A THREE-DIMENSIONAL NONLINEAR SYSTEM WITH A SINGLE HETEROCLINIC TRAJECTORY

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What’s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.


Introduction
During past two decades, the investigations on singular trajectories and bifurcations have been turned out to be the lands of great promise for nonlinear dynamical systems. This is because these investigations not only characterize the complex dynamics [13,25,36,37] of those models themselves, but also involve with numerous applications [3,4,9,10,38,39], such as electrophysics, heart tissue, neurons, cell signalling, planetary field and so on.
Homoclinic and heteroclinic trajectories are an important part of global bifurcation of dynamical system, to which many scientists and engineers [25,36,37] have paid a great deal of attention. Their occurrence often leads to the birth of chaos in corresponding systems. How many kinds of chaos are there under their occurrence? In the sense ofSi'lnikov type [23,[25][26][27], it is traditionally suggested that chaos occurring in 3D quadratic autonomous differential systems may be mainly classified into the following four cases: (i) chaos of theSi'lnikov homoclinic-trajectory type; (ii) chaos of theSi'lnikov heteroclinic-trajectory type; (iii) chaos of the hybrid type with bothSi'lnikov homoclinic and heteroclinic trajectories; (iv) chaos of other types.
Meanwhile, many important theories [25,36,37] and effective methods [12,14,28] have been developed and formulated in the course of the continuous research for detecting homoclinic and heteroclinic trajectories, for example, Poincaré map, Melnikov method, Lyapunov function, Fishing principle, a method of tracing the stable and unstable manifolds, etc.
Here, what we emphasize is the method Li et al formulated in [14] for proving the existence of heteroclinic trajectories: Lyapunov function combining the definitions of the α− and ω-limit set. Moreover, it has been demonstrated that this method can be effectively applied to Lorenz-type systems family: the Chen system [14], the Yang -Chen system or Yang system [18,40] ( in the sense of similar Vanecek and Celikovsky [26] , Yang-Chen system also connects the original Lorenz system and the original Chen system and represents a transition from one to the other), the T and Lü system [29], the general Lorenz family [19], other Lorenz-type systems [2,[15][16][17][30][31][32], the unified Lorenz-type system [33], the complex Lorenz system [34] and the 5D hyperchaotic system [35]. Precisely speaking, it is noticed that those aforementioned models have two symmetric or infinitely many heteroclinic trajectories to origin and the non-zero symmetric equilibria. In addition, as Fishing principle, this method itself has the advantage: one need not consider the mutual disposition of stable and unstable manifolds of a saddle equilibrium in contrast with another technique of proving the existence of homoclinic and heteroclinic trajectories. For related work, see also [1, 6-8, 20-22, 41-45].
However, in neighboring Lorenz-type systems, the scenario for an asymmetric heteroclinic trajectory has not been considered in any publications to the best of our knowledge. Therefore, the following questions naturally arise: (1) Whether does there exist such a model with a single heteroclinic trajectory or not?
(2) If there is such a model, whether or not is the aforementioned technique (combining the definitions of the α− and ω-limit set, and Lyapunov function) applicable to prove the existence of its heteroclinic trajectory?
(3) Except for the heteroclinic trajectory, whether do there exist other rich dynamics like the Lorenz-type system family? For example, chaotic attractors, Hopf bifurcation, invariant algebraic surface, etc.
In the present work, we devote to solving these problems one after another. Indeed, the new system proposed is found to have some other interesting dynamics, which are the essential differences with most of Lorenz or Lorenz-like systems.
1. Most of Lorenz or Lorenz-like systems possess singularly degenerate heteroclinic cycles and chaotic attractors whereas the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed.
2. The local dynamics of non-isolate equilibria (0, 0, z) for most of Lorenz or Lorenz-like systems are related to the variable z whereas the local dynamics of nonisolate equilibria (0, 0, z) of the new system has nothing with the variable z.
These discoveries make the new system proposed more interesting. The rest of this paper is organized as follows. Section 2 introduces the new 3D non-symmetric nonlinear system. Section 3 performs the local stability and bifurcation analysis of this system. Section 4 studies its global bifurcation problem of this system-the existence of homoclinic and heteroclinic trajectories. It is found that the system has a single heteroclinic trajectory but no homoclinic trajectories when b ≥ 3a > 0 and c = 0. Finally, some conclusions are drawn in Section 5.

New 3D nonlinear system
In [40], the authors studied the Lorenz-type system Obviously, in this system, the change rates of y and z are linear functions for the variable x. How about replacing x into x 2 ? Are there new dynamics to occur? Based on this idea, we introduce the following new 3D nonlinear system where a, b, c ∈ R with a = 0. Different from most Lorenz-type system family with only quadratic terms, system (2.2) evidently has two cubic terms and is not symmetric with respect to either x-axis, or y-axis or z-axis.
It is easy to introduce system (2.2). Anyway, what one is really concerned with is whether or not system (2.2) possesses some new unique natures. Fortunately, one finds that system (2.2) indeed possesses some unique characters, which is our real purpose to write this paper. Actually, one will see in the sequel that system (2.2) possesses a unique heteroclinic orbit, which different from most of known Lorenztype or Lorenz-like systems. Moreover, system (2.2) has no singularly degenerate heteroclinic cycles and chaotic attractors have not been observed under numerical simulation so far.
In order to show in detail those unique characters of system (2.2), in the following, one studies the local and global dynamical behaviors of system (2.2) by using respectively the center manifold theory, bifurcation theory and Lyapunov function, mainly including the distribution of equilibria, the stability, the expression of locally unstable manifold, Hopf bifurcation, the invariant algebraic surface, and the homoclinic and heteroclinic trajectories, etc.
3. Dynamical analysis of singular points of system (2.2) First of all, the following assertion holds for the equilibria of system (2.2).
Theorem 3.1. The distribution of the equilibria of system (2.2) is as follows.
Proof. The proof follows easily from the algebraic structure equation for equilibrium points of system (2.2). The details for the proof are omitted here.

Analysis of S 0
The Jacobian matrix of the linearized system (2.2) at the equilibrium S 0 has the eigenvalues and the corresponding eigenvectors given by: Therefore, the equilibrium S 0 is non-hyperbolic and unstable for a < 0 or b < 0. In the following, one considers the stability of S 0 for the case a > 0 and b > 0 by using the center manifold theorem. Lemma 3.1. Suppose a > 0, b > 0 and c = 0, then the equilibrium S 0 is unstable. (3.1) In light of the center manifold theory, there exists a center manifold for (3.1) which can be locally represented as follows We now compute W c loc (S 0 ). Assume that h 1,2 (u) have the following forms where h 1,2 (u) must satisfy the center manifold. We then equate powers of u, and in that way we can compute h 1,2 (u) to any desired order of accuracy. In practice, computing only a few terms is usually sufficient to answer the question for the stability of S 0 , and it is easy to obtain where O(u 4 ) expresses those terms with orders being equal or greater than four. Substituting the expression (3.4) into system (3.1), the vector field restricted to the center manifold is given byu So, from (3.5) one can see that, for u sufficient small, u = 0 is unstable for c = 0. Hence, by the center manifold theorem, when a > 0, b > 0 and c = 0, the equilibrium S 0 is unstable. Considering a > 0, b > 0 and c = 0, it follows from Lemma 3.1 that the equilibrium S 0 is non-hyperbolic, and has a one-dimensional unstable manifold W u loc (S 0 ) and a two-dimensional stable manifold W s loc (S 0 ). The tangent unstable subspace T W u loc (S 0 ) is given by According to the proof of Lemma 3.1, we get Note that W u loc (S 0 ) is indeed tangent to T W u loc (S 0 ) since x (0) = 1, z (0) = 0 and the vector (x (0), 1, 0) is collinear to the direction vector ξ 2 of the line T W u loc (S 0 ). Note also that the z-axis is included in the stable manifold W s loc (S 0 ). The following consequence is easily derived.

Analysis of equilibria S z and S
First let us consider the dynamics of equilibria S z = 0. The matrix associated with the vector field (2.2) linearized at every one of S z has the same eigenvalues λ 1 = −a, λ 2,3 = 0. Correspondingly, S z has a 2D W c loc and a 1D W u loc (resp. W s loc ) when a < 0 (resp. a > 0).

Remark 3.2.
It is easy to notice that the dynamics of the non-isolated equilibria S z has nothing with the values of the variable z, which is different from most of Lorenztype system family. Furthermore, the singularly degenerate heteroclinic cycle does not exist at all for system (2.2). Consequently, the route to chaos for most chaotic systems, i.e., the collapse of singularly degenerate heteroclinic cycles leads to the birth of chaotic attractors, can not be found from system (2.2).
Next, one investigates the dynamical behaviors of the equilibrium S , which implies that the parameters a, b and c belong to the set W = {(a, b, c) ∈ R 3 |abc = 0}. Notice the characteristic equation for the Jacobian matrix of the linearized system (2.2) at S to take this form For convenience of next discussion, one first divides the set W into W 21 is continued to make the following divisions: And W 212 is written as a union of the following three subsets W 1 212 , W 2 212 and W 3 212 Concerning with the stability of S , the following assertion holds.
According to the Routh-Hurwitz stability criterion [24, p.58], the necessary and sufficient condition for the roots of Eq. (3.7) to have negative real parts is Namely, (a, b, c) ∈ W 211 ∪W 1 212 . So, the proof follows and the other detail is omitted here.
Theorem 3 implies a birth of bifurcation at S for (a, b, c) ∈ W 2 212 . It is a Hopf bifurcation. So, finally, let us study the Hopf bifurcation of equilibrium point S . Combining the Hopf bifurcation theory and symbolic computation [11], one arrives at the following result.
Theorem 3.4. For (a, b, c) ∈ W 2 212 , system (2.2) undergoes a Hopf bifurcation at the equilibrium S . Further, the first Lyapunov coefficient of system (2.2) at S is given by where Noticing D 1,2 > 0 for (a, b, c) ∈ W 2 212 , the sign of l 1 (a, b) depends on the one of N 1 . The periodic trajectory bifurcating from the equilibrium S is stable (resp. unstable) for N 1 > 0 (resp. N 1 < 0). When N 1 = 0, one has to compute the second or the third or even more higher order Lyapunov coefficient to determine the stability of the bifurcated periodic trajectory.
Proof. The proof can be carried out according to two cases: (1) c = c * , (2) c = −c * . For each case, one only verifies the following two conditions for Hopf bifurcation to occur: (i) transversality, (ii) nondegeneracy.
Firstly, one considers the transversality. It follows from the relation between roots and coefficients of an algebraic equation with order 3 that the Eq. (3.7) has one negative real root λ 1 = −(a + b) and a pair of conjugate purely imaginary roots λ 2,3 = ±ωi with ω = Hence, the transversal condition holds. Next, it is time for one to verify the nondegeneracy of S by employing the project method in [11]. Fellow the following steps.
First of all, take the following linear transformation with v = 3 √ bc * so that system (2.2) can be converted into the following equivalent system (3.9) And the equilibrium S of system (2.2) is transformed into the origin S 0 of system (3.9). For c = c * , the Jacobian matrix of system (3.9) at S 0 is given by and the corresponding eigenvalues are Some tedious calculations display that It is also easy to derive The numerical simulation agrees with the theoretical analysis, see Fig. 1.

Homoclinic and heteroclinic trajectories
This section aims to deal with the global bifurcation of system (2.2) on the existence of homoclinic and heteroclinic trajectories. The main conclusions in this section are as follows:    The procedure for the proof of Theorem 4.1 can be divided into the following two parts. Firstly, one considers the nonexistence of homoclinic trajectories of system (2.2).

Nonexistence of homoclinic trajectory
for b > 3a > 0, and the second one for b = 3a > 0 ( At this time, Q = 0, i.e., x 3 = 3az. For the proof, see (18) in the sequel). Proof. For any one solution (x, y, z) of system (2.2), it follows from the Lyapunov fucntions U 1,2 and Eqs. (4.1)-(4.2) thaṫ is an equilibrium point when b ≥ 3a > 0. In fact, all solutions tend to S (resp. S 0 ) when c = 0 (resp. c = 0) according to the LaSalle theorem [5]. Therefore, system (2.2) has no homoclinic trajectories. In light of the simulation result (see Fig. 3) that has not been theoretically proved yet, we present the following problem.   Next, one devotes to investigating the existence of the heteroclinic trajectory of system (2.2) by employing the concepts of both α, ω-limit set and Lyapunov function.

Existence of heteroclinic trajectory
In order to detect the existence of heteroclinic trajectory of system (2.2), one recalls the following fact and makes some numerical simulations with MATLAB software.  When c = 0 and (a, b, c) ∈ W 211 ∪ W 1 212 , it follows from Lemma 3.1 and Theorem 3.3 that S 0 is an unstable non-hyperbolic equilibrium point with 1D W u loc and 2D W s loc but S is locally asymptotically stable. Furthermore, the 1D W u (S 0 ) of S 0 tends to the stable manifolds W s (S ) of S as t → ∞, forming a heteroclinic trajectory to S 0 and S , see Figs. 4-8. Denote in the following by φ t (q 0 ) = (x(t; q 0 ), y(t; q 0 ), z(t; q 0 )) a solution of the system (2.2) through the initial point q 0 = (x 0 , y 0 , z 0 ) and by W u + (resp. W u − ) the positive (resp. negative) branch of the unstable manifold W u (S 0 ) corresponding to x > 0 (resp. x < 0). Set c = 0. Define the first Lyapunov function for b > 3a > 0, and the second one for b = 3a > 0 ( At this time x 3 ≡ 3az ). Some lengthy computations display that and respectively.
(c) System (2.2) has only one heteroclinic trajectory to S 0 and S .
(c) By statement (a), every one-dimensional branch of the unstable manifold W u + (resp. W u − ) has ω-limit, which is an equilibrium point p. Noticing V 1,2 (S 0 ) > V 1,2 (S ), the equilibrium point p has to be S , obtaining a single heteroclinic trajectory. This completes the proof of the theorem.
A numerical case with a single heteroclinic trajectory is illustrated in Figs. 4-5.
Remark 4.2. Figs. 6-8 also illustrate that there exists a heteroclinic trajectory to S 0 and S in a parameter space other than the one D := {(a, b, c)|b ≥ 3a > 0, c = 0}. Theoretically, we can not rigorously prove this case yet. Therefore, this is still an open problem. We hope that interested readers consider it in the future work.

Conclusion
Motivated by coining a single heteroclinic trajectory from the Lorenz system family, this paper introduces a new 3D autonomous Lorenz-like system with two cubic terms. Indeed, it is rigorously proved that there exists a single heteroclinic trajectory to S 0 and S in the proposed system when b ≥ 3a > 0 and c = 0 by employing the tools of Lyapunov function and concepts of α-limit set and ω-limit set. Numerical simulation illustrates that there may exist a homoclinic trajectory to S 0 when a, b, c satisfy (a, b, c) ∈ W 1 212 . Moreover, it is found that the system has no homoclinic (resp. heteroclinic) trajectories when b ≥ 3a > 0 (resp. (a, b, c) ∈ W 211 W 1 212 , or ab = 0 and c = 0). In addition, its other complex dynamical behaviors have been investigated by utilizing the center manifold theory and bifurcation theory, such as the stability and local expression of unstable manifold of S 0 , the stability of non-isolated equilibria S z , and the stability and Hopf bifurcation of S , etc. Different from most Lorenz-like systems, numerical simulation illustrates neither any singularly degenerate heteroclinic cycles nor chaotic attractors in the proposed system. These differences make the new system more interesting, and also indicate that the system deserves deeper investigations in the future.