On the homoclinic orbits of the Lü system

In this paper we consider the Lü system [Lü & Chen , 2002] ẋ = a(y − x), ẏ = cy − xz, (1) ż = −bz + xy where a, b, c ∈ R+ are parameters. For the parameter set given by a = 36, b = 3 and c = 20 this system (1) is a chaotic system which bridges the gap between the Lorenz and the Chen attractors [Lü & Chen , 2002]. They all have the same symmetry, stability of equilibrium points, and similar bifurcations and topological structures, in among other things [Lü & et. al. , 2002]. In the study of the chaotic systems a particular role is played by the homoclinic orbits because the existence of these suggests the existence of chaotic dynamics, see [Guckenheimer & Holmes , 1983]. The system (1) has a saddle point at (0, 0, 0), for all positive parameter values. This paper concerned with the problem of finding conditions to ensure the existence of homoclinic orbits (joins the saddle equilibrium point to itself) of Lü system at origin. Since the introduction of the Lü system, several articles related to this system of equations have been published, see for instance [Leonov, 2012], [Leonov , 2013], [Leonov & Kuznetsov , 2015], [Leonov , 2016] and references therein. One central aspect concerns to the existence of homoclinic orbits related to (0, 0, 0).

For the parameter set given by a = 36, b = 3 and c = 20 this system (1) is a chaotic system which bridges the gap between the Lorenz and the Chen attractors [Lü & Chen , 2002].They all have the same symmetry, stability of equilibrium points, and similar bifurcations and topological structures, in among other things [Lü & et. al. , 2002].
In the study of the chaotic systems a particular role is played by the homoclinic orbits because the existence of these suggests the existence of chaotic dynamics, see [Guckenheimer & Holmes , 1983].The system (1) has a saddle point at (0, 0, 0), for all positive parameter values.This paper concerned with the problem of finding conditions to ensure the existence of homoclinic orbits (joins the saddle equilibrium point to itself) of Lü system at origin.

Equilibrium points and their stability
The system (1) is invariant under the transformation S(x, y, z) = (−x, −y, z).One easily verifies that the system (1) has three equilibrium points, E 0 = (0, 0, 0), which exists for any parameter values and a pair of symmetrically located equilibria ) in the case that bc > 0.
Lemma 1.If parameters a, b and c are positive, then the following statements about system (1) hold.
(ii) For c < a + b 3 , the equilibrium points E + and E − are stable.
Proof.The linearization of the vector field associated to (1) about an equilibrium point is given by the Jacobian matrix It easy to see that at (0, 0, 0) the linearization matrix has two negative eigenvalues λ 1 = −a and λ 2 = −b, while the other one is positive, λ 3 = c, so this equilibrium is unstable.
The linearization of (1) at the equilibrium points E ± has the characteristic polynomial given by As a consequence of the Routh-Hurwitz criterion we get that all of the roots of p(λ) are negative or have negative real part if only if So, all the roots have negative real parts when 0 < c < a + b 3 .The lemma is thus proved.
Since the equilibrium point at the origin is a saddle, then there are two invariant manifolds associated to it.A stable manifold W s , which is two-dimensional and a one-dimensional unstable manifold W u .This unstable manifold consists of (0, 0, 0) and a pair of orbits (separatrices), which are symmetrical to each other under the symmetry S(x, y, z) → (−x, −y, z).Linearization of the system at (0, 0, 0) yields that the tangent stable space T W s 0 is given by T W s = {y + z = 0}, and for the tangent unstable space T W u 0 ,

Existence of homoclinic orbits
In this section we transform the system (1) into a set of new equations which have already been studied by Belykh [1984] and then show the existence of the separatrices loops of the saddle point.Firstly, we recall that Zhang & et. al. [2012] proved that by assuming 2a > b > 2c > 0 and taking a Lyapunov-like function of the form where k 1 > 0, b 2a > k 2 > c a and τ = ak 1 + ak 2 − ck 2 + ak 2 2 the solutions of the Lü system (1) are globally bounded for all positive time.
February 27, 2017 13:37 ijbc˙mar-revision On the homoclinic orbits of the Lü system 3 Here we are considering the case 2a > b > 0 and a > c > 0. By non-linear space change of coordinates the system (1) can be reduced to the form Observe that the spatial transformation (3) is similar to the one used by Tigan & Llibre [2016] when they study the existence of homoclinic orbits in the Chen system.However, the transformation used here is simpler than the one adopted in this last reference since they use also a time reparametrization added to the spatial transformation.
The statement means that for any set of parameters (a, b, c) ∈ (R + ) 3 for which ac = 1, 2a > b > 0 and a > c > 0, the Lü system has two symmetrical homoclinic orbits to the origin equilibrium point.
In addition to this, we emphasize that Leonov [2013] considered the linear scaling, a = 0, , c → c a when studying Lü and Chen systems, such that the system (1) depends only on two parameters, namely b = b/a and c = c/a and, by using the Fishing principle (see [Leonov, 2012]), he was able to prove this result.
It is worth to observe that under the conditions set in Theorem 2, it follows that 2 − b/a > 0 and 1 − c/a > 0. Therefore, Theorem 1 is taking into account the case ac = 1 not considered in Theorem 2, obtaining a more complete picture for the Lü system.
According to Theorem 1, Lü system admits homoclinic orbits if the values of the parameters satisfy ac = 1, 2a − b > 0 and a − c > 0, so it means that a > 1, 0 < b < 2 and 0 < c < 1.In order to check out numerically our theoretical results, we were chosen the initial conditions very close to the origin.
The results are depicted in Figure 1.

Conclusion
The existence of homoclinic orbits of the Lü system has been examined with special attention to the parameter region with ac = 1, 2a − b > 0 and a − c > 0. To achieve this target we use a method based on comparison of differential equation systems given in [Belykh , 1984].We found that Lü system has homoclinic orbits at (0, 0, 0) for some parameter values, which are cases not covered in [Leonov, 2012].We perform numerical experiments to verify our theoretical results.The simulation results were produced using Mathematica R program.