Global dynamics of the Rössler system with conserved quantities

The subject of this paper concerns with a class of Rössler systems that admits conserved quantities. For this class of systems a complete description of the global dynamics in the Poincaré sphere is provided.


Introduction
In general, it is not easy to detect when a differential system has some kind of integrability. This question can cover up an entire book, see for instance [9]. The integrability of a differential system exhibits some recognizable features such as the existence of conserved quantities, or the presence of algebraic invariants, or the ability to give explicit solutions, see for instance [3].
In this paper, we work with the notion of integrability associated with conserved quantities, either with the existence of first integrals or conserved quantities independent of the time, or with the existence of invariants or conserved quantities depending on the time. Clearly, when a system presents first integrals or invariants, they strongly help to understand the dynamics of the system.
In this paper, we study the Rössler system [6]. This system is a three-dimensional system given byẋ = −y − z = P 1 (x, y, z), y = x + ay = P 2 (x, y, z), where a, b, c are real parameters and the dot denotes the derivative with respect to the independent variable t that we call the time.
The Rössler system is a famous dynamical model studied by many authors. Thus, related with this system there are more than 300 papers published (see MathSciNet) in which mainly the dynamical chaos is investigated.
In applications, the parameters a, b and c are considered positive real numbers and it is known that this system exhibits, for some values of the parameters, chaotic behavior. On the other hand, in [8] (see also [4]) a necessary and sufficient condition for the complete integrability of the system (1) is stated. More precisely, it is shown that the system (1) is completely integrable if and only if a = b = c = 0 (see theorem 1, part (iii) of [8]). In this case, the system (1) becomeṡ x = −y − z = P 1 (x, y, z), y = x = P 2 (x, y, z), z = xz = P 3 (x, y, z).
(2) This particular Rössler system has a Darboux first integral given by H 1 = z e −y and a polynomial first integral which are functionally independent.
Also in [8], it is shown that if a = c = 0 and b ∈ R, then the system (1) has as an invariant under the flow (see theorem 1, part (ii) of [8]).
In what follows, we will give a complete description of the dynamics of the integrable Rössler system (2) in the Poincaré sphere using the Poincaré compactification. We also give a description of the phase portrait of the system (1) for a = c = 0 and b = 0.
First, we study the general non-integrable system at the infinity that corresponds to analyze the system restricted to the boundary S 2 of an open ball diffeomorphic to R 3 , via the Poincaré compactification.
Second, we describe the dynamics of the integrable system in R 3 using the expression of the first integrals H 1 and H 2 . We also describe the asymptotic behavior of the orbits that go to or come from the infinity.
Finally, we analyze the behavior of the orbits of the system (1) with a = c = 0 and b = 0. The main results of this paper are the following.

Theorem 1.
On the Poincaré sphere at infinity represented by S 2 , the differential system (1) has two great circles (the ends of the planes x = 0 and z = 0) filled of singular points, and these are all the singular points at infinity. Moreover, the phase portrait on the infinity sphere S 2 is represented in figure 1.
Theorem 1 is proved in section 3.

Theorem 2.
The following statements hold for the system (2).
(a) There is a line (0, y, −y) with y ∈ R of singular points. Moreover, the plane z = 0 is invariant and on this plane the system has a linear center. (b) On the invariant paraboloid H 2 = h 2 with h 2 > −1, there is a homoclinic orbit to the equilibrium point (0, y 1 , −y 1 ) ∈ H −1 2 (h 2 ) with y 1 > 0 and a family of periodic orbits connecting this homoclinic orbit with the equilibrium point (0, y 2 , −y 2 ) ∈ H −1 2 (h 2 ) with y 2 < 0, see figure 2.  Theorem 2 is proved in section 4.
In particular, the system (1) for a = c = 0 and b = 0 has no equilibrium points or periodic orbits.
Theorem 4 is proved in section 5.
As far as we know, all the results of the previous theorems are new, with the exception of statements (a) and (b) of theorem 2 which were described in R 3 without proving them in [5]. Here they are proved and extended to infinity, i.e. to the Poincaré sphere.
For this purpose, we briefly present in section 2 the Poincaré compactification for the particular case considered in this paper, that is the Poincaré compactification in R 3 .
We must mention that the system (2) exhibits a Hamiltonian structure. Moreover, it admits a bi-Hamiltonian structure meaning that there are infinitely different ways of proving this Hamiltonian structure. This result follows from the result of section 5 of [1]. In fact, this result was known before, but in [1] its presentation is very clear. Of course, this Hamiltonian structure cannot be extended to the Poincaré sphere because there are stable equilibrium points at infinity.

Poincaré compactification in R 3
A polynomial vector field X in R n can be extended to a unique analytic vector field on the sphere S n . The technique for making such an extension is called the Poincaré compactification and allows us to study a polynomial vector field in a neighborhood of infinity, which corresponds to the equator S n−1 of the sphere S n . Poincaré introduced this compactification for polynomial vector fields in R 2 . Its extension to R n for n > 2 can be found in [2]. In this section, we describe the Poincaré compactification for polynomial vector fields in R 3 following closely what is made in [2].
In R 3 we consider the polynomial differential systeṁ or equivalently its associated polynomial vector field X = (P 1 , ; y = 1} be the unit sphere in R 4 , S + = {y ∈ S 3 ; y 4 > 0} and S − = {y ∈ S 3 ; y 4 < 0} be the northern and southern hemispheres of S 3 , respectively. The tangent space to S 3 at the point y is denoted by T y S 3 . Then the tangent plane We consider the central projections f + : T (0,0,0,1) S 3 → S + and f − : Through these central projection R 3 is identified with the northern and southern hemispheres. The equator of S 3 is S 2 = {y ∈ S 3 ; y 4 = 0}. Clearly, S 2 can be identified with the infinity of R 3 .
The maps f + and f − define two copies of X on S 3 : one D f + •X in the northern hemisphere and the other D f − • X in the southern one. Denote byX the vector field on S 3 \ S 2 which restricted to S + coincides with D f + • X and restricted to S − coincides with D f − • X.
The expression forX (y) on S + ∪ S − is where P i = P i (y 1 /|y 4 |, y 2 /|y 4 |, y 3 /|y 4 |). Written in this way,X (y) is a vector field in R 4 tangent to the sphere S 3 . Now, we can analytically extend the vector fieldX (y) to the whole sphere S 3 by considering where m is the degree of X. This extended vector field p(X ) is called the Poincaré compactification of X on S 3 .
As S 3 is a differentiable manifold, in order to compute the expression for p(X ) we can consider the eight local charts for i = 1, 2, 3, 4 are the inverses of the central projections from the origin to the tangent planes 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 2 , y 3 , y 4 ) ∈ S 3 and the point (1, z 1 , z 2 , z 3 ) in the tangent plane to S 3 at (1, 0, 0, 0) are collinear. Then we have 1/y 1 = z 1 /y 2 = z 2 /y 3 = z 3 /y 4 , and consequently F 1 (y) = (y 2 /y 1 , and y m−1 In a similar way, we can deduce the expression of p(X ) in U 2 and U 3 . These are When we work with the expression of the compactified vector field p(X ) in the local charts, we shall omit the factor 1/ ( z) m−1 . We can do that through a rescaling of the time variable.
We remark that all the points on the sphere at infinity S 2 in the coordinates of any local chart have z 3 = 0.
In what follows, we shall work with the orthogonal projection of p(X ) from the closed northern hemisphere to y 4 = 0 and we continue denoting this projected vector field by p(X ). Note that the projection of the closed northers hemisphere is a closed ball B of radius 1, whose interior is diffeomorphic to R 3 and whose boundary S 2 corresponds to the infinity of R 3 . Of course, p(X ) is defined in the whole closed ball B in such a way that the flow on the boundary is invariant. The new vector field on B is called the Poincaré compactification of X, and B is called the Poincaré ball, and ∂B = S 2 is called the Poincaré sphere at infinity.

Dynamical behavior of the Rössler system at infinity
Here, we will study the Poincaré compactification of the system (1) in the local charts U i and V i for i = 1, 2, 3 in order to understand the global behavior of the solutions at infinity.

Local chart U 1
Using the results obtained in section 2 we have that the Poincaré compactification Z 1 = p(X ) of the system (1) in the local chart U 1 is given bẏ In the points of the sphere S 2 that correspond to the points at infinity, we have z 3 = 0 and so the system (3) becomeṡ The last equation reflects the fact that the infinity (z 3 = 0) is invariant under the flow. From this system, we see that the system (1) has at infinity continuous singular points given by (z 1 , 0, 0). Moreover A z 1 = DZ 1 (z 1 , 0, 0) has, for each z 1 , the eigenvalues 1 and 0 with multiplicity 2. The orbits of the system in the local chart U 1 at infinity have the phase portrait given in figure 5. Note that the equilibria (z 1 , 0, 0) for z 1 ∈ R represent in the Poincaré sphere S 2 part of the equator.

Local chart U 2
In the same way using the results obtained in section 2 the Poincaré compactification Z 2 = p(X ) of the system (1) in the local chart U 2 iṡ In the points of S 2 (z 3 = 0) we havė Here, we have two lines of singular points given by (z 1 , 0, 0) and (0, z 2 , 0) with z 1 , z 2 ∈ R and the dynamics restricted to z 3 = 0 is given in figure 6. Moreover A z 1 = DZ 2 (z 1 , 0, 0) has for each z 1 the eigenvalues z 1 and 0 with multiplicity 2, and A z 2 = DZ 2 (0, z 2 , 0) has for each z 2 , the eigenvalue 0 with multiplicity 3.
As before we observe that the points (z 1 , 0, 0) with z 1 ∈ R represent half of equator of S 2 with endpoints (1, 0, 0) and (−1, 0, 0) and the points (0, z 2 , 0) with z 2 ∈ R represent half of the great circle connecting the north pole with the south pole.

Local chart U 3
The Poincaré compactification Z 3 = p(X ) of the system (1) in the local chart U 3 iṡ Restricted to S 2 , we obtaiṅ which implies that Z 3 admits the line (0, z 2 , 0), z 2 ∈ R of singular points. These singular points are such that A z 2 = DZ 3 (0, z 2 , 0) has for each z 2 the eigenvalue 0 with multiplicity 3. The phase portrait of Z 3 restricted to z 3 = 0 is given in figure 7.

Remark 5.
We observe that the flows in the V i charts for i = 1, 2, 3 are the same as the ones in the respective U i charts for i = 1, 2, 3 but with the time reversed because the compactified vector field p(X ) in V i coincides with the vector field in U i multiplied by −1 for each i = 1, 2, 3.

Dynamics of the Rössler system for the integrable case a = b = c = 0
In this section, we describe the dynamics of the Rössler system in R 3 . We restrict the analysis to the integrable case where the system admits two independent first integrals.
The orbits of the system are given by the intersection of the level surfaces H −1 We observe that H 1 = z e −y does not depend on x. So the graph of the level surface H 1 = h 1 is given by the translation of the curve z = h 1 e y in the yz-plane along the x-axis. Moreover, the level surfaces of H 2 = h 2 are paraboloids given by the rotation of the curve y 2 + 2z = h 2 in the yz-plane around the z-axis.
From the above observations to study the intersection between the level surfaces H 1 = h 1 and H 2 = h 2 , it is sufficient to study the intersection between the curves z e −y = h 1 and y 2 + 2z = h 2 in the yz-plane.   (0, 0). Now suppose that h 1 < 0. We study the intersection between C 1 h 1 and C 2 h 2 . So we have to solve the system  Observe that to solve the system (4) is equivalent to find the zeros of the function f (y) = 2h 1 e y + y 2 − h 2 with y ∈ R.
We have that Moreover, If −1 < h 1 − 1 e we have y * > 0 and f is strictly increasing in the interval (0, y * ) and strictly decreasing in the interval (y * , ∞).
Solving the system (5), we obtain λ = 1/2, z = −y = −(1 ± √ 1 + h 2 ). As y > 0 we take z = −y = −(1 + √ 1 + h 2 ). From equation (5), we obtain Now the case h 1 > 0 can be obtained as follows. In the same way as that in the previous case, if we take the tangency point given by , and in this case there is a unique zero of the function f . Also, if 0 < h 1 < h * 1 there are two zeros of the function f , and if h 1 > h * 1 there is no zero of f (see figure 9). This concludes the proof.

Proposition 7.
Suppose that h 2 < 0. The relative position between C 1 h 1 and C 2 h 2 in the yz-plane depends on h 1 and h 2 in the following way.  for h 2 < 0 and h 1 < 0. In these three pictures for h 2 < 0 and h 1 < 0. In these three pictures h 1 h 2 /2 Proof. As in the case h 2 0 we will study the zeros of the function f (y) = 2h 1 e y + y 2 − h 2 . However, first we consider a trivial case when h 1 0. In this case it is easy to see that So from now on we will restrict the analysis to the case h 1 < 0.
Case 1: Suppose that h 2 > −1. Here we will divide the study into two subcases. Subcase 1.1: 2h 1 < h 2 . In this case we have So there is a unique zero of f for y 0. A similar analysis as the one done in the previous proposition shows that the possible configuration of the curves C 1 h 1 and C 2 h 2 are given in figure 10, where the tangency point occurs when So there is no zero of f in (−∞, 0). Now, for y 0, we have f (0) 0 and lim y→∞ f (y) = −∞. This implies that there is at least one zero of f in (0, ∞). Moreover, Now note that the points of tangency between C 1 h 1 and C 2 h 2 (when they exist) occur at (y, −y) with y ∈ R. They are associated with the equilibria (0, y, −y), y ∈ R of the system (2). The intersection {H 1 = h 1 } ∩ {H 2 = h 2 } containing the point (0, y, −y) with y > 1 is formed by a closed curve and two other no-limited curves. For each y > 1, the closed curve corresponds to a homoclinic orbit and the no-limited ones to separatrices of the equilibrium.
On the other hand, when the intersection of C 1 h 1 and C 2 h 2 is not tangent we have two possibilities: either the curves are secant in a unique point and the corresponding intersection between the level curves represents a no-limited solution of the system (2), or the curves intersect into two points that correspond to a periodic solution. Note that in this case, we have a family of periodic solutions connecting the singular point (0, y, −y) with y < 1 with a homoclinic solution to the equilibrium point (0, y, −y) with y > 1.