UNIVERSIDADE DE SÃO PAULO

UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação ISSN 010-577 GLOBAL DYNAMICAL ASPECTS OF A GENERALIZED SPROTT E DIFFERENTIAL SYSTEM REGILENE OLIVEIRA CLAUDIA VALLS N o 41 NOTAS DO ICMC SÉRIE MATEMÁTICA São Carlos SP Set./015

Global dynamical aspects of a generalized Sprott E differential system Regilene Oliveira 1 and Claudia Valls 1 Departamento de Matemática, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, C.P 668, 1.560-970, São Carlos, SP, Brazil regilene@icmc.usp.br Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049 001, Lisboa, Portugal cvalls@math.ist.utl.pt Abstract In this paper we consider some global dynamical aspects of the generalized Sprott E differential system ẋ = ayz + b, ẏ = x y, ż = 1 4x, where a, b R are parameters and a 0. This is a very interesting chaotic differential system with one equilibria for all values of the parameters. We show that for b sufficiently small it can exhibit two limit cycles emerging from the classical Hopf bifurcation at the equilibrium point p = 1/4, 1/16, 0. We use the Poincaré compactification for a polynomial vector field in R to do a global analysis of the dynamics on the sphere at infinity. To show how the solutions reach the infinity we study the existence of invariant algebraic surfaces and its Darboux integrability. Keywords: Hopf bifurcation; Poincaré compactification; invariant algebraic surface; Sprott E system 1 Introduction and statement of the main results Chaos, an interesting phenomenon in nonlinear dynamical systems, has been developed and intensively studied in the past decades. A chaotic system is a nonlinear deterministic system that presents a complex and unpredictable behavior. The now-classic Lorenz system has motivated a great deal of interest and investigation of D-autonomous chaotic systems with simple nonlinearities, such as the Lorenz system [8], the Rössler system [9] and the Chen system[]. All these system have seven terms and have two or one quadratic nonlinearities. Sprott [10] found 19 simple chaotic systems called Sprott systems with no more than three equilibria, one or two quadratic nonlinearities and with less than seven terms. In [11], Wang and Chen proposed a generalization of the Sprott E system that we will call the Wang Chen system with a single stable node focus equilibria and 1 scroll chaotic attractor. This new system is very interesting because for a -dimensional autonomous quadratic system with a single stable node focus equilibrium one typically would expect the non existence of a chaotic attractor. We recall that the Lorenz system and the Rössler system are all of hyperbolic type, while the Wang Chen system is not hyperbolic. In fact, they proposed a system which is the original Sprott E system with a new parameter that causes a delay feedback. Since the stability of the single equilibrium changes from one system to the other the Wang Chen system is not topologically equivalent to the Sprott E system. In the investigation of chaos theory and its applications it is very important to generate new chaotic systems or enhance complex dynamics and topological structure based on the existing of chaotic attractors. It is also very important to study the stability of the equilibria of the system. This is a very challenging task, but one of the mechanisms could be the addition of new parameters to a given system. 1

Figure 1: The chaotic atractor of system 1 when a = 1.1 and b = 0.006: D views on the xy-plane, yz-plane, xz-plane and the Dview. In this paper we modify the Wang Chen system by considering more parameters in the nonlinear part, expecting to cause more chaotic behavior. Although the most natural way would be to add two parameters in the quadratic part the system has only two quadratic terms it is easy to see that with a linear change of coordinates the system with two additional parameters can be reduced to a system with only one additional parameter. More precisely, we study the following generalization of the Sprott E system x = ayz + b, y = x y, 1 z = 1 4x. where a, b R are parameters and a 6= 0. This system for a 6= 1 when a = 1 system 1 is the Wang Chen system also generates a 1-scroll chaotic attractor as shown in Figure 1. The first goal is to study all possible bifurcations which occurs at the equilibrium point p = 1/4, a/16, 16b/a of system 1. The first bifurcations that we may study are the codimension one bifurcations. Three elementary static bifurcations are associated with a simple zero eigenvalue of the Jacobian matrix at the equilibrium point: saddle-node, transcritical and pitchfork bifurcations. Easy calculations show that for any value of a, b R with a 6= 0 system 1 has never a zero eigenvalue. So we will study the other codimension one bifurcation: the Hopf bifurcation. We recall that for the arguments above we will not have a zero Hopf bifurcation. For the same reason our system 1 will not exhibit the well known codimension two bifurcation of Bogdanov Takens. We will study analytically all possible classical and degenerate Hopf bifurcations that occurs at the equilibrium point p of system 1. For this, we will use the classical projection method to compute the

Lyapunov coefficients associated to the Hopf bifurcations. More precisely, our first main result, concerning Hopf bifurcations is the following. Theorem 1. The following statements hold a Let C = {a, b R : a > 0, b = 0}. If a, b C then the Jacobian matrix of system 1 in p has one real eigenvalue 1 and a pair of purely imaginary eigenvalues ±i a/. b The first Lyapunov coefficient at p for the parameter values in C is given by a 16 40a + a l 1 a, 0 = 1 + a4 + a56 + 69a + a. If 16 40a + a 0 then system 1 has a transversal Hopf point at p for b = 0 and a > 0. More precisely, if a < 45 6 or a > 45 + 6 then the Hopf point at p is asymptotic stable weak attractor focus and for each b > 0 but sufficiently close to zero there exists a stable limit cycle near the unstable equilibrium point p. If 45 6 < a < 45+ 6 then the Hopf point at p is unstable weak repelling focus and for each b < 0 but sufficiently close to zero there exists an unstable limit cycle near the asymptotically stable equilibrium point p. c The second Lyapunov coefficient at p for a = 45 6 and b = 0 is given by l 45 6, 0 = 56 6781759746 + 1095848414 6 48101859918 Since l 45 6, 0 > 0 system 1 has a transversal Hopf point of codimension at p for the parameters a = 45 6 and b = 0 which is unstable. d The second Lyapunov coefficient at p for a = 45 + 6 and b = 0 is given by l 45 + 6, 0 = 566781759746 + 1095848414 6 48101859918 Since l 45 + 6, 0 < 0 system 1 has a transversal Hopf point of codimension at p for the parameters a = 45 + 6 and b = 0 which is stable. In Theorem 1 we have analyzed the Hopf and degenerate Hopf bifurcations of system 1. We have analytically proved that there exist two points in the parameter space for which the equilibrium point p is a codimension Hopf point. With the analytical data provided in the analysis of the proof of Theorem 1 we will conclude a qualitative information of the dynamical aspects of system 1. There are regions in the parameter space where system 1 has two limit cycles bifurcating from the equilibrium point p which are described as follows: for l 1 < 0 and b > 0 with l 1 b > 0 for the parameters where l > 0 and for l 1 > 0 and b < 0 with l 1 b > 0 for the parameters where l < 0. Theorem 1 will be proved in Section. For a review of the projection method described in [5] and the calculation of the first and second Lyapunov coefficients, see Section.1. We note that system Sprott E has no parameters so it can not present any bifurcation and the Wang Chen system has no codimension two transversal Hopf points. Now we continue the study of the global dynamics of system 1 by studying its behavior at infinity. For that we shall use the Poincaré compactification for a polynomial vector field in R see Section. for a brief description of this technique and for the definition of Poincaré sphere. Theorem. For all values of a R \ {0} and b R, the phase portrait of system 1 on the Poincaré sphere S is topologically equivalent to the one shown in Figure. > 0. > 0.

Figure : Global phase portrait of system 1 on the Poincaré sphere at infinity. Theorem is proved in Section 4. Note that the dynamics at infinity do not depend on the value of the parameter b because it appears in the constant terms of system 1. It depends on the parameter a but the global phase portraits at the Poincaré sphere for different values of a are topologically equivalent. From Figure we have four equilibrium points at infinity, two nodes and two cusps and there are no periodic orbits on the sphere. The Poincaré sphere at infinity is invariant by the flow of the compactified systems. A good way to understand how the solutions approach the infinity is studying the behavior of the system along of invariant algebraic surfaces, if they exist. More precisely, if system 1 has an invariant algebraic surface S, then for any orbit γ not starting on S either αγ S and ωγ S, or αγ S and αγ S for more details see Theorem 1. of [1] and, αγ and ωγ are the α limit and ω limit of γ, respectively for more details on the ω and α limit sets see for instance Section 1.4 of [4]. This property is the key result which allows to describe completely the global flow of our system when it has an invariant algebraic surface and, consequently, how the dynamics approach the infinity. Guided by this we will study the existence of invariant algebraic surfaces for system 1. The knowledge of the algebraic surfaces and the so called exponential factors see again Section. for definitions allow us to characterize the existence of Darboux first integrals. It is worth mentioning that the existence of one or two first integrals for system 1 will much contribute to understand its dynamics and so its chaotic behavior. Theorem. The following statements holds for system 1 a It has neither invariant algebraic surfaces, nor polynomial first integrals. b The unique exponential factor is expz and linear combinations of it. Moreover the cofactor of expz is 1 4x. c It has no Darboux first integrals. The paper is organized as follows. In Section we present some preliminaries. In Section we prove Theorem 1. The dynamics at infinity is studied in Section 4. In Section 5 we prove Theorem. Preliminaries.1 Hopf bifurcation. 4

In this section we present a review of the projection method used to compute the Lyapunov constants associated to Hopf bifurcations described in [5]. Consider the differential system ẋ = fx, µ, where x R and µ R are respectively vectors and parameters. Assume that f is quadratic and that x, µ = x 0, µ 0 is an equilibrium point of the system. Denoting the variable x x 0 by x we write where A = f x 0, µ 0 and, for i = 1,, F x = fx, µ 0 as F x = Ax + 1 Bx, x, B i x, y = j,k=1 F i µ µ j µ k µ=0 x j y k. 4 Suppose that x 0, µ 0 is an equilibrium point of where the Jacobian matrix A has a pair of purely imaginary eigenvalues λ, = ±iω 0, ω 0 > 0, and admits no other eigenvalues with zero real part. Let T c be the generalized eigenspace of A corresponding to λ,, i.e., the largest subspace invariant by A on which the eigenvalues are λ,. Let p, q C be vectors such that Aq = iω 0 q, A t p = iω 0 p, < p, q >= p i q i = 1, where A T denotes the transposed matrix of A and p is the conjugate of p. Note that any y T c can be represented by y = wq + wq, where w =< p, q > C. The -dimensional center manifold associated to the eigenvalues λ, = ±iω 0 can be parameterized by the variables w and w by the immersion of the form x = Hw, w, where H : C R has a Taylor expansion of the form Hw, w = wq + w q + j+k 6 i=1 1 j!k! h jkw j w k + O w 7, with h jk C and h jk = h kj. So substituting this expression in we get the following equation H w ẇ + H w ẇ = F Hw, w 5 where F is given by. The vectors h jk are obtained solving the linear systems defined by the coefficients of 5, taking into account the coefficients of F in the expressions and 4. So system 5 on the chart ω for a central manifold, is writing as with G jk C. More precisely, we have ẇ = iω 0 w + 1 w w + 1 1 G w w 4 + 1 144 G 4w w 6 + O w 8, h 11 = A 1 Bq, q, h 0 = iω 0 I A 1 Bq, q, where I is the identity matrix. For the cubic terms, the coefficients of the terms w in 5, we have h 0 = iω 0 I A 1 Bq, h 0. 5

From the coefficients of the terms w w in 5, in order to solve h 1 must take We define the first Lyapunov coefficient l 1 by G 1 =< p, Bq, h 0 + Bq, h 11 >. l 1 = 1 ReG 1. The complex vector h 1 can be found by solving the 4-dimensional system iω0 I A q h1 Bp, h0 + Bq, h = 11 G 1 q, 6 q 0 s 0 with the condition < p, h 1 >= 0. From the coefficients of the terms w 4, w w and w w in 5 one obtain respectively Defining h 40 = 4iω 0 I A 1 Bh 0, h 0 + 4Bq, h 0, h 1 = iω 0 I A 1 Bq, h 1 + Bq, h 0 + Bh 0, h 11 G 1 h 0, h = A 1 Bh 11, h 11 + Bq, h 1 + Bq, h 1 + Bh 0, h 0. H =6Bh 11, h 1 + Bh 0, h 0 + Bh 1, h 0 + Bq, h + Bq, h 1 6G 1 h 1 G 1 h 1, we have that the second Lyapunov coefficient l is where G =< p, H >.. Poincaré compactification Consider in R the polynomial differential system l = 1 ReG, ẋ = P 1 x, y, z, ẏ = P x, y, z, ż = P x, y, z, or equivalently its associated polynomial vector field X = P 1, P, P. The degree n of X is defined as n = max {degp i : i = 1,, }. Let S = {y = y 1, y, y, y 4 : y = 1} be the unit sphere in R 4 and S + = {y S : y 4 > 0} and S = {y S : y 4 < 0} be the northern and southern hemispheres of S, respectively. The tangent space of S at the point y is denoted by T y S. Then the tangent plane can be identified with R. by T 0,0,0,1 S = {x 1, x, x, 1 R 4 : x 1, x, x R } Consider the identification R = T 0,0,0,1 S and the central projection f ± : T 0,0,0,1 S S ± defined f ± x = ± x 1, x, x, 1, where x = 1 + x 1/ x i. Using these central projections R is identified with the northern and southern hemispheres. The equator of S is S = {y S : y 4 = 0}. The maps f ± define two copies of X on S, one Df + X in the northern hemisphere and the other Df X in the southern one. Denote by X the vector field on S \ S = S + S, which restricted to S + i=1 6

coincides with Df + X and restricted to S coincides with Df X. Now we can extend analytically the vector field Xy to the whole sphere S by px = y4 n 1 Xy. This extended vector field px is called the Poincaré compactification of X on S. As S is a differentiable manifold in order to compute the expression for px, we can consider the eight local charts U i, F i, V i, G i, where U i = {y S : y i > 0} and V i = {y S : y i < 0} for i = 1,,, 4. The diffeomorphisms F i : U i R and G i : V i R for i = 1,,, 4 are the inverse of the central projections from the origin to the tangent hyperplane at the points ±1, 0, 0, 0, 0, ±1, 0, 0, 0, 0, ±1, 0 and 0, 0, 0, ±1, respectively. Now we do the computations on U 1. Suppose that the origin 0, 0, 0, 0, the point y 1, y, y, y 4 S and the point 1, z 1, z, z in the tangent hyperplane to S at 1, 0, 0, 0 are collinear. Then we have and, consequently 1 y 1 = z 1 y = z y = z y 4 F 1 y = y /y 1, y /y 1, y 4 /y 1 = z 1, z, z defines the coordinates on U 1. As y /y1 1/y 1 0 0 DF 1 y = y /y1 0 1/y 1 0 y 4 /y1 0 1/y 1 0 and y n 1 4 = z / z n 1, the analytical vector field px in the local chart U 1 becomes where P i = P i 1/z, z 1 /z, z /z. z n z n 1 z1 P 1 + P, z P 1 + P, z P 1, In a similar way, we can deduce the expressions of px in U and U. These are where P i = P i z 1 /z, 1/z, z /z, in U and z n z n 1 z1 P + P 1, z P + P, z P, z n z n 1 z1 P + P 1, z P + P, z P, with P i = P i z 1 /z, z /z, 1/z, in U. The expression for px in U 4 is z n+1 P 1, P, P and the expression for px in the local chart V i is the same as in U i multiplied by 1 n 1, where n is the degree of X, for all i = 1,,, 4. Note that we can omit the common factor 1/ z n 1 in the expression of the compactification vector field px in the local charts doing a rescaling of the time variable. From now on we will consider only the orthogonal projection of px from the northern hemisphere to y 4 = 0 which we will denote by px again. Observe that the projection of the closed northern hemisphere is a closed ball of radius one denoted by B, whose interior is diffeomorphic to R and whose boundary S corresponds to the infinity of R. Moreover, px is defined in the whole closed ball B in such way that the flow on the boundary is invariant. The vector field induced by px on B is called the Poincaré compactification of X and B is called the Poincaré sphere. All the points on the invariant sphere S at infinity in the coordinates of any local chart U i and V i have z = 0. 7

. Integrability theory We start this subsection with the Darboux theory of integrability. As usual C[x, y, z] denotes the ring of polynomial functions in the variables x, y and z. Given f C[x, y, z] \ C we say that the surface fx, y, z = 0 is an invariant algebraic surface of system 1 if there exists k C[x, y, z] such that ayz + b f x + x y f + 1 4x f y z = kf. 7 The polynomial k is called the cofactor of the invariant algebraic surface f = 0 and it has degree at most 1. When k = 0, f is a polynomial first integral. When a real polynomial differential system has a complex invariant algebraic surface, then it has also its conjugate. It is important to consider the complex invariant algebraic surfaces of the real polynomial differential systems because sometimes these forces the real integrability of the system. Let f, g C[x, y, z] and assume that f and g are relatively prime in the ring C[x, y, z], or that g = 1. Then the function expf/g C is called an exponential factor of system 1 if for some polynomial L C[x, y, z] of degree at most 1 we have ayz + b expf/g x + x y expf/g y + 1 4x expf/g z = L expf/g. 8 As before we say that L is the cofactor of the exponential factor exp f/g. We observe that in the definition of exponential factor if f, g C[x, y, z] then the exponential factor is a complex function. Again when a real polynomial differential system has a complex exponential factor surface, then it has also its conjugate, and both are important for the existence of real first integrals of the system. The exponential factors are related with the multiplicity of the invariant algebraic surfaces, for more details see [], Chapter 8 of [4], and [6, 7]. Let U be an open and dense subset of R, we say that a nonconstant function H : U R is a first integral of system 1 on U if Hxt, yt, zt is constant for all of the values of t for which xt, yt, zt is a solution of system 1 contained in U. Obviously H is a first integral of system 1 if and only if ayz + b H x + x y H + 1 4x H y z = 0, for all x, y, z U. A first integral is called a Darboux first integral if it is a first integral of the form f λ1 1 f λp p F µ1 1 F µq q, where f i = 0 are invariant algebraic surfaces of system 1 for i = 1,... p, and F j are exponential factors of system 1 for j = 1,..., q, λ i, µ j C. The next result, proved in [4], explain how to find Darboux first integrals. Proposition 4. Suppose that a polynomial system 1 of degree m admits p invariant algebraic surfaces f i = 0 with cofactors k i for i = 1,..., p and q exponential factors expg j /h j with cofactors L j for j = 1,..., q. Then, there exist λ i and µ j C not all zero such that if and only if the function f λ1 1... f p λp is a Darboux first integral of system 1. p λ i K i + i=1 exp g1 h 1 q µ j L j = 0, 9 j=1 µ1... exp µq gq h q 8

. The following result whose proof is given in [6, 7] will be useful to prove statement b of Theorem Lemma 5. The following statements hold. a If expf/g is an exponential factor for the polynomial differential system 1 and g is not a constant polynomial, then g = 0 is an invariant algebraic surface. b Eventually expf can be an exponential factor, coming from the multiplicity of the infinity invariant plane. Hopf bifurcation In this section we prove Theorem 1. We will separate each of the statements in Theorem 1 in different subsections. Proof of Theorem 1a. System 1 has the equilibrium point p = 1/4, a/16, 16b/a with a 0. The proof is made computing directly the eigenvalues at the equilibrium point. The characteristic polynomial of the linear part of system 1 at the equilibrium point p is pλ = a 4 a 4 + 8b λ λ λ. As pλ is a polynomial of degree, it has either one, two then one has multiplicity, or three real zeros. Imposing the condition pλ = λ kλ + β 10 with k, β R, k 0 and β > 0 we obtain a system of three equations that correspond to the coefficients of the terms of degree 0, 1 and in λ of the polynomial in 10. This system has only the solution k = 1, b = 0, β = a/, with a > 0. This completes the proof. Proof of Theorem 1b. We will compute the first Lyapunov coefficient at the equilibrium point p of system 1 with a, b C. We will use the projection method described in Section.1 with ω 0 = a/, x 0 = p, µ = a, b and µ 0 = a, 0. The linear part of system 1 at the equilibrium point p is 0 0 a/16 A = 1/ 1 0. 4 0 0 The eigenvalues of A are ±i a/ and 1. In this case, the bilinear form B evaluated at two vectors u = u 1, u, u and v = v 1, v, v is given by a Bu, v = u v + u v, u 1 v 1, 0. The normalized eigenvector q of A associated to the eigenvalue i a/ normalized so that q q = 1 is q = a4 + a a4 + a i,, 8 a. 56 + 69a + a a i The normalized adjoint eigenvector p such that A T p = ip, where A T is the transpose of the matrix A, so that p q = 1 is p = 56 + 69a + a i, 0,. a4 + a 8 9

The vectors h 11 and h 0 are a h 11 = 56 + 69a + a 0, 4 + a, 18, 1 h 0 = 56 + 69a + a ai a + i, a16 a + 44 + ai, 18 a a + i. a i Moreover, The first Lyapunov coefficient is given by G 1 = a/ 1 a + 15a / + 8 46a + 6a i a i. a i56 + 69a + a a 16 40a + a l 1 a, 0 = a + 1a + 456 + 69a + a. If 16 40a + a 0 then system 1 has a transversal Hopf point at p for b = 0 and a > 0. Note that the denominator of l 1 a, 0 is positive because a > 0. If 16 40a+a > 0 which corresponds to a < 45 6 or a > 45+ 6 then l 1 a, 0 is negative, so the Hopf point at p is a asymptotic stable weak attractor focus. If 16 40a + a < 0 which corresponds to 45 6 < a < 45 + 6 then l 1 a, 0 is positive, so the Hopf point at p is unstable weak repelling focus. This completes the proof. Proof of Theorem 1c and d. We will only proof statement c of Theorem 1 because the proof os statement d is analogous. We consider system 1 with b = 0 and a = 45 6. Guided by Section.1, first we compute h 1, solving system 6. Doing that we get s = 0 and h 1 = 608 18098178879 + 7885474 6 i, 0546 1815 6 + i965 + 4189 114 The vectors h 0, h 40, h 1 and h are, respectively 8 507 + 49 6 h 0 = h 1 0, h 0, h 0, 8088161446 508 6 + 5500 6769 i, 4. where 56 h 40 = 16585947155 h1 40, h 40, h 40, 64 h 1 = 5165189407 h1 1, h 1, h 1, 51 h = 5887565449 0, 5471691 10079780 6, 16 56419784 + 14671 6, h 1 0 = 94944 8760 6 + 1164 5 6 576 65 6i, h 0 = 19785 8077 6 7 5 6 + 967 65 6i, h 0 = 1755 + 7168 6 + 1800 5 6 516 65 6i, h 1 40 = 16 19510614 + 80446490 6 + 7017610 804605 i, h 40 = 654176161 744540 + 607980 + 18740 + 505780 6i 195577006 4714916 + 651169 6780, 6i h 40 = 161001195 + 1004895 6 + 45746 461144 i, 10

and so Finally h 1 1 = 8 796775675 + 5775 6 + 550156156 + 448974156 i, h 1 = 99758797 + 771064 6 + 1074914550 8710697748 i, 19 h 1 = 16 407670851 + 1668987 6 + 1946 17597119 i. G = 5195158 + 74110 + 10505 + 4600 6i 87506749 + 8607179, l 45 6, 0 = 56 6781759746 + 1095848414 6. 48101859918 Since l 45 6, 0 > 0, system 1 has a transversal Hopf point of codimension at p. This completes the proof. 4 Compactification of Poincaré In this section we investigate the flow of system 1 at infinity by analyzing the Poincaré compactification of the system in the local charts U i, V i for i = 1,,. From the results of Section. the expression of the Poincaré compactification px of system 1 in the local chart U 1 is given by ż 1 = 1 az 1z z 1 z bz 1 z, ż = az 1 z 4z + z bz z, ż = z az 1 z + bz. 11 For z = 0 which correspond to the points on the sphere S at infinity system 11 becomes ż 1 = 1 az 1z,, ż = az 1 z. This system has no equilibria. It follows from the Flow Box Theorem that the dynamics on local chart U 1 is equivalent to the one shown in Figure??, whose the solutions are given by parallel straight lines. The flow in the local chart V 1 is the same as the flow in the local chart U 1 because the compacted vector field px in V 1 coincides with the vector field px in U 1 multiplied by 1. Hence the phase portrait on the chart V 1 is the same as the one shown in the Figure reserving in an appropriate way the direction of the time. In order to obtain the expression of the Poincaré compactification px of system 1 in the local chart U we use again the results given in Section. From there we get the system ż 1 = z 1 + az + z 1 z + bz, ż = z 1z 4z 1 z + z z + z, ż = z 1 z z. 1 System 1 restricted to z = 0 becomes ż 1 = z 1 + az, ż = z 1z. 1 11

Figure : Phase portrait of system 1 on the Poincaré sphere at infinity in the local charts U 1 on the left-hand side, U on the center and U on the right-hand side. The origin is the unique equilibria of system 1 and it is a nilpotent point. Applying Theorem.5 in [4] we conclude that the origin is an stable node. It local dynamics on the local chart U is topologically equivalent to the one shown in Figure. Again the flow in the local chart V is the same as the flow in the local chart U shown in Figure by reversing in an appropriate way the direction of the time. Finally, the expression of the Poincaré compactification px of system 1 in the local chart U is ż 1 = az + 4z 1z + bz z 1 z, ż = z 1 z z + 4z 1 z z z z, ż = 4z 1 z z. Observe that system 14 restricted to the invariant z 1 z -plane reduces to ż 1 = az, ż = z 1. The solutions of this system behave as in Figure which corresponds to the dynamics of system 1 in the local chart U. Indeed the origin is a nilpotent equilibrium point and from Theorem.5 in [4] we conclude that the origin is a cusp in this case fx 0, F x = Bx, 0 = x and Gx 0, with m even. The flow at infinity in the local chart V is the same as the flow in the local chart U reversing appropriately the time. Proof of Theorem. Considering the analysis made before and gluing the flow in the three studied charts, taking into account its orientation shown in Figure 4, we have a global picture of the dynamical behavior of system 1 at infinity. The system has four equilibrium points on the sphere, two nodes and two cusps and there are no periodic orbits. We observe that the description of the complete phase portrait of system 1 on the sphere at infinity was possible because of the invariance of these sets under the flow of the compactified system. This proves Theorem. We remark that the behavior of the flow at infinity does not depend on the parameter b and the global phase portrait at the sphere for different values of a are topologically equivalent. 14 5 Darboux integrability In this section we prove Theorem. We first prove statement a proceeding by contradiction. Let f C[x, y, z]\c be an invariant algebraic surface of system 1 with cofactor k. Then k = k 0 +k 1 x+k y +k z for some k 0, k 1, k, k C. 1

Figure 4: Orientation of the local charts U i, i = 1,, in the positive endpoints of coordinate axis x, y, z, used to drawn the phase portrait of system 1 on the Poincaré sphere at infinity Figure. The charts V i, i = 1,, are diametrically opposed to U i, in the negative endpoints of the coordinate axis. First we will show that k = k = 0. Expanding f in powers of the variable y we get f = m j=0 f jx, zy j where f j are polynomials in the variables x, z and m N {0}. Computing the terms of y m+1 in 7 we obtain az f m x = k f m. Solving this linear differential equation we get f m = g m z exp k x, az where g m is an arbitrary smooth function in z. Since f m must be a polynomial we must have that either f m = 0 or k = 0. If k 0 then f = fx, z and so by 7 it must satisfy The linear terms in y in 15 satisfy ayz + b f + 1 4x f x z = k 0 + k 1 x + k y + k zf. 15 az f x = k k x f, that is, f = gz exp, az for some arbitrary smooth function g. Since f must be a polynomial and k 0 we must have that f = 0 in contradiction with the fact that f is an invariant algebraic surface. In short, k = 0. Expanding f in powers of the variable z we get f = m j=0 f jx, yz j where f j are polynomials in the variables x, y and m N {0}. Computing the terms of z n+1 in 7 we get ay f m x = k k x f m and so f m = g m y exp, ay where g m is an arbitrary smooth function in the variable y. Since f m must be a polynomial we must have that either f m = 0 or k = 0. If k 0 then f = fx, y and so by 7 ayz + b f x + x y f y = k 0 + k 1 x + k zf. 16 1

The linear terms in the variable z in 16 satisfy ay f x = k k x f, that is, f = gy exp ay for some arbitrary smooth function g. Since f must be a polynomial and k 0 we must have that f = 0 in contradiction with the fact that f is an invariant algebraic surface. So, k = 0. Let n be the degree of f. Expanding the invariant algebraic surface f in sum of its homogeneous parts we get f = n j=0 f jx, y, z where each f j x, y, z is a homogeneous polynomial in x, y, z of degree j. Without loss of generality we can assume that f n 0 and n 1. Computing the terms of degree n + 1 in 7 we have ayz f n x + x f n y = k 1xf n 17 or in other words, if we consider the linear partial differential operator of the form then equation 17 can be written as M = ayz x + x y, 18 Mf n = k 1 xf n. 19 The characteristic equations associated with the linear partial differential equation in 19 are dz dy = 0, dx dy = ayz x. This system of equations has the general solution z = d 1, x az y = d, where d 1 and d are constants of integration. According with the method of characteristics, we make the change of variables Its inverse transformation is x = u = x y az, v = y, w = z. 0 1/ u + aw v, y = v, z = w. 1 Under changes 0 and 1, equation 19 becomes the following ordinary differential equation for fixed u, w: 1/ u + aw v d f n dv = k f 1 n, where f n is f n, written in the variables u, v and w. In what follows, we always use θ to denote a function θx, y, z written in terms of the variables u, v and w. Using that dv 1/ = 1/ 1 1/ vu 1/ F 1, 1,, w av, u u + aw v 14

where F 1 is the hypergeometric function defined as a k b k F 1 a, b, c, z = z k, with x k = xx + 1 x + k 1 c k k! k=0 we get that 1 f n = k 1 1/ ḡ n u, wvu 1/ F 1, 1,, w av, u being ḡ n an arbitrary smooth function in u and w. So, f n x, y, z = f x y n u, v, w = ḡ n az, z k 1 1/ y 1 x y 1/ F 1 aw, 1,, ay z. x y az 1 Note that it follows from that F 1, 1,, ay z is never a polynomial. So, in order that f x az y n is a homogeneous polynomial of degree n we must have that k 1 = 0 and f n is a polynomial in the variables u and w. Consequently, the cofactor of every invariant algebraic surface of system 1 is constant, i.e, k = k 0 and f n = [n/] a l z n l x where [ ] stands for the integer part function of a real number. The terms of degree n in 7 are Mf n 1 = k 0 f n + y f n y + 4x f n z [n/] = k 0 a l z n l x [n/] + 4x a l n lz n l 1 x [n/] y l az azy a l lz n l x y l, az 4 [n/] y l az axy y l 1 az a l lz n l x y l 1. az Using transformations 0 and 1 and working in a similar way to solve f n we get the following ordinary differential equation for fixed u and w: / u + aw v d f n 1 dv [n/] = k 0 [n/] a l w n l u l awv a l lw n l u l 1 + 4 u + aw v [n/] a l n lw n l 1 u l a u + aw v [n/] v a l lw n l u l 1. Integrating this equation with respect to v and using the formula in together with dv / = / 1 / vu / F 1,,, w av, u u + aw v v dv u + aw v v dv u + aw v 1/ = 1/ / 7aw vu + av w / 1/ / 7aw / = / 1/ 5aw vu + av w 1/ / 1/ 5aw 1 vu/ F 1, 1,, w av, u 1 vu1/ F 1,,, w av, u 5 15

we obtain f n 1 = ḡ n 1 u, w + / / vf 1 / 1/ 5 [n/] vu + av w 1/ 1,,, w av u + 4/ / vf 1 1, 1,, av w u [n/] a l k 0 + l 5 a l lw n l u l 1 4/ / [n/] 7 w n l u l / [n/] vu + av w / a l 4n l + 6 7 l w n l 1 u l 1/, a l lw n l 1 u l 1 where ḡ n 1 is an arbitrary smooth function in u and w. Since f n 1 must be a homogeneous polynomial 1 of degree n 1 and neither F 1,,, av w 1 u nor F 1, 1,, av w u are polynomials we must have Note that the second condition in 6 implies that a l k 0 + l for l = 0,..., [n/], 5 a l 4n l + 6 7 l for l = 0,..., [n/]. 6 a l = 0 for l = 0,..., [n/] because l 0 and n l 0 with n 1. Therefore, from 4 we get that f n = 0 which is not possible. Note that a polynomial first integral is an algebraic invariant surface with cofactor k = 0. Proceeding as above with k 0 = k 1 = k = k = 0 we also obtain that there are no polynomial first integrals. This concludes the proof of statement a. Now we prove statement b. Let E = expf/g C be an exponential factor of system 1 with cofactor L. Then L = L 0 + L 1 x + L y + L z, where f, g C[x, y, z] with f, g = 1. From Theorem a and Lemma 5, we have that E = expf with f = fx, y, z C[x, y, z] C. It follows from equation 8 that f satisfies ayz + b f x + x y f + 1 4x f y z = L 0 + L 1 x + L y + L z, 7 where we have simplified the common factor expf. Let n be the degree of f. We write f = n i=0 f ix, y, z, where f i is a homogeneous polynomial of degree i. Without loss of generality we can assume that f n 0. Assume n > 1. Computing the terms of degree n + 1 in 8 we obtain ayz f n x + x f n y = 0 or using the operator M in 18 we have Mf n = 0. Proceeding as we did to solve 19 we obtain that f n becomes as in 4. Computing the terms of degree n in 7 we obtain Mf n 1 = y f [n/] n y + 4x f n z = azy a l lz n l x [n/] + 4x a l n lz n l 1 x y l 1 az [n/] y l az axy a l lz n l x y l 1. az which is 5 with k 0 = 0. Proceeding exactly in the same way as we did to solve 5 we get that f n = 0, which is not possible. So n = 1. 16

We can write f = a 1 x + a y + a z with a i C. Imposing that f must satisfy 7 we get f = a z with cofactor a 1 4x. This concludes the proof of statement b of Theorem. It follows from Proposition 4 and statements a and b that if system 1 has a Darboux first integral then there exist µ C \ {0} such that 9 holds, that is, such that µ1 4x = 0. But this is not possible. Hence, there are no Darboux first integrals for system 1 and the proof of statement c of Theorem is completed. Acknowledgements The first author is partially supported by the project FP7-PEOPLE-01-IRSES number 168, a CAPES grant number 88881.00454/01-01 and Projeto Temático FAPESP number 014/0004-. The second author is supported by FCT/Portugal through the project UID/MAT/04459/01. References [1] J. Cao and X. Zhang, Dynamics of the Lorenz system having an invariant algebraic surface, J. Math. Phys. 48 007, 1 1. [] G. Chen and T. Ueta,Yet another chaotic attractor. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 1999, 1465 1466. [] C. Christopher, J. Llibre and J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. of Math. 9 007, 6 117. [4] F. Dumortier, J. Llibre and J. C. Artés, Qualitative theory of planar differential systems, Universitext, Springer, New York, 006. [5] Yu. A. Kuznetsov, Elements of applied bifurcation theory, third ed., Applied Mathematical Sciences, vol. 1, Springer Verlag, New York, 004. [6] J. Llibre and X. Zhang, Darboux theory of integrability in C n taking into account the multiplicity, J. Differential Equations 46 009, 541 551. [7] J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in R n taking into account the multiplicity at infinity, Bull. Sci. Math. 1 009, 765 778. [8] E. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 0 196, 10 141. [9] O. E. Rössler, An equation for hyperchaos Phys. Lett. A 71 1979, 155 157. [10] J. C. Sprott, Some simple chaotic flows. Physical review E 50, 1994, part A, R647 R650. [11] X. Wang and G. Chen, Constructing a chaotic system with any number of equilibria, Nonlinear Dyn. 71 01, 49 46. 17