Integrability analysis of a simple model for describing convection of a rotating fluid
aa r X i v : . [ m a t h . D S ] F e b Vol. XX (XXXX) No. X
INTEGRABILITY ANALYSIS OF A SIMPLE MODEL FORDESCRIBING CONVECTION OF A ROTATING FLUID
Jia JiaoCollege of Science, Dalian Minzu University, Dalian 116600, P R ChinaE-mail: [email protected] ZhouCollege of Science, Dalian Minzu University, Dalian 116600, P R ChinaE-mail: [email protected] YangSchool of Mathematics, Jilin University, Changchun 130012, P R ChinaE-mail: [email protected] (Received 2018)
We study the Darboux integrability of a simple system of three ordinary dif-ferential equations called the Glukhovsky-Dolzhansky system, which describes athree-mode model of rotating fluid convection inside the ellipsoid. (1) Our resultsshow that it has no polynomial, rational, or Darboux first integrals for any value ofparameters in the physical sense, that is, positive parameters. (2) We also providesome integrable cases of this model when parameters are allowed to be non-positive.(3) We finally give some links between the Glukhovsky-Dolzhansky system and othersimilar systems in R , which admits rotational symmetry and has three nonlinearcross terms. Keywords:
Glukhovsky-Dolzhansky system; Darboux integrability; Darboux polyno-mials; Exponential factor
1. Introduction
Consider the convection of viscous incompressible fluid motion inside the ellipsoid( x a ) + ( x a ) + ( x a ) = 1 , a > a > a > , under the condition of stationary inhomogeneous external heating. Glukhovsky andDolzhansky [1] assumed that the ellipsoid rotates with the constant velocity Ω alongthe axis a , and the axis a has a constant angle α with the gravity vector g . They alsoassumed that the temperature difference is generated along the axis a and its gradient S. Yang, corresponding author [1]
Integrability analysis of a simple model for describing convection of a rotating fluid] q . Denote by λ, µ, β the coefficients of viscosity, heat conduction,and volume expansion, respectively. Then they proposed the following system of ordinarydifferential equations ˙ x = Ayz + Cz − σx := P ( x, y, z ) , ˙ y = − xz + Ra − y := Q ( x, y, z ) , ˙ z = − z + xy := R ( x, y, z ) , (1)which can be interpreted as one of the models of ocean flows. Here σ = λµ , T α = Ω λ , R a = gβa q a a λµ ,A = a − a a + a cos αT − α , C = a − a a + a cos αT − α ,x ( t ) = µ − ω ( t ) , y ( t ) = gβa a a λµ q ( t ) , z ( t ) = gβa a a λµ q ( t ) ,ω ( t ) is the projection of the vector of angular velocity on the axis a , q ( t ) and q ( t )are the projections of temperature gradients on the axes a and a . The parameters σ , T α and R a are the Prandtl, Taylor and Rayleigh numbers, respectively. Clearly, theparameters A, C, σ and Ra of system (1) are positive.In spite of its simple form, the Glukhovsky-Dolzhansky system (1) admits rich dy-namics and has been intensively investigated, see [2, 3, 4, 5] and the references therein.However, this system has never been studied from the integrability point of view. Themain purpose of this work is to cover this gap and to investigate the integrability ofsystem (1). The investigation of integrability for systems of differential equations hasbeen one of significant topics not only in mathematics but also in physics. Generally,a system of differential equations is integrable if it possesses a sufficient number of firstintegrals (and/or other tensor invariants) such that we can solve this system explicitly.Hence we could obtain its global information and understand its topological structure[8, 9]. Furthermore, non-integrability of the system also seems necessary for better un-derstanding of the complex phenomenon [6, 7]. However, to study the integrability of agiven system is not an easy task, since there is no any effective approach to determinethe existence or nonexistence of first integrals.In this paper, we aim to study the integrability of system (1) in the framework ofDarboux integrability theory. Darboux integrability theory plays an important role inthe integrability of the polynomial differential systems [10, 11, 12, 13, 14], which helpsus find first integrals by knowing a sufficient number of algebraic invariant surfaces (theDarboux polynomials) and of the exponential factors, see [15, 16, 17, 18, 19] for instance.Moreover, it can also help us make a more precise analysis of the global dynamics of theconsidered system topologically [20, 8]. In addition, let us mention that the Darbouxintegrability of the Rabinovich system, 3D forced-damped system and D2 vector field,which are similar but different from the Glukhovsky-Dolzhansky system (1), have beenstudied in [28, 29, 30, 31]. These systems admit a common structure: the original is Integrability analysis of a simple model for describing convection of a rotating fluid] (a) (b) Fig. 1: (a) chaotic attractor for the system (1) with parameter values (
A, C, σ, Ra ) =(0 . , , , x, y ). (a) (b) Fig. 2: Projections of the chaotic attractor into the plane ( x, z ) (a) and into the plane( y, z ) (b).
Integrability analysis of a simple model for describing convection of a rotating fluid] { yz, xz, xy } . In the appendix, we provide some linear rescaling of timeand coordinate to show the Glukhovsky-Dolzhansky system (1) can be transformed intothe Rabinovich system or 3D forced-damped system or D2 vector field only when theparameters satisfies some conditions: C = − AR a or σ = 1 or Ra = C = 0 respectively.The main result of this paper is as following. Theorem 1.
The following statements hold for the Glukhovsky-Dolzhansky system(1).(a.) It has no polynomial first integrals.(b.) It has no Darboux polynomials with non-zero cofactors.(c.) It has no exponential factors.(d.) It admits no Darboux first integrals.
Remark 1.
The Glukhovsky-Dolzhansky system admits complex dynamics for alarger range of its parameters. For example, it has a strange attractor with (
A, C, σ, Ra ) =(0 . , , , • Ra = 0 and σ = 1, system (1) has a Darboux polynomial f = y + z with aconstant cofactor k = − • C = 0 and σ = 1, system (1) has a Darboux polynomial f = x − Az with aconstant cofactor k = − • Ra = C = 0 and σ = 1, system (1) has a rational first integral Φ = ( y + z ) / ( x − Az ).But we do not have a clear physical understanding of the above integrable results for theGlukhovsky-Dolzhansky system (1). Remark 2.
Another tool to study the non-integrability of non-Hamiltonian systemsis the differential Galois theory [21, 22, 23]. Observing system (1) has a straight linesolution ( x ( t ) , y ( t ) , z ( t )) = (0 , Ra − e − t , Integrability analysis of a simple model for describing convection of a rotating fluid]
2. Preliminary results
Let R [ x, y, z ] be the ring of the real polynomials in the variables x , y and z . We saythat f ( x, y, z ) ∈ R [ x, y, z ] is a Darboux polynomial of system (1) if it satisfies ∂f∂x P + ∂f∂y Q + ∂f∂z R = Kf, (2)for some polynomial K , called the cofactor of f ( x, y, z ). If f ( x, y, z ) is a Darboux polyno-mial, then the surface f ( x, y, z ) = 0 is an invariant manifold of system (1). Particularly,if K = 0, f ( x, y, z ) satisfies the following equation ∂f∂x P + ∂f∂y Q + ∂f∂z R = 0 , (3)then polynomial f ( x, y, z ) is called a polynomial first integral of system (1) .Let g, h ∈ R [ x, y, z ] be coprime. We say that a nonconstant function E = exp( g/h )is an exponential factor of system (1) if E satisfies ∂E∂x P + ∂E∂y Q + ∂E∂z R = LE, for some polynomial L ∈ R [ x, y, z ] with the degree at most one, called the cofactor of E .A first integral G of system (1) is called Darboux type if it is a first integral of theform G = f λ · · · f λ p p E µ · · · E µ q q , where f , · · · , f p are Darboux polynomials, E , · · · , E q are exponential factors and λ i , µ j ∈ R , for i = 1 , · · · , p and j = 1 , · · · , q .To prove Theorem 1, we need the following results. Proposition 1.
Suppose that a polynomial vector field X defined in R n of degree m admits p Darboux polynomials f i with cofactor K i for i = 1 , · · · , p, and q exponentialfactors E j = exp( g j /h j ) with cofactors L j for j = 1 , · · · , q . If there exist λ i , µ j ∈ R notall zero such that p X i =1 λ i K i + q X j =1 µ j L j = 0 , then the following real (multivalued) function of Darboux type f λ · · · f λ p p E µ · · · E µ q p , substituting f λ i i by | f i | λ i if λ i ∈ R , is a first integral of the vector field X .The proof can be seen in [10]. Proposition 2.
The following statements hold.(a) If e g/h is an exponential factor for the polynomial differential system (1) and h is nota constant polynomial, then h is a Darboux polynomial(b) Eventually e g can be an exponential factor, coming from the multiplicity of theinfinite invariant plane. Integrability analysis of a simple model for describing convection of a rotating fluid]
Proposition 3.
Let f be a polynomial and f = Q sj =1 f α j j be its decomposition intoirreducible factors in R [ x, y, z ]. Then f is a Darboux polynomial of system (1) if andonly if all the f j are Darboux polynomials of system (1). Moreover, if K and K j are thecofactors of f and f j , then K = P sj =1 α j K j .The proof of Proposition 3 can be found in [12].
3. Proof of Theorem 1
We separate the proof of Theorem 1 into the following different propositions.
Propositon 4.
System (1) has no polynomial first integrals.
Proof:
Suppose f ( x, y, z ) = n X i =0 f i ( x, y, z ) (4)is a polynomial first integral of system (1), where f i are the homogeneous polynomials ofdegree i and f n = 0. Firstly, substituting (4) into (3) and identifying the homogeneouscomponents of degree n + 1, we get Ayz ∂f n ∂x − xz ∂f n ∂y + xy ∂f n ∂z = 0 . (5)The characteristic equations associated with (5) are dxdy = Ayz − xz , dzdy = xy − xz . Its general solution are x + Ay = c , y + z = c , where c and c are arbitrary constants. We make the change of variables u = x + Ay , w = y, v = y + z . Correspondingly, the inverse transformation is x = ± p u − Aw , y = w, z = ± p v − w . (6)Without loss of generality, we set x = p u − Aw , y = w, z = − p v − w , (7) Integrability analysis of a simple model for describing convection of a rotating fluid] d ¯ f n dw = 0 , (8)where ¯ f n ( u, v, w ) = f n ( x, y, z ). In the following, unless otherwise specified, we alwaysdenote the function R ( x, y, z ) by ¯ R ( u, v, w ). Hence we obtain f n ( x, y, z ) = f m ( x, y, z ) = m X i =0 a mi ( x + Ay ) m − i ( y + z ) i , a mi ∈ R , where n = 2 m must be an even number. Secondly, substituting (4) into (3) and identi-fying the homogeneous components of degree n yields Ayz ∂f n − ∂x − xz ∂f n − ∂y + xy ∂f n − ∂z = ( σx − Cz ) ∂f n ∂x + y ∂f n ∂y + z ∂f n ∂z (9)= m X i =0 a mi σ ( m − i )( x + Ay ) m − i − ( y + z ) i x − m X i =0 a mi C ( m − i )( x + Ay ) m − i − ( y + z ) i xz + m X i =0 a mi A ( m − i )( x + Ay ) m − i − ( y + z ) i y + m X i =0 a mi i ( x + Ay ) m − i ( y + z ) i − y + m X i =0 a mi i ( x + Ay ) m − i ( y + z ) i − z . Using the transformations (7) again, the above equation becomes p u − Aw p v − w d ¯ f m − dw = m X i =0 a mi σ ( m − i ) u m − i − v i ( u − Aw )+ m X i =0 a mi C ( m − i ) u m − i − v i p u − Aw p v − w + m X i =0 a mi A ( m − i ) u m − i − v i w + m X i =0 a mi iu m − i v i − w + m X i =0 a mi iu m − i v i − ( v − w ) . Integrability analysis of a simple model for describing convection of a rotating fluid] f m − = m X i =0 a mi σ ( m − i ) u m − i − v i Z √ u − Aw √ v − w dw + m − X i =0 a mi A ( m − i ) + a mi +1 ( i + 1)] u m − i − v i Z w dw √ u − Aw √ v − w + m X i =0 a mi iv i − Z √ v − w √ u − Aw dw + m X i =0 a mi C ( m − i ) u m − i − v i w + A n − ( u, v ) , where A n − ( u, v ) is an arbitrary smooth function in u and v . It is easy to check Z w dw √ u − Aw √ v − w = − Z √ u − Aw √ v − w dw + u Z dw √ u − Aw √ v − w . (10)Since Z √ u − Aw √ v − w dw, Z dw √ u − Aw √ v − w dw are elliptic integrals of the second and first kinds, respectively, in order that f m − ( x, y, z ) =¯ f m − ( u, v, w ) is a homogeneous polynomial of degree 2 m −
1, we must have a mi σ ( m − i ) = 0 , i = 0 , , · · · , ma mi C ( m − i ) = 0 , i = 0 , , · · · , ma mi i = 0 , i = 0 , , · · · , ma mi A ( m − i ) + a mi +1 ( i + 1) = 0 , i = 0 , , · · · , m − . (11)By (11) and A, C, σ being positive, we obtain a mi = 0 , i = 0 , · · · , m . This leads to acontradiction. So system (1) has no polynomial first integrals. Proposition 5.
System (1) has no Darboux polynomial with nonzero cofactors.
Proof:
Suppose f ( x, y, z ) = n X i =0 f i ( x, y, z ) (12)is a Darboux polynomial of the system (1) with a non-cofactor K ( x, y, z ), where f i is ahomogeneous polynomial of degree i for i = 0 , , · · · , n . Comparing the degree on bothsides of (2) yields deg K ≤
1. Without loss of generality, we can assume that the cofactoris of the form K ( x, y, z ) = k x + k y + k z + k , k i ∈ R , i = 0 , , , . (13) Integrability analysis of a simple model for describing convection of a rotating fluid]
Ayz ∂f n ∂x − xz ∂f n ∂y + xy ∂f n ∂z = ( k x + k y + k z ) f n , (14) Ayz ∂f n − ∂x − xz ∂f n − ∂y + xy ∂f n − ∂z =( k x + k y + k z ) f n − + ( σx − Cz ) ∂f n ∂x + y ∂f n ∂y + z ∂f n ∂z + k f n ,Ayz ∂f i ∂x − xz ∂f i ∂y + xy ∂f i ∂z =( k x + k y + k z ) f i + ( σx − Cz ) ∂f i +1 ∂x (15)+ y ∂f i +1 ∂y + z ∂f i +1 ∂z − Ra ∂f i +2 ∂y + k f i +1 ,i = n − , . . . , . We claim that the cofactor is a constant, i.e., k = k = k = 0. Indeed, under thechange of (6), we can transform (14) into an ordinary differential equation if we fixe u and v − ( ± p u − Aw )( ± p v − w ) d ¯ f n dw = [ k ( ± p u − Aw ) + k w + k ( ± p v − w )] ¯ f n , where ¯ f n ( u, v, w ) = f n ( x, y, z ). In the following proof, we only consider the case of xz <
0. For the case of xz >
0, the proof is similar. Solving the last equation we findthat for xz < f n = ¯ A ( u, v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A p ( u − Aw )( v − w ) + 2 Aw − ( u + Av )2 √ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − k √ A exp (cid:18) − k arcsin w √ v (cid:19) exp (cid:18) − k √ A arcsin w √ u (cid:19) . In order for f n ( x, y, z ) = ¯ f n ( u, v, w ) to be a homogeneous polynomial of degree n in x, y, z , we have k = k = 0 and the function ¯ A is a homogeneous polynomial in x + Ay and y + z . Then f n = ¯ A ( x + Ay , y + z )( x + √ Az ) k √ A . Therefor f is a Darboux polynomial of degree n (= 2 m + k √ A ) with the cofactor K = k y + k . Set ¯ k = k / √ A , we get f = P m +¯ k i =0 f i and f m +¯ k = ( x + √ Az ) ¯ k m X i =0 a mi ( x + Ay ) m − i ( y + z ) i . Integrability analysis of a simple model for describing convection of a rotating fluid] f m +¯ k into (15) and performing some calculations, we obtain Ayz ∂f m +¯ k − ∂x − xz ∂f m +¯ k − ∂y + xy ∂f m +¯ k − ∂z − k yf m +¯ k − = ( x + √ Az ) ¯ k m X i =0 a mi [2 σ ( m − i ) + k + 2 i ]( x + Ay ) m − i ( y + z ) i + ( x + √ Az ) ¯ k − m X i =0 a mi ¯ k σ ( x + Ay ) m − i ( y + z ) i x − ( x + √ Az ) ¯ k m X i =0 a mi C ( m − i )( x + Ay ) m − i − ( y + z ) i xz + ( x + √ Az ) ¯ k − m X i =0 a mi ¯ k ( √ A − C )( x + Ay ) m − i ( y + z ) i z + ( x + √ Az ) ¯ k m X i =0 a mi A ( m − i )(1 − σ )( x + Ay ) m − i − ( y + z ) i y . Using the transformations (7), the above equation becomes p u − Aw p v − w d ¯ f m +¯ k − dw − ¯ k w ¯ f m +¯ k − (16)= ( p u − Aw − √ A p v − w ) ¯ k m X i =0 a mi [2 σ ( m − i ) + k + 2 i ] u m − i v i + ( p u − Aw − √ A p v − w ) ¯ k − m X i =0 a mi ¯ k σu m − i v i p u − Aw + ( p u − Aw − √ A p v − w ) ¯ k m X i =0 a mi C ( m − i ) u m − i − v i p u − Aw p v − w + ( p u − Aw − √ A p v − w ) ¯ k − m X i =0 a mi ¯ k ( C − √ A ) u m − i v i p v − w + ( p u − Aw − √ A p v − w ) ¯ k m X i =0 a mi A ( m − i )(1 − σ ) u m − i − v i w . which is a non-homogeneous linear ordinary differential equation in ¯ f m +¯ k − , The cor-responding homogeneous equation p u − Aw p v − w d ¯ f ∗ m +¯ k − dw − ¯ k w ¯ f ∗ m +¯ k − = 0has a general solution¯ f ∗ m +¯ k − = ( p u − Aw − √ A p v − w ) ¯ k ¯ A ∗ m − ( u, v ) , Integrability analysis of a simple model for describing convection of a rotating fluid] A ∗ m − ( u, v ) is an arbitrary smooth function in u and v . In order to use the methodof variation of constants, we assume that¯ f m +¯ k − = ( p u − Aw − √ A p v − w ) ¯ k ¯ A m − ( u, v, w )is a solution of (16), then ¯ A m − ( u, v, w ) satisfies d ¯ A m − dw = m X i =0 a mi [2 σ ( m − i ) + b + 2 i ] u m − i v i √ u − Aw √ v − w + m X i =0 a mi ¯ k σu m − i v i √ u − Aw − √ A √ v − w ) √ v − w + m X i =0 a mi C ( m − i ) u m − i − v i + m X i =0 a mi ¯ k ( C − √ A ) u m − i v i √ u − Aw − √ A √ v − w ) √ u − Aw + m X i =0 a mi A ( m − i )(1 − σ ) u m − i − v i w √ u − Aw √ v − w . Some easy computations lead to Z dw ( √ u − Aw − √ A √ v − w ) √ u − Aw = wu − v + Z √ v − w √ u − Aw dw (17)Since Z dw √ u − Aw √ v − w , Z √ u − Aw √ v − w dw are elliptic integrals of the first and second kind, by (10) and (17), in order for A m − ( x, y, z ) =¯ A m − ( u, v, w ) to be a homogeneous polynomial of degree 2 m −
1, we must have a mi [2 σ ( m − i ) + k + 2 i ] = 0 ,a mi ¯ k σ = 0 ,a mi ¯ k ( C − √ A ) = 0 ,a mi A ( m − i )(1 − σ ) = 0 , i = 0 , , · · · , m, (18)which implies ¯ k = 0, that is to say, k = 0. So the cofactor K = k is a constant. Then(18) becomes ( a mi [2 σ ( m − i ) + 2 i + k ] = 0 ,a mi A ( m − i )(1 − σ ) = 0 , i = 0 , , · · · , m. Integrability analysis of a simple model for describing convection of a rotating fluid]
Case I . σ = 1, k = − m . In this case, we have f n = f m = m X i =0 a mi ( x + Ay ) m − i ( y + z ) i , (19) f n − = f m − = m X i =0 a mi C ( m − i )( x + Ay ) m − i − ( y + z ) i y. (20)Substituting (19) and (20) into (15) with i = n −
2, we get
Ayz ∂f n − ∂x − xz ∂f n − ∂y + xy ∂f n − ∂z = − m X i =0 a mi C ( m − i )( m − i − x + Ay ) m − i − ( y + z ) i xyz − m X i =0 a mi Rai ( x + Ay ) m − i ( y + z ) i − z − m X i =0 a mi AC ( m − i )( x + Ay ) m − i − ( y + z ) i y . By (7), the above equation becomes d ¯ f n − dw = m X i =0 a mi C ( m − i )( m − i − u m − i − v i w + m X i =0 a mi Raiu m − i v i − √ u − Aw − m X i =0 a mi C ( m − i ) u m − i − v i w √ u − Aw √ v − w , that is, ¯ f n − = m X i =0 a mi C ( m − i )( m − i − u m − i − v i w + m X i =0 a mi Raiu m − i v i − √ A arctan √ Aw √ u − Aw ! − m X i =0 a mi C ( m − i ) u m − i − v i ln |√ u − Aw + p A ( v − w ) |√ A .
We must have ( a mi iRa = 0 ,a mi C ( m − i ) = 0 , i = 0 , , · · · , m, Integrability analysis of a simple model for describing convection of a rotating fluid] a mi = 0 for i = 0 , , · · · , m. This is a contradiction to f n = 0. Case II . σ = 1, a mi = 0, i = 0 , , · · · , m − k = − σm . In this case, we have f n = f m = a mm ( y + z ) m , f n − = f m − = 0 . Working in a similar way to the previous case, we get − p u − Aw p v − w d ¯ f n − dw = − a mm mRav m − p v − w , and ¯ f n − = 2 a mm mRav m − √ A arctan √ Aw √ u − Aw ! . Therefore, a mm = 0 and f n = 0, which is a contradiction. Proposition 6.
System (1) has no exponential factors.
Proof:
Let E = exp( g/h ) be an exponential factor of system (1) with a cofactor L = l + l x + l y + l z, l i ∈ R , i = 0 , , , , where g, h ∈ R [ x, y, z ] with f, g being prime. From Proposition 2, 4 and 5, E = exp( g )with g = g ( x, y, z ) ∈ R [ x, y, z ] / R . By definitions, g satisfies( Ayz + Cz − σx ) ∂g∂x + ( − xz + Ra − y ) ∂g∂y + ( − z + xy ) ∂g∂z = l + l x + l y + l z. (21)If g is a polynomial of degree n ≥
3. We write g as g = P nj =0 g j ( x, y, z ), where each g j is a homogeneous polynomial of degree j and g n = 0. Observing the right hand sideof (21) has degree at most one, we compute the terms of degree n + 1 in (21) and get Ayz ∂g n ∂x − xz ∂g n ∂y + xy ∂g n ∂z = 0 , which is (5) replacing f n by g n . Then the arguments used in the proof of Proposition 4imply g n = L n ( y + z ) m with L n ∈ R . Now computing the terms of degree n in (21)leads to Ayz ∂g n − ∂x − xz ∂g n − ∂y + xy ∂g n − ∂z = ( σx − Cz ) ∂g n ∂x + y ∂g n ∂y + z ∂g n ∂z , which is (9) with f n replaced by g n and f n − replaced by g n − . Again, the argumentsused in the proof of Proposition 4 imply that g n = 0, which is a contradiction.Hence, g is a polynomial of degree at most two satisfying (21). So we can write g as g = b + b x + b y + b z + b x + b y + b z + b xy + b xz + b yz. (22) Integrability analysis of a simple model for describing convection of a rotating fluid] g i = 0 for i =1 , · · · ,
15, where g = Rab − l , g = − σb + Rab − l ,g = 2 Rab − b − l , g = Cb + Rab − b − l ,g = − σb − b + b , g = − σb − b − b ,g = Ab + Cb − b , g = − σb ,g = − b , g = 2 Cb + Cb − b , g = − b + b ,g = b , g = b , g = b , g = 2 Ab + Ab . From the above equations, we obtain that l = l = l = l = 0, b = b = b = 0, b = b = b = b = b = b = 0, which implies g is a constant. This completes theproof. Proof of Theorem 1 : By Proposition 1 and 4-6, it is obvious.
4. Appendix
In this section, we provide a relationship between the Glukhovsky-Dolzhansky system(1) and other similar quadric systems including the Rabinovich system, the 3D forced-damped system and the D2 vector field.The Rabinovich system [28, 29] is a three-wave interaction model and is given by ˙ X = hY − v X + Y Z ˙ Y = hX − v Y − XZ ˙ Z = − v Z + XY. (23)where h, v , v , v are parameters. When C = − ARa , by a family of linear scaling S α : ( x, y, z, t ) → ( X, Y, Z, T ) x = Xα , y = Ra − Zα √− A , z = Yα √− A , t = αT, we transform the Glukhovsky-Dolzhansky system (1) into dXdT = α √− ARaY − ασX + Y Z dYdT = α √− ARaX − αY − XZ dZdT = − αZ + XY. (24)Clearly, system (24) corresponds to the Rabinovich system (23) with parameters ( h, v , v , v ) =( α √− ARa, ασ, α, α ). Hence, if C = − ARa , the Glukhovsky-Dolzhansky system (1) isequivalent to the Rabinovich system (23) in the particular case of the parameter region { ( h, v .v .v ) | v = v } , that is, two systems are homothetic copies. In addition, the linearscaling S α forces parameter A is less than zero, which cannot hold in the physical regionof parameter A . Integrability analysis of a simple model for describing convection of a rotating fluid] ˙ X = − aX + Y + Y Z ˙ Y = X − aY + bXZ ˙ Z = cZ − bXY (25)where a, b, c are parameters. Compared with the Glukhovsky-Dolzhansky system (1),the 3D forced-damped system (25) has less parameters. When σ = 1, the linear scaling x = ( ARa + C ) / A √ Ra X, y = Ra + ( ARa + C ) ZA , z = (
ARa + C ) YA , t = T p ( ARa + C ) Ra , transforms the Glukhovsky-Dolzhansky system (1) into dXdT = − √ ( ARa + C ) Ra X + Y + Y Z dYdT = X − √ ( ARa + C ) Ra Y + ARa + CARa XZ dZdT = − √ ( ARa + C ) Ra Z − ARa + CARa
XY, (26)which corresponds to the 3D forced-damped system (25) with parameters( a, b, c ) = ( 1 p ( ARa + C ) Ra , ARa + CARa , − p ( ARa + C ) Ra ) . Therefore, when σ = 1, the Glukhovsky-Dolzhansky system (1) is equivalent to the 3Dforced-damped system (25) in the particular case of the parameter region { ( a, b, c ) | a + c =0 } . Consider the vector field equivariant under the D2 symmetry group, called the D2 vector field [31] ˙ X = aX + Y Z ˙ Y = bY + XZ ˙ Z = Z − XY. (27)where a, b are parameters. Similar to above, when Ra = C = 0, we make a linear scaling x = X, y = Z √ A , z = − Y √ A , t = − T and transform (1) into dXdT = σX + Y Z dYdT = Y + XZ dZdT = Z − XY. (28)which corresponds to the D2 vector field (27) with parameters ( a, b ) = ( σ, Ra = C = 0, the Glukhovsky-Dolzhansky system (1) is equivalent to the D2 vector field(27) in the particular case of the parameter region { ( a, b ) | b = 1 } . Integrability analysis of a simple model for describing convection of a rotating fluid] [1] A.B. Glukhovsky, F.A. Dolzhansky, Three-component geostrophic models of convection ina rotating fluid,
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