Linearizability and critical period bifurcations of a generalized Riccati system
Valery G. Romanovski, Wilker Fernandes, Yilei Tang, Yun Tian
LLINEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OFA GENERALIZED RICCATI SYSTEM
VALERY G. ROMANOVSKI , , , , WILKER FERNANDES , YILEI TANG , AND YUN TIAN Abstract.
In this paper we investigate the isochronicity and linearizability problem for acubic polynomial differential system which can be considered as a generalization of the Ric-cati system. Conditions for isochronicity and linearizability are found. The global structureof systems of the family with an isochronous center is determined. Furthermore, we findthe order of weak center and study the problem of local bifurcation of critical periods in aneighborhood of the center. Introduction
A classical problem in the qualitative theory of ordinary differential equations is to char-acterize the existence of centers and isochronous centers. A singular point of a planar au-tonomous differential system is called a center if all solutions sufficiently closed to it areperiodic, that is, all trajectories in a small neighborhood of the singularity are ovals. If allperiodic solutions inside the period annulus of the center have the same period it is said thatthe center is isochronous .Poincar´e and Lyapunov have shown that the existence of an isochronous center at theorigin of a system of the form(1.1) ˙ x = − y + P ( x, y ) , ˙ y = x + Q ( x, y ) , where P ( x, y ) and Q ( x, y ) are real polynomials without constant and linear terms, is equiv-alent to the linearizability of the system. This equivalence has made the studies of theisochronicity problem simpler, since the linearizability problem can be extended to the com-plex field, where the computational methods are more efficient.The investigation on isochronicity of oscillations started in the 17th century, when Huygensstudied the cycloidal pendulum [27]. However, only in the second half of the last century theisochronicity problem began to be intensively studied. In 1964 Loud [34] found the necessaryand sufficient conditions for isochronicity of system (1.1) with P and Q being quadratichomegeneous polynomials. Later on, the isochronicity problem was solved for system (1.1)when P and Q are homogeneous polynomials of degree three [40] (see also [29]) and degreefive [41]. However in the case of the linear center perturbed by homogeneous polynomialsof degree four the problem is still unsolved, although some partial results were obtained[7, 23]. The reason is that linearizability quantities (which are polynomials in the parametersof system (1.1) defined at the beginning of Section 2) have more complicate expressionsin the case of homogeneous perturbations of degree four, than in the case of homogeneousperturbations of degree five. There are also many works devoted to the investigation of Key words and phrases.
Linearizability, isochronicity, global structure, weak center, local bifurcation ofcritical periods. a r X i v : . [ m a t h . D S ] J un V.G. ROMANOVSKI, W. FERNANDES, Y. TANG, Y. TIAN particular families of some other polynomial systems, see e.g. [1, 6, 9, 14, 35, 43] and referencestherein. Many works also deal with investigation of isochronicity of Hamiltonian systems, seee.g. [12, 15, 24, 28, 31] and references given there.The problem of critical period bifurcations is tightly related to the isochronicity problem.In a neighborhood of a center the so-called period function T ( r ) gives the least period of theperiodic solution passing through the point with coordinates ( x, y ) = ( r,
0) inside the periodannulus of the center. For a center that is not isochronous any value r > T (cid:48) ( r ) = 0is called a critical period. The problem of critical period bifurcations is aimed on estimatingof the number of critical periods that can arise near the center under small perturbations. In1989, Chicone and Jacobs [11] introduced for the first time the theory of local bifurcations ofcritical periods and solved the problem for the quadratic system. Local bifurcations of criticalperiods have been investigated for cubic systems with homogeneous nonlinearities [45], thereduced Kukles system [46], the Kolmogorov system [10], the Z -equivariant systems [8] andsome other families (see e.g. [16, 22, 49] and references therein). In [20] a general approachto studying bifurcations of critical periods based on a complexification of the system wasdescribed, and some upper bounds on the number of critical periods of several cubic systemswere obtained.In this paper we are interested in the family of Riccati systems. The classic Riccati system is written in the form(1.2) ˙ x = 1 , ˙ y = g ( x ) y + g ( x ) y + g ( x ) , where each g j ( x ) is a C function with respect to x and g ( x ) g ( x ) (cid:54)≡
0. System (1.2) becomesa special case of Berouilli system if g ( x ) ≡
0, and it obviously is a linear differential systemif g ( x ) ≡ x = f ( y ) , ˙ y = g ( x ) y + g ( x ) y + g ( x ) , which is called the generalized Riccati system , since it becomes the classic Riccati systemwhen f ( y ) ≡
1. In this paper we study a subfamily of the generalized Riccati system, cubicsystems of the form(1.3) ˙ x = − y + a y + a y , ˙ y = ( b + b x ) y + ( b x + b x ) y + ( x + b x + b x )= x + b x + b xy + b y + b x + b x y + b xy , where x, y are unknown real functions and a ij , b ij are real parameters. Note that system (1.3)is the so-called reduced Kukles system when a = a = 0.The aims of our study are to obtain conditions on parameters a ij and b ij for the lineariz-ability of system (1.3), to study the global structures of trajectories when the system has an INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 3 isochronous center, and to investigate the local bifurcations of critical periods at the origin.In Section 2 we present our main result on linearizability, Theorem 2.1, which gives condi-tions for the linearizability of system (1.3). We also describe an approach for deriving suchconditions which is based on making use of modular computations which are performed inthe systems of computer algebra
Singular [17] and
Mathematica [48]. The approach canbe applied to investigate many problems involving solving systems of algebraic polynomials.In Section 3 we study the global dynamics of system (1.3) when the origin is an isochronouscenter. The last section is devoted to the investigation of local bifurcations of critical periodsin a neighborhood of the center.2.
Linearizability of system (1.3)We first briefly remind an approach for studying the isochronicity and linearizability prob-lems for polynomial differential systems of the form(2.1) ˙ x = − y + n (cid:88) p + q =2 a p,q x p y q , ˙ y = x + n (cid:88) p + q =2 b p,q x p y q , where x, y and a p,q , b p,q are in R .System (2.1) is linearizable if there is an analytic change of coordinates(2.2) x = x + (cid:88) m + n ≥ c m,n x m y n , y = y + (cid:88) m + n ≥ d m,n x m y n , which reduces (2.1) to the canonical linear system ˙ x = − y , ˙ y = x .Obstacles for existence of a transformation (2.2) are some polynomials in parameters ofsystem (2.1) called the linearizability quantities and denoted by i k , j k ( k = 1 , , ... ).Differentiating with respect to t both sides of each equation of (2.2) we obtain(2.3) ˙ x = ˙ x + (cid:88) m + n ≥ mc m,n x m − y n ˙ x + (cid:88) m + n ≥ nc m,n x m y n − ˙ y, ˙ y = ˙ y + (cid:88) m + n ≥ md m,n x m − y n ˙ x + (cid:88) m + n ≥ nd m,n x m y n − ˙ y. Substituting in the above equations the expressions from (2.2) and (2.1), one computes thelinearizability quantities i k , j k step-by-step (see e.g. [21] for more details).From (2.3) it is easy to see that the linearizability quantities i k , j k are polynomials inparameters a p,q , b p,q of system (2.1). We denote by ( a, b ) the s -tuple ( s is the number of pa-rameters a p,q , b p,q in system (2.1)) of parameters of (2.1), so ( a, b ) = ( a , , a , , . . . , b ,n ), andby R [ a, b ] and C [ a, b ] the rings of polynomials in a p,q , b p,q with real and complex coefficients,respectively.Thus, the simultaneous vanishing of all linearizability quantities i k , j k provides conditionswhich characterize when a system of the form (2.1) is linearizable. The ideal defined by thelinearizability quantities, L = (cid:104) i , j , i , j , ... (cid:105) ⊂ R [ a, b ], is called the linearizability ideal andits affine variety, V L = V ( L ) is called the linearizability variety . V.G. ROMANOVSKI, W. FERNANDES, Y. TANG, Y. TIAN
In order to find a linearizing change of coordinates explicitly one can look for Darbouxlinearization. To construct a Darboux linearization for system (2.1) it is convenient to com-plexify the system using the substitution(2.4) z = x + iy, w = x − iy. Then, after a time rescaling by i we obtain from (2.1) a system of the form(2.5) ˙ z = z + X ( z, w ) , ˙ w = − w − Y ( z, w ) . System (2.1) is linearizable if and only if system (2.5) is linearizable.A
Darboux factor of system (2.5) is a polynomial f ( z, w ) satisfying ∂f∂z ˙ z + ∂f∂w ˙ w = Kf, where polynomial K ( z, w ) is called the cofactor of f . A Darboux linearization of system (2.5)is an analytic change of coordinates z = Z ( z, w ), w = W ( z, w ), such that Z ( z, w ) = m (cid:89) j =0 f α j j ( z, w ) = z + ˜ Z ( z, w ) ,W ( z, w ) = n (cid:89) j =0 g β j j ( z, w ) = w + ˜ W ( z, w ) , which linearizes (2.5), where f j , g j ∈ C [ z, w ], α j , β j ∈ C , and ˜ Z and ˜ W have neither constantterms nor linear terms.It is easy to see that system (2.5) is Darboux linearizable if there exist s + 1 ≥ f , ..., f s with corresponding cofactors K , ..., K s , and t + 1 ≥ g , ..., g t with corresponding cofactors L , ..., L t with the following properties:(i) f ( z, w ) = z + · · · but f j (0 ,
0) = 1 for j ≥ g ( z, w ) = w + · · · but g j (0 ,
0) = 1 for j ≥
1; and(iii) there are s + t constants α , ..., α s , β , ..., β t ∈ C such that(2.6) K + α K + · · · + α s K s = 1 and L + β L + · · · + β t L t = − . The Darboux linearization is then given by the transformations z = H ( z, w ) = f f α · · · f α s s , y = H ( z, w ) = g g β · · · g β t t . The readers can consult [13, 35, 43] for more details.Before passing to the results of our paper we remind some fact about solutions of systemsof nonlinear polynomial equations which we will need for our study.Denote by k [ x , . . . , x n ] the ring of polynomials with coefficients in a field k and considera system of polynomials of k [ x , . . . , x n ]: f ( x , . . . , x n ) = 0 , ...(2.7) f m ( x , . . . , x n ) = 0 . We recall that the ideal I in k [ x , . . . , x n ] generated by polynomials f , . . . , f m , denotedby I = (cid:104) f , . . . , f m (cid:105) , is the set of all polynomials of k [ x , . . . , x n ] expressed in the form f h + f h + · · · + f m h m , where h , h , . . . , h m are polynomials of k [ x , . . . , x n ]. The variety INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 5 of the ideal I = (cid:104) f , . . . , f m (cid:105) ⊂ k [ x , . . . , x n ] in k n , denoted by V ( I ), is the zero set of allpolynomials of I , V ( I ) = { A = ( a , . . . , a n ) ∈ k n | f ( A ) = 0 for all f ∈ I } . The situation when the variety of a polynomial ideal consists of a finite number of pointsarises very rarely. In a generic case, the variety consists of infinitely many points, so generallyspeaking,“to solve” system (2.7) means to find a decomposition of the variety of the ideal intoirreducible components. More precisely, an affine variety V ⊂ k n is irreducible if, whenever V = V ∪ V for affine varieties V and V , then either V = V or V = V . Let I be an idealand V = V ( I ) its variety. Then V can be represented as a union of irreducible components, V = V ∪ · · · ∪ V m , where each V i is irreducible. The radical of I denoted by √ I is the set of allpolynomials f of k [ x , . . . , x n ] such that for some non-negative integer p f p is in I . Clearly, I and √ I have the same varieties. It is known that √ I can be expressed as an intersectionof prime ideals, √ I = ∩ sj =1 Q j . Prime ideals Q i are called the minimal associate primes of I .Let V i ( i = 1 , . . . , s ) be the variety of Q i . Since the variety of an intersection of some idealsis equal to the union of the varieties of the ideals, we have that V ( I ) = V ( √ I ) = ∩ sj =1 V j .For example, if I = (cid:104) x y , xz (cid:105) , then √ I = (cid:104) xy, xz (cid:105) = (cid:104) x (cid:105) ∩ (cid:104) y, z (cid:105) , that is, the variety of I is the union of two irreducible components: the plane x = 0 and the line y = z = 0. Inthe computer algebra system Singular [17] one can compute the minimal associate primesof a given polynomial ideal and, thus, the irreducible decomposition of its variety using theroutine minAssGTZ .Proceeding now to the results of our paper we first state the following theorem on thelinearizability of system (1.3).
Theorem 2.1.
System (1.3) is linearizable at the origin if one of the following conditionsholds: (1) b = a = b = b = a = b + b = b + 4 b = 0 , (2) b = a = b = b = b = a = 9 b − b = 0 , (3) b = a = b = b = b = b = 9 a + 4 b = 0 , (4) b = b = b = a = 2 b + 5 b = 10 a − b = 4 b + 25 b = 0 .Proof. Using the computer algebra system
Mathematica and the standard procedure men-tioned above for system (1.3) we have computed the first eight pairs of the linearizabilityquantities i , j , ..., i , j . Their expressions are very large, so we only present the first twopairs in the Appendix. The reader can easily compute the other quantities using any availablecomputer algebra system .The next computational step is to compute the irreducible decomposition of the variety V ( L ) = V ( (cid:104) i , j , ..., i , j (cid:105) ).Performing the computations by the routine minAssGTZ [18] of Singular [17] over thefield of characteristic 32452843 we obtain that V ( L ) is equal to the union of the varieties offour ideals. After lifting these four ideals to the ring of polynomials with rational coefficients One can download linearizability quantities i , j , . . . , i , j and the Singular code toperform the decomposition of the variety from http://teacher.shnu.edu.cn/ upload/article/files/79/14/f36e87e342b8b0d6977e6debdeb3/3b818cf4-a6f7-4f07-8669-f4b78e48f733.txt.
V.G. ROMANOVSKI, W. FERNANDES, Y. TANG, Y. TIAN using the rational reconstruction algorithm of [47] we obtain the ideals J = (cid:104) b , a , b , b , a , b + b , b + 4 b (cid:105) ,J = (cid:104) b , a , b , b , b , a , b − b (cid:105) ,J = (cid:104) b , a , b , b , b , b , a + 4 b (cid:105) ,J = (cid:104) b , b , b , a , b + 5 b , a − b , b + 25 b (cid:105) . The varieties of J , J , J and J provide conditions (1), (2), (3) and (4) of the theorem,respectively.To check the correctness of the obtained conditions we use the procedure described in[42]. First, we computed the ideal J = J ∩ J ∩ J ∩ J , which defines the union of all foursets given in the statement of the theorem. Then we check that V ( J ) = V ( L ). Accordingto the Radical Membership Test, to verify the inclusion V ( J ) ⊃ V ( L ) it is sufficient tocheck that the Groebner bases of all ideals (cid:104) J, − wi k (cid:105) , (cid:104) J, − wj k (cid:105) (where k = 1 , . . . , w is a new variable) computed over Q are { } . The computations show that this is thecase. To check the opposite inclusion, V ( J ) ⊂ V ( L ), it is sufficient to check that Groebnerbases of the ideals (cid:104)L , − wf i (cid:105) (where the polynomials f i ’s are the polynomials of a basisof J ) computed over Q are equal to { } . Unfortunately, we were not able to perform thesecomputations over Q however we have checked that all the bases are { } over few fields offinite characteristic. It yields that the list of conditions in Theorem 2.1 is the complete listof linearizability conditions for system (1.3) with high probability [3].We now prove that under each of conditions (1)–(4) of the theorem the system is lineariz-able. Condition (1) . In this case b = ± b i . We consider only the case b = 2 b i , since when b = − b i the proof is analogous. After the change of variables (2.4) system (1.3) becomes(2.8) ˙ z = z + b z , ˙ w = − w − b z , which is a quadratic system. By Theorem 3.1 of [13] and Theorem 4.5.1 of [43] system (2.8)is Darboux linearizable and, therefore, system (1.3) is linearizable if condition (1) holds. Condition (2) . After substitution (2.4) system (1.3) becomes(2.9) ˙ z = z + 172 ( − ib z + 18 ib w + b z + 3 b z w + 3 b zw + b w ) , ˙ w = − w + 172 (18 ib z − ib w − b z − b z w − b zw − b w ) . It has the Darboux factors l = z + ib z + ib zw + ib w ,l = w − ib z − ib zw − ib w ,l =1 − ib z + b z + ib w + b zw + b w ,l =1 − ib z + b z + ib w + b zw + b w with the respective cofactors k = 1 − ib z − ib w, k = − − ib z − ib w,k = − ib z − ib w, k = − ib z − ib w. It is easy to verify that (2.6) is satisfied with α = 1 , α = − , β = 1 and β = −
1. Hencethe Darboux linearization for system (2.9) is given by the analytic change of coordinates z = l l α l α , w = l l β l β . Thus, system (2.9) is linearizable and therefore the corresponding system (1.3) is linearizableas well.
Condition (3) . In this case after substitution (2.4) the corresponding system (1.3) is changedto(2.10) ˙ z = z + 136 ( − b z + 18 b zw − b w − b z + 6 b z w − b zw + 2 b w ) , ˙ w = − w + 136 (9 b z − b zw + 9 b w − b z + 6 b z w − b zw + 2 b w ) . System (2.10) has the Darboux factors l = z − b z + b zw − b w ,l = w − b z + b zw − b w ,l =1 − b z + 2 b z − b w − b zw + 2 b w ,l =1 − b z + b z − b w − b zw + b w , which allow to construct the Darboux linearization z = l l α l α , w = l l β l β , where α = 1 , α = − , β = 1 and β = − Condition (4) . For this condition it is easy to see that b = ± b i/
2. We consider only thecase b = 5 b i/
2, since when b = − b i/ z = z + 116 (21 b z − b zw + b w ) , ˙ w = − w + 116 ( − b z + 18 b zw − b w ) . V.G. ROMANOVSKI, W. FERNANDES, Y. TANG, Y. TIAN
System (2.11) has the Darboux factors l = z + 116 3 b z + 18 b zw + 148 b w ,l = w + 116 9 b z + 3 b zw + b w ,l =1 + 3 b z + 27 b z + b w − b zw + 3 b w ,l =1 + 3 b z + 9 b z + b w + 3 b zw + b w , yielding the Darboux linearization z = l l α l α , w = l l β l β , where α = 1 , α = − , β = 1 and β = − (cid:3) Global dynamics of system (1.3) having an isochronous center
Global phase portrait of a planar autonomous system is usually plotted on the Poincar´edisc, which is obtained using the Poincar´e compactification. We remind the procedure briefly,for more details see for instance [2, 19].Consider the planar vector field X = ˜ P ( x, y ) ∂∂x + ˜ Q ( x, y ) ∂∂y , where ˜ P ( x, y ) and ˜ Q ( x, y ) are polynomials of degree n . Let S = { y = ( y , y , y ) ∈ R : y + y + y = 1 } , S be the equator of S and p ( X ) be the Poincar´e compactification of X on S . On S \ S there are two symmetric copies of X , and once we know the behaviour of p ( X ) near S , we know the behaviour of X in a neighbourhood of the infinity. The Poincar´ecompactification has the property that S is invariant under the flow of p ( X ). The projectionof the closed northern hemisphere of S on y = 0 under ( y , y , y ) (cid:55)→ ( y , y ) is called the Poincar´e disc , and its boundary is S .Because S is a differentiable manifold, we consider the six local charts U i = { y ∈ S : y i > } and V i = { y ∈ S : y i < } for computing the expression of p ( X ) where i = 1 , ,
3. The dif-feomorphisms F i : U i → R and G i : V i → R for i = 1 , , , , , ( − , , , (0 , , , (0 , − , , (0 , , , , −
1) respectively. We denote by ( u, v ) the value of F i ( y ) or G i ( y ) for any i = 1 , , p ( X ) in the local chart ( U , F ) is given by˙ u = v n (cid:20) − u ˜ P (cid:18) v , uv (cid:19) + ˜ Q (cid:18) v , uv (cid:19)(cid:21) , ˙ v = − v n +1 ˜ P (cid:18) v , uv (cid:19) , for ( U , F ) is ˙ u = v n (cid:20) ˜ P (cid:18) uv , v (cid:19) − u ˜ Q (cid:18) uv , v (cid:19)(cid:21) , ˙ v = − v n +1 ˜ Q (cid:18) uv , v (cid:19) , and for ( U , F ) is ˙ u = ˜ P ( u, v ) , ˙ v = ˜ Q ( u, v ) . INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 9
The expressions for V i ’s are the same as that for U i ’s but multiplied by the factor ( − n − .In these coordinates v = 0 always denotes the points of S . When we study the infinitesingular points on the charts U ∪ V , we only need to verify if the origin of these charts aresingular points.It is said that two polynomial vector fields X and Y on R are topologically equivalent ifthere exists a homeomorphism on S preserving the infinity S carrying orbits of the flowinduced by p ( X ) into orbits of the flow induced by p ( Y ), preserving or not the sense of allorbits.In this section, we study the global structures of system (1.3) in Poincar´e discs for the casewhen it has an isochronous center listed in Theorem 2.1. Theorem 3.1.
The global phase portrait of system (1.3) possessing an isochronous centerlisted in Theorem 2.1 is topologically equivalent to one of phase portraits in Fig. 1. More pre-cisely, there exists only one equilibrium of system (1.3) in the plane, which is an isochronouscenter at the origin. The neighborhood of equilibrium at infinity consists of one elliptic sec-tor and three hyperbolic sectors (or two hyperbolic sectors and two parabolic sectors) underconditions (2) and b (cid:54) = 0 (or under conditions (3) and b (cid:54) = 0 ); otherwise, the isochronouscenter is global. Figure 1.
Global phase portraits of system (1.3) possessing an isochronouscenter listed in Theorem 2.1.
Proof.
From Theorem 2.1 we have that under conditions (1)–(4) system (1.3) is linearizable.Under conditions (1) and (4) real systems (1.3) becomes the linear system ˙ x = − y, ˙ y = x and its phase portrait is presented in Figure 1.A.Under conditions (2) and (3) system (1.3) becomes(3.1) ˙ x = − y, ˙ y = x + b xy + b x , and(3.2) ˙ x = − y − b y , ˙ y = x + b y , respectively.Note that if b = 0 in (3.1) and b = 0 in (3.2), then both systems are the canonic linearsystems and have a global center shown in Figure 1.A. Thus, we consider the cases when b (cid:54) = 0 and b (cid:54) = 0. In both cases by a linear change of coordinates we can reduce systems(3.1) and (3.2) to systems(3.3) ˙ x = − y, ˙ y = x + xy + x , and(3.4) ˙ x = − y − y , ˙ y = x + y , respectively.System (3.3) has only the isochronous center at (0 ,
0) as a finite singular point. Now weanalyze its singular points at infinity. In the local chart U system (3.3) becomes˙ u = 19 (1 + 9 uv + 9 v + 9 u v ) , ˙ v = uv . This system has no real singular points. So the unique possible infinite singular point is theorigin of the local chart U . In the local chart U system (3.3) becomes(3.5) ˙ u = 19 ( − u − u v − v − u v ) , ˙ v = − uv ( u + 9 v + 9 v ) . It is clear that (0 ,
0) is a singular point of (3.5) and the linear part of (3.5) at (0 ,
0) is thenull matrix, i.e, (cid:18) (cid:19) . Applying the directional blow-up in the v -axis twice we obtainthat the behaviour of the orbits close to the origin of U is as in Figure 2. Therefore, theglobal phase portrait of system (3.3) is topologically equivalent to the one in Figure 1.B. Figure 2.
Behaviour ofthe orbits close to theorigin of system (3.5).
Figure 3.
Behaviour ofthe orbits close to theorigin of system (3.6).Now we study system (3.4). This system has only the isochronous center at (0 ,
0) as a finitesingular point. For the infinite singular points, in the local chart U system (3.4) becomes(3.6) ˙ u = 19 (4 u + 9 u v + 9 v + 9 u v ) , ˙ v = 19 uv (4 u + 9 v ) . This system has only (0 ,
0) as a singular point, and the linear part of (3.6) at (0 ,
0) is the nullmatrix. Applying the directional blow-up in the v -axis twice we obtain that the behaviourof the orbits close to the origin of U is as showing in Figure 3. INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 11
In the local chart U system (3.4) becomes(3.7) ˙ u = 19 ( − − uv − v − u v ) , ˙ v = − v (1 + uv ) . As it is mentioned above, we need to study only the origin of this chart, but (0 ,
0) is not asingular point for system (3.7). Thus, the global phase portrait of system (3.4) is topologicallyequivalent to the portrait in Figure 1.C. (cid:3) Weak center and local bifurcation of critical periods
Let α = ( a , a , ..., b , b , ... ) be the string of parameters of real system (2.1) with acenter at the origin. Changing the system to the polar coordinates x = r cos θ , y = r sin θ and eliminating t , we obtain(4.1) drdθ = r x ˙ x + y ˙ yx ˙ y − y ˙ x = rH ( r, θ, α )1 + G ( r, θ, α ) , where H ( r, θ, α ) and G ( r, θ, α ) are polynomials of r, α, cos θ and sin θ . The solution r = r ( θ, α )of equation (4.1) satisfying the initial condition r (0 , α ) = r > r ,(4.2) r ( θ, α ) = ∞ (cid:88) k =1 v k ( θ, α ) r k . Substituting (4.2) into (4.1), one can find coefficients v k ( θ, α ) ( k >
1) by successive integra-tion.Assuming that Γ r is the closed trajectory through ( r , T ( r , α ) = (cid:73) Γ r dt = (cid:90) π dθ G ( r, θ, α ) = ∞ (cid:88) k =0 p k ( α ) r k . The period function is even and has the Taylor series expansion(4.3) T ( r , α ) = 2 π + ∞ (cid:88) k =1 p k ( α ) r k , where r < δ and coefficients p k ’s are polynomials in parameters of system (2.1) (see e.g.[1, 11, 35, 43]).If p = ... = p k = 0 and p k +2 (cid:54) = 0, then the origin of system (2.1) is a weak center oforder k . If p k = 0 for each k ≥
1, then the origin is an isochronous center . For a centerwhich is not isochronous, a local critical period is any value ˜ r < δ for which T (cid:48) (˜ r ) = 0.By classical results of local critical period bifurcations [11], at most k local critical periodscan bifurcate from the period function related to a weak center of order k . In order to provethat there are perturbations with exactly k local critical periods, we remind Theorem 2 of[49] as follows. Theorem 4.1.
Assume that the period constants p j ( j = 1 , , ..., k ) of system (2.1) dependon k independent parameters a , a , ..., a k . Suppose that there exists ˜ a = (˜ a , ˜ a , ..., ˜ a k ) suchthat p j (˜ a ) = p j (˜ a , ˜ a , ..., ˜ a k ) = 0 , j = 1 , , ..., k,p k +2 (˜ a ) (cid:54) = 0 and det (cid:16) ∂ ( p , p , ..., p k ) ∂ ( a , a , ..., a k ) (˜ a ) (cid:17) (cid:54) = 0 , then k critical periods bifurcate from the center at the origin of system (2.1) after smallappropriate perturbations.Remark . The proof that k critical periods can bifurcate after perturbations of system (2.1)corresponding to parameters ˜ a is derived using the Implicit Function Theorem, and the proofthat the bound k is sharp can be derived either using the Mean Values Theorem [26] orRolle’s Theorem [4]. In practice k critical periods can be obtained choosing perturbationssuch that for some system a ∗ close to | p ( a ∗ ) | (cid:28) | p ( a ∗ ) | (cid:28) · · · (cid:28) | p k ( a ∗ ) | (cid:28) | p k +2 ( a ∗ ) | and the signs in the sequence p ( a ∗ ) , p ( a ∗ ) , . . . p k ( a ∗ ) , p k +2 ( a ∗ ) alternate (see e.g. [25, 30,43] for more details).Because bifurcations of critical periods are bifurcations from centers, to study them forsystem (1.3) we need to know the center variety of the system. Due to computational diffi-culties the center variety of system (1.3) has been found only in the case when a = 0 [51].So, from now on we assume that in system (1.3) a = 0 and consider the system(4.4) ˙ x = − y + a y , ˙ y = x + ( b x + b xy + b y ) + ( b x + b x y + b xy ) . The centers of system (4.4) are identified in the following theorem.
Theorem 4.2 ([51]) . System (4.4) has a center at the origin if the 7-tuple of its parametersbelongs to the variety of one of the following prime ideals: (1) I = (cid:104) b , b , b (cid:105) , (2) I = (cid:104) b , b , b , b b − b (cid:105) , (3) I = (cid:104) b , b , b , − b b + 4 b b − b b , a b + b − b b , a b − b b − b b , a − b − b (cid:105) , (4) I = (cid:104) b , b , a (cid:105) , (5) I = (cid:104) a , b b + b b , b b + b b + b b , b b + b b − b , b + b b + b , b b b − b b b − b b , b b b − b − b b , b b − b b − b b b − b , b b b − b b + b b b − b b , − ( b b ) + b b b + b b (cid:105) , (6) I = (cid:104) b , b , b , b (cid:105) , (7) I = (cid:104) b , b , b , b + 5 b , a − b , b + 25 b (cid:105) .Remark. Like in the proof of our Theorem 2.1 modular computations were used in orderto determine centers of system (4.4), so it can happen that the list of centers of the systemgiven in Theorem 4.2 is incomplete. For this reason it stands in the theorem ”if” but not ”ifand only if”.
INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 13
We consider the local bifurcations of critical periods for system (4.4) when all param-eters are real. Because in R the variety of I consists of one point which is the origin(0 , , , , , , ∈ R , we only need to consider varieties of first six ideals I − I . Theorem 4.3.
Suppose that the origin O : (0 , of system (4.4) is a weak center of a finiteorder.(1) Then the order is at most . More precisely, the order is at most (resp. , , , , )when parameters belong to the variety of the ideal I (resp. I − I ).(2) Moreover, at most (resp. , , , , ) critical periods can be bifurcated from the weakcenter O of system (4.4) and there exists a perturbation with exactly (resp. , , , , )critical periods bifurcated from O when parameters belong to the variety of the ideal I (resp. I − I ).Proof. When the parameter α = ( a , b , b , b , b , b , b ) belongs to the variety of theideal I , we found that the first four period coefficients of (4.3) are p , ( α ) = 10 a − a b + b − b − b ,p , ( α ) = 1540 a + 700 a b + 21 a b − a b + b + 84 a b + 300 a b +6 a b b + 18 a b b − b b − b b + 9 b + 54 b b + 513 b ,p , ( α ) = 3403400 a + 3303300 a b + 690690 a b − a b + 281340 a b b − b + 1261260 a b + 1455300 a b + 346500 a b b + 11935 a b +7263 a b b − a b b − a b b − a b b + 417 a b +1251 b b + 1377 b b + 6966 a b + 31860 a b b − a b +13878 a b b b − a b b − b b + 2025 b + 5265 b b − b b b + 41391 b b − b b − b + 7209 a b b . We omit the expression of p , ( α ), since it is long and the number of its terms is 55.We compute the decomposition of (cid:104) p , , p , , p , , p , (cid:105) with minAssGTZ and obtain (cid:104) a , b − b , b (cid:105) . That is, the condition p , = p , = p , = p , = 0 yields that b = a = b = b = b = a = 9 b − b = 0, showing that the origin is an isochronous center of system(4.4) in this case by Theorem 2.1.Solving the equation p , ( α ) = 0 we get b = ˜ b := (10 / a − (1 / a b + (1 / b − b . (4.5)Substituting (4.5) in p , ( α ), we obtain432 b + 48( a − b ) b + 1920 a + 672 a b + 48 a b = 0 . Thus, when − a − a b − a b + b < O is a weak center of order1. When − a − a b − a b + b ≥
0, from p , ( α ) = 0 we find that b = ˜ b := 118 (cid:16) − a + b + (cid:113) − a − a b − a b + b (cid:17) . We now employ the procedure
Reduce of computer algebra system
Mathematica for theset of equalities and inequalities { b = ˜ b , b = ˜ b , − a − a b − a b + b ≥ , p , ( α ) = 0 , p , ( α ) (cid:54) = 0 } , and find that this semi-algebraic system is fulfilled if and only if a (cid:54) = 0 and − a − a b − a b − a b − a b − a b − a b − a b − a b + 1650 a b + 125 b = 0 . (4.6)Assuming that a = 1 /
2, we can calculate one of solutions b ≈ − . a and b in real field. Moreover, computing with Mathematica the rank of thematrix ∂ ( p , , p , , p , ) ∂ ( a , b , b , b ) , we find that it is equal to 3 when b = ˜ b , b = ˜ b , a (cid:54) = 0 and (4.6) holds. From Theorem4.1 there exists a perturbation of system (4.4) with exactly 3 critical periods bifurcated fromweak center O of order 3 when α belongs to the variety of I .When the parameter α = ( a , b , b , b , b , b , b ) belongs to the variety of the ideal I , we have the first period coefficient in (4.3): p , ( α ) = 10 a − a b + b + 10 b , which cannot be equal to zero in the real field, since 10 a − a b + b > a = b = 0. That is, the center at the origin is of order 0 in this case.When the parameter α = ( a , b , b , b , b , b , b ) lies in the variety of the ideal I ,we compute the first period coefficient in (4.3): p , ( α ) = 10 a − a b + b + 4 b + 10 b b + 10 b , finding that p , ( α ) (cid:54) = 0 unless all parameters vanish. Thus the center O is of order 0 in thiscase.When the parameter α = ( a , b , b , b , b , b , b ) belongs to the variety of the ideal I or I , we can see that system (4.4) is a reduced Kukles system. The variety of ideal I (resp. I ) for center conditions corresponds to the center type K III (resp. K II or K IV ) in[46]. Applying Theorems 3.3, 3.4 and 3.7 of [46] we obtain that the order at the origin isat most 3 (resp. 2), and there exists a perturbation with exactly 3 (or 2) critical periodsbifurcated from O when parameters belong to the variety of the ideal I (resp. I ).When the parameter α = ( a , b , b , b , b , b , b ) belongs to the variety of the ideal I , we found that the first three period coefficients in (4.3) are p , ( α ) = 10 a + 10 b − b ,p , ( α ) = 1540 a + 200 a b + 1540 b + 300 a b − b b + 513 b ,p , ( α ) = 136136 a + 38808 a b + 13080 a b + 165704 b + 58212 a b − a b b − b b − a b + 341334 b b − b . Eliminating b from p , ( α ) = 0 we find b = ˆ b := (10 a + 10 b ) / . INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 15
Letting b = ˆ b we obtain from p , = 0 that47 a − a b − b = 0 , yielding a = ˆ a := ± (cid:113) /
94 + (3 / √ b . Eliminating b and a by substituting b = ˆ b and a = ˆ a into p , ( α ), we obtain b (3578681 + 142373 √ , which does not vanish if b (cid:54) = 0. Therefore, the order of the weak center is at most 2, andthere exists a perturbation with exactly 2 critical periods bifurcated from O when parametersbelong to the variety of the ideal I by Theorem 4.1, since the rank of the matrix ∂ ( p , , p , ) ∂ ( a , b , b ) , is equal to 2 when b = ˆ b , a = ˆ a and b (cid:54) = 0. Notice that when b = 0 and a (cid:54) = 0the center O is a weak center of order 1, and when b = a = 0 the center O is either thelinear isochronous center or the order is 0. (cid:3) Conclusion
For cubic generalized Riccati system (1.3), we derived conditions on parameters of thesystem for the linearizability of the origin, see conditions (1)-(4) of Theorem 2.1.For the study we have used the approach based on the modular calculations of the set ofsolutions of polynomial systems, which was used for the first time in [41] and described indetails in [42]. The approach can be considered as one between precise symbolic computationsand numerical computations since it produces a result which is not completely correct, butcorrect with high probability – in the sense that it is easily verified if the obtained solutionsof a given system of polynomials are correct, but it can happen, that some solutions are lost.Recently an efficient algorithm to verify if the list of solutions obtained with the approach iscomplete was proposed in [38] however it is not yet implemented in freely available computeralgebra systems. The approach can be efficiently applied to study various mathematicalmodels where arises the problem of solving polynomial equations.When the origin is an isochronous center, we found that system (1.3) has at most threetopologically equivalent global structures, which are the global center at the origin, the neigh-borhood of equilibrium at infinity consists of one elliptic sector and three hyperbolic sectors,and the neighborhood of equilibrium at infinity consists of two hyperbolic sectors and twoparabolic sectors, as shown in Theorem 3.1. The last result is the investigation of local bi-furcations of critical periods in a neighborhood of the center. We proved that the order ofweak center at the origin is at most 3 when parameters belong to the center variety and atmost 3 critical periods can be bifurcated from the weak center of system (4.4), as shown inTheorem 4.3.
Acknowledgements
The first author acknowledges the financial support from the Slovenian Research Agency(research core funding No. P1-0306). The second author is partially supported by a CAPESgrant. The third author has received funding from the European Union’s Horizon 2020research and innovation programme under the Marie Sklodowska-Curie grant agreement No655212, and is partially supported by the National Natural Science Foundation of China (No.11431008) and the RFDP of Higher Education of China grant (No. 20130073110074). Thefirst, second and third authors are also supported by Marie Curie International Research StaffExchange Scheme Fellowship within the 7th European Community Framework Programme,FP7-PEOPLE-2012-IRSES-316338. The forth author is partially supported by the NationalNatural Science Foundation of China (No. 11501370). The first author thanks ProfessorMaoan Han for fruitful discussions on the work.
Appendix
Here are listed the first two pairs of the linearizability quantities of system (1.3). i =10 a + 9 a + 4 b − a b + b − b + 10 b b + 10 b − b ,j =2 a b − b b − b b + b ,i =168 a a − a b − a b − b − a b − a a b + 40 a b b − a b − b b − a b − b + 12 a b − b b + 72 a b b − b b + 18 b − a b b − a b b − b b − a b b b + 47 b b b − b b b − a b − a b + 12 b b − a b b + 61 b b − b b + 260 b b + 200 b − a b b − b b b + 84 a b b − b b b + 27 b + 132 a b + 81 a b − b b + 207 a b b − b b − a b + 81 b b − b b b − b b + 135 b ,j =224 a b + 240 a a b − a b − a b b + 6 a b b + 40 b b + 124 a b b − b b − a b b + 54 b b b − a b − a a b + 104 a b b + 40 a b b + 27 a b b + 38 b b b + 77 a b b − b b + 24 a b b + 39 b b b + 140 a b b − b b b − a b − b b − a b − a b − b b − a b b − b b − b b b + 6 b b + 30 b b − a b b + 105 b b b + 108 a b b + 84 b b b − b b . References [1] V. V. Amel’kin, N. A. Lukashevich, A. P. Sadovskii,
Nonlinear Oscillations in Second Order Systems (Russian), Belarusian State University, Minsk, 1982.[2] A. A. Andronov, E. A. Leontovitch, I. I. Gordon, A. G. Maier,
Qualitative Theory of Second-OrderDynamic Systems , Israel Program for Scientific Translations, John Wiley and Sons, New York, 1973.[3] E. A. Arnold, Modular algorithms for computing Grobner bases,
J. Symbolic Comput. (2003) 403–419.[4] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from anequilibrium position of focus or center type, Mat. Sb. (1952) 181–196; Amer. Math. Soc. Transl. (1954) 181–196.[5] I. L. Buchbinder, S. D. Odintsov, I. L. Shapiro,
Effective Action in Quantum Gravity , p. 282, IOPPublishing Ltd, 1992.[6] J. Chavarriga, I. A. Garc´ıa, J. Gin´e, Isochronicity into a family of time-reversible cubic vector fields,
Appl. Math. Comput. (2001) 129-145.
INEARIZABILITY AND CRITICAL PERIOD BIFURCATIONS OF RICCATI SYSTEM 17 [7] J. Chavarriga, J. Gin´e, I. A. Garc´ıa, Isochronous centers of a linear center perturbed by fourth degreehomogeneous polynomial,
Bull. Sci. Math. (1999) 77-96.[8] T. Chen, W. Huang, D. Ren, Weak centers and local critical periods for a Z -equivariant cubic system, Nonlinear Dyn. (2014) 2319–2329.[9] X. Chen, W. Huang, V. G. Romanovski, W. Zhang, Linearizability conditions of a time-reversible quartic-like system, J. Math. Anal. Appl. (2011) 179–189.[10] X. Chen, W. Huang, V. G. Romanovski, W. Zhang, Linearizability and local bifurcation of critical periodsin a cubic Kolmogorov system,
J. Comput. Appl. Math. (2013) 86–96.[11] C. Chicone, M. Jacobs, Bifurcation of critical periods for planar vector fields,
Trans. Amer. Math. Soc. (1989) 433–486.[12] C. Christopher, C. Devlin, Isochronous centers in planar polynomial systems,
SIAM J. Math. Anal. (1997) 162–177.[13] C. Christopher, C. Rousseau, Nondegenerate linearizable centres of complex planar quadratic and sym-metric cubic systems in C , Publ. Mat. (2001) 95–123.[14] A. Cima, A. Gasull, V. Ma˜nosa, F. Ma˜nosas, Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math. (1997) 471-501.[15] A. Cima, A. Gasull, F. Ma˜nosas, Period function for a class of Hamiltonian systems, J. DifferentialEquations (2000) 180–199.[16] A. Cima, A. Gasull, P. Silvab, On the number of critical periods for planar polynomial systems,
NonlinearAnal. (2008) 1889–1903.[17] W. Decker, G.-M. Greuel, G. Pfister, H. Sh¨onemann, Singular
SINGULAR , 2010.[19] F. Dumortier, J. Llibre, J. C. Art´es,
Qualititive theory of planar differential systems , Springer–Verlag,Berlin, 2006.[20] B. Ferˇcec, V. Levandovskyy, V. G. Romanovski, D. S. Shafer, Bifurcation of critical periods of polynomialsystems,
J. Differential Equations (2015) 3825–3853.[21] W. Fernandes, V. G. Romanovski, M. Sultanova, Y. Tang, Isochronicity and linearizability of a planarcubic system,
J. Math. Anal. Appl. (2017) 795–813.[22] A. Gasull, Y. Zhao, Bifurcation of critical periods from the rigid quadratic isochronous vector field,
Bull.Sci. Math. (2008) 291–312.[23] J. Gin´e, Zh. Kadyrsizova, Y.-R. Liu, V. G. Romanovski, Linearizability conditions for Lotka-Volterraplanar complex quartic systems having homogeneous nonlinearities,
Comput. Math. Appl. (2011)1190–1201.[24] K. M. Al. Haider, On isochronicity of Hamiltonian second order differential systems with polynomials ofdegree 3, Vestnik BSU. Ser. 1 (1983) 63-67.[25] M. Han, Bifurcation Theory of Limit Cycles , Science Press, Beijing, 2013.[26] M. Han, Liapunov constants and Hopf cyclicity of Li´enard systems,
Ann. Differential Equations (1999)113-126.[27] C. Huygens, Horologium Oscillatorium , 1673.[28] X. Jarque, J. Villadelprat, Nonexistence of isochronous centers in planar polynomial Hamiltonian systemsof degree four,
J. Differential Equations (2002) 334–373.[29] J. Li, Y. Lin, Normal form of planar autonomous system and periodic critical points of closed orbits,
Acta Math. Sinica (1991) 490-501.[30] N. Li, M. Han, V. G. Romanovski, Cyclicity of some Li´enard Systems, Communications on pure andapplied analysis (2015) 2127-2150.[31] J. Llibre, V. G. Romanovski, Isochronicity and linearizability of planar polynomial Hamiltonian systems, J. Differential Equations (2015) 1649–1662.[32] J. Llibre, C. Valls, Algebraic invariant curves and first integrals for Riccati polynomial differential systems,
Proceedings of the American Mathematical Society (2014) 3533–3543.[33] J. Llibre, C. Valls, Liouvillian first integrals for generalized Riccati polynomial differential systems,
Ad-vanced Nonlinear Studies (2015) 951-961. [34] W. S. Loud, Behaviour of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations. (1964) 21-36.[35] P. Mardeˇsi´c, C. Rousseau, B. Toni, Linearization of isochronous centers, J. Differential Equations (1995) 67-108.[36] V. B. Matveev, M. A. Salle,
Darboux Transformations and Solitons , Springer, Berlin, 1991.[37] Z. Nehari,
Conformal mapping , Dover Publications Inc., New York, 1975.[38] N. Noro, K. Yokoyama, Verification of Gr¨obner basis candidates, in: H. Hong, C. Yap, (eds.) MathematicalSoftware ICMS 2014,
Lecture Notes in Computer Science , vol. 8592, 419-424, Springer, Berlin, 2014.[39] M. Nowakowski, H. C. Rosu, Newton’s laws of motion in the form of a Riccati equation,
Phys Rev E StatNonlin Soft Matter Phys. (2002) (4 Pt 2B):047602.[40] I. I. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, Dokl. Akad. Nauk SSSR (1968) 768-771;
Soviet Math. Dokl. (1968) 1205-1209.[41] V. G. Romanovski, X. Chen, Z. Hu, Linearizability of linear systems perturbed by fifth degree homoge-neous polynomials, J. Phys. A. (2007) 5905-5919.[42] V. G. Romanovski, M. Preˇsern, An approach to solving systems of polynomials via modular arithmeticswith applications, J. Comput. Appl. Math. (2011) 196-208.[43] V. G. Romanovski, D. S. Shafer,
The Center and cyclicity Problems: A computational Algebra Approach ,Birkhauser, Boston, 2009.[44] H. C. Rosu, F. Aceves de la Cruz, One-parameter Darboux-transformed quantum actions in Thermody-namics,
Physica Scripta (2002) 377–382.[45] C. Rousseau, B. Toni, Local bifurcations of critical periods in vector fields with homogeneous nonlinearitiesof the third degree, Can. J. Math. (1993) 473–484.[46] C. Rousseau, B. Toni, Local bifurcation of critical periods in the reduced Kukles system, Canad. Math.Bull. (1997) 338–358.[47] P. S. Wang, M. J. T. Guy, J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull. (1982) 2–3.[48] Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017.[49] P. Yu, M. Han, Critical periods of planar revertible vector field with third-degree polynomial functions, Int. J. Bifurc. Chaos. (2009) 419-433.[50] M. I. Zelekin, Homogeneous Spaces and Riccati Equation in Variational Calculus (in Russian), Factorial,Moskow, 1998.[51] Z. Zhou, V. G. Romanovski, J. Yu, Centers and limit cycles of a generalized cubic Riccati system, preprint,http://arxiv.org/abs/1706.00099. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China
E-mail address : [email protected] (V.G. Romanovski), [email protected] (Y. Tian) Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova17, Maribor, SI-2000 Maribor, Slovenia Faculty of Natural Science and Mathematics, University of Maribor, Koroˇska c.160, Mari-bor, SI-2000 Maribor, Slovenia Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova2, Maribor, SI-2000 Maribor, Slovenia Instituto de Ciˆencias Matem´aticas e de Computac¸˜ao - USP, Avenida Trabalhador S˜ao-carlense, 400, 13566-590, S˜ao Carlos, Brazil
E-mail address : [email protected] (W. Fernandes) School of Mathematical Science, Shanghai Jiao Tong University, Dongchuan Road 800,Shanghai, 200240, P.R. China
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