as the vector fields associated to systems (2) and (5) respectively. We compute the wedge product X ∧ (uY) and we obtain X ∧ (uY) = α1 u6 + α2 u5 v + α3 u4 v 2 + α4 u3 v 3 + α5 u2 v 4 + α6 uv 5 + α7 v 6 + · · · , where α1 = 0, and α2 is given by α2 =
9c − 9c21 − 21c30 λ + (10c − 3c21 )λ2 − 3c30 λ3 + cλ4 . 3(1 + λ2 )(9 + λ2 )
We can vanish α2 by choosing c of the form c=
3(3c21 + 7c30 λ + c21 λ2 + c30 λ3 ) . 9 + 10λ2 + λ4
The next coefficients are α6 = α7 = 0 and α3 =
(3c30 − c21 λ)(3 + λ2 ) , (1 + λ2 )(9 + λ2 )
α4 = −
2λ(3c30 − c21 λ) , (1 + λ2 )(9 + λ2 )
α5 =
λ(3c30 − c21 λ) . (1 + λ2 )(9 + λ2 )
Therefore we have X ∧ (uY) =
u2 v 2 (3c30 − c21 λ)(3u2 + 2v 2 − 2uvλ + u2 λ2 ) + ··· . (1 + λ2 )(9 + λ2 )
The factor 3u2 + 2v 2 − 2uvλ + u2 λ2 has only complex roots, hence is positive definite. Consequently X ∧ (uY) is semidefinite. The values u = 0 and v = 0 can be a problem depending on the higher order terms. However
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as all the tail is multiplied by u and v, due to the form of the vector fields X and Y, when u or v are equal to zero all the tail is zero. Hence applying Lemma 4 we obtain that a necessary condition to have a center is 3c30 − c21 λ = 0, which is also sufficient because if c21 = 3c30 /λ then the vector field X with c12 = c03 = 0 has the form u˙ = v, v˙ = −u − uw,
(6)
3c30 2 u v, w˙ = −λw + c30 u + λ 3
the center manifold is given by w = c30 u3 /λ, and system (6) reduced to the center manifold takes the form u˙ = v,
v˙ = −u −
c30 u4 , λ
(7)
a Hamiltonian system with a center at the origin whose first integral is H(u, v) =
u2 + v 2 c30 5 + u . 2 5λ
Note that this first integral depends analytically on the parameter c30 . □ In fact the approach used in the proof of Lemma 5 was used to characterize the centers of the classical Li´enard systems, see [11]. Lemma 6. System (4) with c21 = 3c30 /λ and c03 = 0 has a center on the center manifold if and only if c12 = 0. Proof . We also construct a polynomial approximation of any center manifold at the origin up to certain degree. In this case coefficients hij are found by equating coefficients in the expression that determines the center manifold, ∂h ∂h 3c30 2 u˙ + v˙ = −λh + c30 u3 + u v + c12 uv 2 . ∂u ∂v λ Next we consider the two-dimensional differential system (2), where w is replaced by the computed approximation up to certain degree, and the differential system (5). We compute the wedge product X ∧(uY) and we obtain X ∧ (uY) = β1 u6 + β2 u5 v + β3 u4 v 2 + β4 u3 v 3 + β5 u2 v 4 + β6 uv 5 + β7 v 6 + · · · , where β1 = 0, and β2 is given by β2 =
9c − 9c30 + (10c − 2c12 − 10c30 )λ2 + (c − c30 )λ4 . λ(1 + λ2 )(9 + λ2 )
We can vanish β2 choosing c of the form c=
9c30 + (2c12 + 10c30 )λ2 + c30 λ4 . 9 + 10λ2 + λ4
The next coefficients are β6 = β7 = 0 and β3 = −
2c12 λ2 , (1 + λ2 )(9 + λ2 )
β4 = −
c12 λ(3 + λ2 ) , (1 + λ2 )(9 + λ2 )
β5 =
c12 (3 + λ2 ) . (1 + λ2 )(9 + λ2 )
J. Giné, C. Valls / Nonlinear Analysis: Real World Applications 39 (2018) 411–417
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Therefore we have X ∧ (uY) = −
u2 v 2 c12 (−3v 2 + 3uvλ + 2u2 λ2 − v 2 λ2 + uvλ3 ) + ··· . (1 + λ2 )(9 + λ2 )
But now factor −3v 2 + 3uvλ + 2u2 λ2 − v 2 λ2 + uvλ3 has the real roots √ −3λ − λ3 ± λ 33 + 14λ2 + λ4 v. u= 4λ2 Therefore in this case we cannot apply Lemma 4. In this case we compute the first non-null center condition given by v8 = −
c12 (54c30 + (29c12 − 5c30 )λ2 − (16c12 + 60c30 )λ4 + (3c12 − c30 )λ6 ) . 128λ(1 + λ2 )2 (4 + λ2 )(9 + λ2 )
From the vanishing of this constant we obtain c12 = 0 or c12 =
c30 (1 + λ2 )(−54 + 59λ2 + λ4 ) , λ2 (29 − 16λ2 + 3λ4 )
where all the roots of 29 − 16λ2 + 3λ4 are complex. Substituting this value in the next non-null center conditions we obtain v14 =
v20 =
c430 (−54 + 59λ2 + λ4 )w14 (λ) , 10240λ7 (9 + λ2 )3 (16 + λ2 )(36 + λ2 )(29 − 16λ2 + 3λ4 )4
1 c630 (−54 + 59λ2 + λ4 )w20 (λ) , 235929600 λ11 (1 + λ2 )(4 + λ2 )(9 + λ2 )5 (16 + λ2 )2 (36 + λ2 )2 (29 − 16λ2 + 3λ4 )6
where w14 (λ) and w20 (λ) are polynomials of λ that we do not write here because their extension and have no common root. Hence the vanishing of these constants for each value of λ implies c30 = 0 which results in c12 = 0. So we always obtain that a necessary condition to have a center is c12 = 0, which is also sufficient because system (4) with c21 = 3c30 /λ, c03 = 0 and c12 = 0 has the associated system (2) Hamiltonian. □ Lemma 7. System (4) with c21 = 3c30 /λ and c12 = 0 has a center on the center manifold if and only if c03 = 0. Proof . The proof of this lemma is similar to the proof of Lemma 6. The first non-null center condition is v8 = −
3c03 w8 (λ) , 128(1 + λ2 )2 (4 + λ2 )(9 + λ2 )
where w8 (λ) = −128c30 − 108c03 λ − 145c30 λ2 + 38c03 λ3 − 18c30 λ4 + 2c03 λ5 − c30 λ6 . The vanish of this constant implies c03 = 0 or c30 =
2c03 λ(−54 + 19λ2 + λ4 ) , (1 + λ2 )(128 + 17λ2 + λ4 )
where the polynomial 128 + 17λ2 + λ4 has all its roots complex. The next non-null center condition is v14 =
10240(1 +
λ2 )4 (9
+
λ 2 )3
9c403 λw14 (λ) (16 + λ2 )(36 + λ2 )(128 + 17λ2 + λ4 )3
where w14 (λ) is a polynomial in the variable λ. We do not write it here because of its extension. However w14 (λ) has all its roots complex. Hence the vanishing of this last constant for each value of λ ∈ R implies c03 = 0. So we always obtain that a necessary condition to have a center is c03 = 0 which is also a sufficient condition. □
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We now assume that system (4) satisfies Property A. We also assume that system (4) has a center on the center manifold and f3 ̸= c1 u3 + λ3 c1 u2 v. By hypothesis, there exists a local analytic first integral in the variables x and y that is continuous in the parameters c30 , c21 , c12 , c03 in a neighborhood of zero in the parameter space. We write it as Hc30 ,c21 ,c12 ,c03 . We observe that, in particular, Hc30 ,c21 ,0,0 must be a local analytic in the variables (x, y) first integral of system (4) restricted to c12 = c03 = 0. It follows from Lemma 5 that c21 = 3c30 /λ. Now note that Hc30 ,3c30 /λ,c12 ,0 must be a local analytic in the variables (x, y) first integral of system (4) restricted to c21 = 3c30 /λ and c03 = 0. It follows from Lemma 6 that c12 = 0. Finally, we observe that Hc30 ,3c30 /λ,0,c03 must be a local analytic in the variables (x, y) first integral of system (4) restricted to c21 = 3c30 /λ and c12 = 0. It follows from Lemma 7 that c03 = 0. Hence f3 = c1 u3 + λ3 c1 u2 v which is not possible. This contradiction leads to the conclusion of the proof of Theorem 1. Acknowledgments The first author is partially supported by a MINECO/FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is supported by Portuguese national funds through FCT–Funda¸c˜ao para a Ciˆencia e a Tecnologia within the project PEst-OE/EEI/LA0009/2013 (CAMGSD). References
[1] F.C. Moon, R.H. Rand, Parametric stiffness control of flexible structures, in: Jet Propulsion Laboratory Publication 85–29, Vol. II, California Institute of Technology, 1985, pp. 329–342. [2] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, third ed., in: Applied Mathematical Sciences, vol. 112, SpringerVerlag, New York, 2004, p. xxii+631. [3] A. Mahdi, V.G. Romanovski, D.S. Shafer, Stability and periodic oscillations in the Moon-Rand systems, Nonlinear Anal. RWA 14 (1) (2013) 294–313. [4] C. Valls, Center problem in the center manifold for quadratic and cubic differential systems in R3 , Appl. Math. Comput. 251 (2015) 180–191. [5] J. Gin´ e, J. Llibre, C. Valls, Centers for the Kukles homogeneous systems with odd degree, Bull. Lond. Math. Soc. 47 (2) (2015) 315–324. [6] J. Gin´ e, J. Llibre, C. Valls, Centers for the Kukles homogeneous systems with even degree. J. Appl. Anal. Comput. (in press), ISSN: 2156-907X. [7] V. Edneral, A. Mahdi, V.G. Romanovski, D.S. Shafer, The center problem on a center manifold in R3 , Nonlinear Anal. A 75 (4) (2012) 2614–2622. [8] J. Gin´ e, On some open problems in planar differential systems and Hilbert’s 16th problem, Chaos Solitons Fractals 31 (5) (2007) 1118–1134. [9] A. Gasull, J. Gin´ e, J. Torregrosa, Center problem for systems with two monomial nonlinearities, Commun. Pure Appl. Anal. 15 (2) (2016) 577–598. [10] J. Gin´ e, C. Valls, Center problem in the center manifold for quadratic differential systems in R3 , J. Symbolic Comput. 73 (2016) 250–267. [11] C.J. Christopher, An algebraic approach to the classification of centers in polynomial Li´ enard systems, J. Math. Anal. Appl. 229 (1) (1999) 319–329.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
The generalized polynomial Moon–Rand system Jaume Ginéa, *, Claudia Vallsb a
Departament de Matemàtica, Inspires Research Centre, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain b Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal
article
abstract
info
Article history: Received 8 March 2016 Received in revised form 14 July 2017 Accepted 15 July 2017 Available online 10 August 2017
The Moon–Rand systems, developed to model control flexible space structures, are systems of differential equations in R3 of the form u˙ = v,
v˙ = −u − uw,
w˙ = −λw + f (u, v).
We give a partially positive answer to a recently conjecture for a special class of such systems, called the generalized polynomial Moon–Rand systems in the case when λ ∈ (0, ∞) and f is either a homogeneous cubic, quartic, quintic or sextic polynomial. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Moon–Rand system Center–focus problem Center manifolds First integrals
1. Introduction and statement of the main results The Moon–Rand systems are systems of differential equations on R3 with polynomial or rational right hand sides having an isolated singularity at the origin at which the linear part has one negative and one pair of purely imaginary eigenvalues for all choices of the parameters. These systems were developed to model control of flexible space structures. More specifically, Moon and Rand [1] (see also Exercise 5 of Section 5.5 of [2]) introduced the following system of differential equations u˙ = v,
v˙ = −u − uw,
w˙ = −λw + f (u, v),
(1)
which we shall call the Moon–Rand system and where f (u, v) = c20 u2 + c11 uv + c02 v 2 or f (u, v) = c11 uv/(1 + ηu2 ). Here λ, η, c20 , c11 and c02 are real numbers and λ > 0 and η > 0. The authors showed that in the polynomial case the origin is asymptotically stable for the flow restricted to the center manifold if 2c20 − 2c02 − λc11 < 0. Recently in [3] the authors have given a complete analysis of the flow restricted to a neighborhood of the origin in any center manifold for arbitrary values of η and strictly positive values of λ so that the singularity at the origin becomes isolated.
*
Corresponding author. E-mail addresses: [email protected] (J. Gin´ e), [email protected] (C. Valls).
http://dx.doi.org/10.1016/j.nonrwa.2017.07.006 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
412
J. Giné, C. Valls / Nonlinear Analysis: Real World Applications 39 (2018) 411–417
In [3] is given a natural generalization of system (1) in the polynomial case. More precisely, the authors in [3] consider that ∑ f (u, v) = cjk uj v k , j+k=n
and the resulting systems with such f are called the generalized polynomial Moon–Rand systems. In this paper, we study a conjecture posed in [3] related to these generalized polynomial Moon–Rand systems. More concretely, in [3] the authors conjectured that in the case of f being a homogeneous polynomial of degree n, called fn , then system (1) has a center on the local center manifold at the origin if and only if n fn (u, v) = c un + c un−1 v. λ They proved this for the case λ ∈ (0, ∞) and f = f2 , i.e., f being a homogeneous polynomial of degree two and also in the case λ = 1 and f = f3 , that is, f being a homogeneous polynomial of degree three. In [4] the author proves the conjecture for the case λ = 2 and f = f3 a homogeneous polynomial of degree three. Due to the enormous computation restrictions the authors in [3,4] state that they do not consider the general case of λ ∈ (0, ∞) and f = f3 . We recall that the case f = f3 is computationally much harder than the case f = f2 due to the presence of an extra parameter. Moreover, the parameter λ appears in the denominator of the focus quantities and growing λ brings computational difficulties. Finally, we want to mention that going from a fixed value of λ to considering it as a parameter provides again enormous computation difficulties. This is the main issue so that the authors either in [3] or in [4] could not consider the general case. In this paper using another smart approach we are able to partially solve the conjecture for the case λ ∈ (0, ∞) and f = f3 , f = f4 , f = f5 or f = f6 . This gives the intuition that the conjecture must be right. More concretely, we prove the following results. We recall that a monodromic system with a nondegenerate singular point has a center if and only if there exists a locally analytic first integral defined in a neighborhood of it. We define Property A: If system (1) with λ ∈ (0, ∞) and f a homogeneous polynomial of degree n has a center on the center manifold at the origin, then either fn = c1 un + nλ c1 un−1 v, or it has a local analytic first integral in the variables x and y and continuous in the parameters of the system in a neighborhood of zero in the parameter space formed by the free parameters once we have a fixed center. Theorem 1. System (1) with λ ∈ (0, ∞) and f a homogeneous polynomial of degree three satisfies Property A if and only if f = c1 u3 + λ3 c1 u2 v. Theorem 1 is proved in Section 3. Theorem 1 is also true in the case of a homogeneous polynomial of degrees four, five or six. The proof in these cases is analogous to the proof when we have a homogeneous polynomial of degree three and we will not do it to save the reader of repetitive developments. Thus, we state the results without proof. The approach used in this work is also used in [5,6]. Theorem 2. System (1) with λ ∈ (0, ∞) and f a homogeneous polynomial of degree n for n = 4, 5 and 6 satisfies Property A if and only if f = c1 un + nλ c1 un−1 v. Note that Theorems 1 and 2 prove the conjecture stated in [3] for the case of f being a cubic, quartic, quintic or sextic homogeneous polynomial and λ ∈ (0, ∞) with the restriction that the system satisfies Property A. We believe that the same method of proof works for the case λ ∈ (0, ∞) and f = fn a homogeneous polynomial of degree n ≥ 7 but it is computationally too hard and we cannot treat it now with our computational facilities. In the next section we provide general techniques and definitions necessary for the proof of our main theorem. We recall that in [3] the authors provide the Mathematica code for automatic computation of the coefficients of the lowest order terms in the expansion of the center manifold in a neighborhood of the origin and of the first focus quantities in an abbreviated form due to its length.
J. Giné, C. Valls / Nonlinear Analysis: Real World Applications 39 (2018) 411–417
413
2. Preliminary results First, we briefly discuss the procedure to study the center problem on a center manifold for the threedimensional system (1). It is well-known that system (1) admits a local center manifold at the origin. We will denote by X the corresponding vector field X =v
∂ ∂ ∂ + (−u − uw) + (−λw + f (u, v)) . ∂u ∂v ∂w
A local analytic first integral of system (1) is a nonconstant differentiable function I that is analytic and constant on the trajectories of (1), that is, X I ≡ 0. A formal first integral for system (1) is a nonconstant formal power series I in the variables u, v, w and satisfying X I = 0. In [7] is given the following theorem for knowing whether a system has a center on a center manifold. Theorem 3. The following statements are equivalent for system (1). (i) The origin is a center for the vector field X restricted to the center manifold. (ii) Admits a local analytic first integral Φ. It can be always chosen of the form Φ = u2 + v 2 + · · · in a neighborhood of R3 (the dots indicate higher order terms) (iii) Admits a formal first integral. Hence the computation of center conditions in [3] is constructing a formal first integral for system (1). Here the computation of the center conditions is done in the following way: We begin by introducing an approximation of the center manifold w = h(u, v) up to certain order. Next we substitute this approximation in system (1) obtaining the two-dimensional differential system u˙ = v,
v˙ = −u − uh(u, v).
(2)
For system (2) we apply the classical method of passing to polar coordinates u = r cos θ, v = r sin θ, and propose a formal first integral of the form H = r2 /2 + · · · to find the obstructions to its existence, see for instance [8]. The following lemma is established in [9]. Here is adapted in the case we use real coordinates. Lemma 4. Let X and Y be two smooth differential systems with a singular point at the origin. If one of the equations has a center at the origin and X ∧ Y = µ(x2 + y 2 )j + O(2j + 1),
µ ∈ R,
(3)
then a necessary condition for the other system to have a center at the origin is µ = 0. Proof . Notice that the wedge product gives the scalar product between the vector field X and the orthogonal of the vector field Y. Therefore, if µ ̸= 0 the above equality implies that, in a neighborhood of the origin, the level curves of the solutions of the system having a center are without contact for the flow associated to the other equation, giving the impossibility of having a center for the second system. Hence µ = 0 is a necessary condition to have a center for the other differential system. □ In fact, in expression (3) for the term multiplied by µ it is only necessary to have a definite term of lower degree with respect to the rest of the terms.
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3. Proof of Theorem 1 We consider system (1) with f = f3 a homogeneous polynomial of degree 3 that we write as f = f3 = c30 u3 + c21 u2 v + c12 uv 2 + c03 v 3 . Hence we consider the system u˙ = v, v˙ = −u − uw, w˙ = −λw + c30 u3 + c21 u2 v + c12 uv 2 + c03 v 3 .
(4)
We will separate the proof of Theorem 1 into different lemmas. Lemma 5. System (4) with c12 = c03 = 0 has a center on the center manifold if and only if c21 = 3c30 /λ. Proof . First we construct a polynomial approximation of any center manifold at the origin up to certain degree. Thus we express the center manifold as w = h(u, v) = h20 u2 + h11 uv + h02 v 2 + · · · , where hij are parameters to be determined. The coefficients hij are found by equating coefficients in the expression that determines the center manifold, ∂h ∂h u˙ + v˙ = −λh + c30 u3 + c21 u2 v. ∂u ∂v Next we consider the two-dimensional differential system (2) where w is replaced by the computed approximation up to certain degree, see for instance [10]. We also consider the polynomial differential system u˙ = v,
v˙ = −u −
c u4 , 3
(5)
where c is an arbitrary parameter. System (5) has a center at the origin because it is Hamiltonian. Now we consider X and Y as the vector fields associated to systems (2) and (5) respectively. We compute the wedge product X ∧ (uY) and we obtain X ∧ (uY) = α1 u6 + α2 u5 v + α3 u4 v 2 + α4 u3 v 3 + α5 u2 v 4 + α6 uv 5 + α7 v 6 + · · · , where α1 = 0, and α2 is given by α2 =
9c − 9c21 − 21c30 λ + (10c − 3c21 )λ2 − 3c30 λ3 + cλ4 . 3(1 + λ2 )(9 + λ2 )
We can vanish α2 by choosing c of the form c=
3(3c21 + 7c30 λ + c21 λ2 + c30 λ3 ) . 9 + 10λ2 + λ4
The next coefficients are α6 = α7 = 0 and α3 =
(3c30 − c21 λ)(3 + λ2 ) , (1 + λ2 )(9 + λ2 )
α4 = −
2λ(3c30 − c21 λ) , (1 + λ2 )(9 + λ2 )
α5 =
λ(3c30 − c21 λ) . (1 + λ2 )(9 + λ2 )
Therefore we have X ∧ (uY) =
u2 v 2 (3c30 − c21 λ)(3u2 + 2v 2 − 2uvλ + u2 λ2 ) + ··· . (1 + λ2 )(9 + λ2 )
The factor 3u2 + 2v 2 − 2uvλ + u2 λ2 has only complex roots, hence is positive definite. Consequently X ∧ (uY) is semidefinite. The values u = 0 and v = 0 can be a problem depending on the higher order terms. However
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as all the tail is multiplied by u and v, due to the form of the vector fields X and Y, when u or v are equal to zero all the tail is zero. Hence applying Lemma 4 we obtain that a necessary condition to have a center is 3c30 − c21 λ = 0, which is also sufficient because if c21 = 3c30 /λ then the vector field X with c12 = c03 = 0 has the form u˙ = v, v˙ = −u − uw,
(6)
3c30 2 u v, w˙ = −λw + c30 u + λ 3
the center manifold is given by w = c30 u3 /λ, and system (6) reduced to the center manifold takes the form u˙ = v,
v˙ = −u −
c30 u4 , λ
(7)
a Hamiltonian system with a center at the origin whose first integral is H(u, v) =
u2 + v 2 c30 5 + u . 2 5λ
Note that this first integral depends analytically on the parameter c30 . □ In fact the approach used in the proof of Lemma 5 was used to characterize the centers of the classical Li´enard systems, see [11]. Lemma 6. System (4) with c21 = 3c30 /λ and c03 = 0 has a center on the center manifold if and only if c12 = 0. Proof . We also construct a polynomial approximation of any center manifold at the origin up to certain degree. In this case coefficients hij are found by equating coefficients in the expression that determines the center manifold, ∂h ∂h 3c30 2 u˙ + v˙ = −λh + c30 u3 + u v + c12 uv 2 . ∂u ∂v λ Next we consider the two-dimensional differential system (2), where w is replaced by the computed approximation up to certain degree, and the differential system (5). We compute the wedge product X ∧(uY) and we obtain X ∧ (uY) = β1 u6 + β2 u5 v + β3 u4 v 2 + β4 u3 v 3 + β5 u2 v 4 + β6 uv 5 + β7 v 6 + · · · , where β1 = 0, and β2 is given by β2 =
9c − 9c30 + (10c − 2c12 − 10c30 )λ2 + (c − c30 )λ4 . λ(1 + λ2 )(9 + λ2 )
We can vanish β2 choosing c of the form c=
9c30 + (2c12 + 10c30 )λ2 + c30 λ4 . 9 + 10λ2 + λ4
The next coefficients are β6 = β7 = 0 and β3 = −
2c12 λ2 , (1 + λ2 )(9 + λ2 )
β4 = −
c12 λ(3 + λ2 ) , (1 + λ2 )(9 + λ2 )
β5 =
c12 (3 + λ2 ) . (1 + λ2 )(9 + λ2 )
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Therefore we have X ∧ (uY) = −
u2 v 2 c12 (−3v 2 + 3uvλ + 2u2 λ2 − v 2 λ2 + uvλ3 ) + ··· . (1 + λ2 )(9 + λ2 )
But now factor −3v 2 + 3uvλ + 2u2 λ2 − v 2 λ2 + uvλ3 has the real roots √ −3λ − λ3 ± λ 33 + 14λ2 + λ4 v. u= 4λ2 Therefore in this case we cannot apply Lemma 4. In this case we compute the first non-null center condition given by v8 = −
c12 (54c30 + (29c12 − 5c30 )λ2 − (16c12 + 60c30 )λ4 + (3c12 − c30 )λ6 ) . 128λ(1 + λ2 )2 (4 + λ2 )(9 + λ2 )
From the vanishing of this constant we obtain c12 = 0 or c12 =
c30 (1 + λ2 )(−54 + 59λ2 + λ4 ) , λ2 (29 − 16λ2 + 3λ4 )
where all the roots of 29 − 16λ2 + 3λ4 are complex. Substituting this value in the next non-null center conditions we obtain v14 =
v20 =
c430 (−54 + 59λ2 + λ4 )w14 (λ) , 10240λ7 (9 + λ2 )3 (16 + λ2 )(36 + λ2 )(29 − 16λ2 + 3λ4 )4
1 c630 (−54 + 59λ2 + λ4 )w20 (λ) , 235929600 λ11 (1 + λ2 )(4 + λ2 )(9 + λ2 )5 (16 + λ2 )2 (36 + λ2 )2 (29 − 16λ2 + 3λ4 )6
where w14 (λ) and w20 (λ) are polynomials of λ that we do not write here because their extension and have no common root. Hence the vanishing of these constants for each value of λ implies c30 = 0 which results in c12 = 0. So we always obtain that a necessary condition to have a center is c12 = 0, which is also sufficient because system (4) with c21 = 3c30 /λ, c03 = 0 and c12 = 0 has the associated system (2) Hamiltonian. □ Lemma 7. System (4) with c21 = 3c30 /λ and c12 = 0 has a center on the center manifold if and only if c03 = 0. Proof . The proof of this lemma is similar to the proof of Lemma 6. The first non-null center condition is v8 = −
3c03 w8 (λ) , 128(1 + λ2 )2 (4 + λ2 )(9 + λ2 )
where w8 (λ) = −128c30 − 108c03 λ − 145c30 λ2 + 38c03 λ3 − 18c30 λ4 + 2c03 λ5 − c30 λ6 . The vanish of this constant implies c03 = 0 or c30 =
2c03 λ(−54 + 19λ2 + λ4 ) , (1 + λ2 )(128 + 17λ2 + λ4 )
where the polynomial 128 + 17λ2 + λ4 has all its roots complex. The next non-null center condition is v14 =
10240(1 +
λ2 )4 (9
+
λ 2 )3
9c403 λw14 (λ) (16 + λ2 )(36 + λ2 )(128 + 17λ2 + λ4 )3
where w14 (λ) is a polynomial in the variable λ. We do not write it here because of its extension. However w14 (λ) has all its roots complex. Hence the vanishing of this last constant for each value of λ ∈ R implies c03 = 0. So we always obtain that a necessary condition to have a center is c03 = 0 which is also a sufficient condition. □
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We now assume that system (4) satisfies Property A. We also assume that system (4) has a center on the center manifold and f3 ̸= c1 u3 + λ3 c1 u2 v. By hypothesis, there exists a local analytic first integral in the variables x and y that is continuous in the parameters c30 , c21 , c12 , c03 in a neighborhood of zero in the parameter space. We write it as Hc30 ,c21 ,c12 ,c03 . We observe that, in particular, Hc30 ,c21 ,0,0 must be a local analytic in the variables (x, y) first integral of system (4) restricted to c12 = c03 = 0. It follows from Lemma 5 that c21 = 3c30 /λ. Now note that Hc30 ,3c30 /λ,c12 ,0 must be a local analytic in the variables (x, y) first integral of system (4) restricted to c21 = 3c30 /λ and c03 = 0. It follows from Lemma 6 that c12 = 0. Finally, we observe that Hc30 ,3c30 /λ,0,c03 must be a local analytic in the variables (x, y) first integral of system (4) restricted to c21 = 3c30 /λ and c12 = 0. It follows from Lemma 7 that c03 = 0. Hence f3 = c1 u3 + λ3 c1 u2 v which is not possible. This contradiction leads to the conclusion of the proof of Theorem 1. Acknowledgments The first author is partially supported by a MINECO/FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is supported by Portuguese national funds through FCT–Funda¸c˜ao para a Ciˆencia e a Tecnologia within the project PEst-OE/EEI/LA0009/2013 (CAMGSD). References
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