Dynamical Analysis and Simulation of a New Lorenz-Like Chaotic System

This work presents and investigates a new chaotic system with eight terms. By numerical simulation, the two-scroll chaotic attractor is found for some certain parameters. And, by theoretical analysis, we discuss the dynamical behavior of the new-type Lorenz-like chaotic system. Firstly, the local dynamical properties, such as the distribution and the local stability of all equilibrium points, the local stable and unstable manifolds, and the Hopf bifurcations, are all revealed as the parameters varying in the space of parameters. Secondly, by applying the way of Poincar´e compactiﬁcation in R 3 , the dynamics at inﬁnity are clearly analyzed. Thirdly, combining the dynamics at ﬁnity and those at inﬁnity, the global dynamical behaviors are formulated. Especially, we have proved the existence of the inﬁnite heteroclinic orbits. Furthermore, all obtained theoretical results in this paper are further veriﬁed by numerical simulations.


Introduction
Chaos, as a magical and charming nonlinear phenomenon, has attracted attentions of many scholars in nonlinear dynamics community. is is not only because it is a mysterious and profound subject but also because it is beneficial to many practical applications. Especially, with the development of computer science, chaos has been applied to secure communication [1], biomedical system analysis [2], power system protection [3], and fluid mixing [4]. Accordingly, due to these influential applications, the synchronization and control of chaotic systems have fascinated numerous research studies; one may see [5][6][7][8] and references therein.
Ever since Lorenz discovered the chaotic phenomena in a simple 3D nonlinear ODE in 1963, more and more specialists devoted themselves to chaotic dynamics. Some of them were interested in finding different kinds of new chaotic models and discussed their dynamical properties, for example, Lü system [9], Chen system [10], Lorenz-type system [11][12][13][14], and other new chaotic system [15][16][17][18][19], while some of them focused their attentions on revisiting the existing chaotic system and explored some new phenomena which had not been found before [20][21][22][23][24][25][26][27][28]. All of these research studies are beneficial to reveal the essence of chaos.
As we know, if we add or change the linear or nonlinear term of existing chaotic system, then a new type chaotic system will be achieved. In [29], the authors constructed a new Lorenz-like chaotic system: ey pointed out that when the parameters satisfy (a, b, l, c, h, k) � (10, 40, 1, 2.5, 2, 2), this new system has a butterfly-shaped attractor. Based on Lü system, Li and Ou [11] presented and considered a new chaotic system: where a > 0, b, c, d ∈ R, g ≥ 0, f ≥ 0, and g + f > 0. Notice that there are eight terms in this system, and it is obvious that it is not topologically equivalent to any form of the generalized Lorenz system. Furthermore, Lü system [9] and system (2) are included in our new system. By taking (a, d, c, b, f, g) � (10, 2, 6, 3, 2, 1, 1) and t � 5000, we can achieve the Lyapunov exponents of system (5) as λ LE 1 � 0.2603, λ LE 2 � − 0.0008, and λ LE 3 � − 7.2595, which means that there is a chaotic phenomena in system (5). And, the numerical simulation in Figure 1 shows that system (5) has a two-scroll chaotic attractor which looks like Lorenz attractor. Hence, this new system can be called as Lorenz-like system. Due to the above discovery, we are inspired to explore some more and new complex dynamical behaviors of system (5). erefore, in this paper, our main purpose is to explore the global dynamics of system (5) in detail. It is worthy pointing out that the dynamics at infinity especially the existence of infinite heteroclinic orbits have not been mentioned in [11].
is paper is constructed as follows. In Section 2, the basic dynamics of system (5), including the existence and their local properties of the equilibrium points, are provided. Particularly, the specific expressions of local stable (resp. unstable) manifolds and the corresponding numerical simulations are also presented. In Section 3, by virtue of the Poincaré compactification in R 3 , we present the dynamical behaviors on the sphere at infinity, where we find the infinite heteroclinic orbits for some certain parameters. In Section 4, we give a brief conclusion and some future work.

Local Dynamical Behavior of System (5)
In the following, we consider the local dynamical properties for system (5), including the existence and the stability of all equilibrium points, Hopf bifurcation, and the local structure of trajectories.
First, we indicate that system (5) satisfies the following properties: (1) Symmetry and invariance.
(3) Dissipativity. ese can be easily proved with similar arguments to [11], so we omit it here.

e Existence of Equilibrium Points.
In order to discuss the existence of equilibrium points for system (5), we need to solve the algebraic equations as follows: By some simple computations, we achieve the following results. Theorem 1. For system (5), we have the following true statements: Here,

Local Dynamical Properties of the Equilibrium Point E 0 .
For the equilibrium point E 0 of system (5), we can get its Jacobian matrix A 0 : en, the characteristic equation of A 0 is obtained: We will discuss the local dynamical properties of equilibrium point E 0 in the following three sections.

Lemma 1.
For c − d > 0 and b < 0, E 0 is a saddle. And, it has a 2D unstable manifold that embraces z-axis and a 1D stable manifold W s loc , which can be expressed by Proof. When c − d > 0 and b < 0, it is obvious that λ 1 > 0, λ 3 > 0, and λ 2 < 0.
is means that E 0 is a saddle, which consists of a 2D unstable manifold that includes z-axis and a 1D stable manifold W s loc , which can be expressed by where . Put them into system (5) and then compare the coefficients of the same term on both sides, and we achieve It follows from the matrix equation, that e proof of Lemma 1 is then finished.
And, it has a 2D stable manifold that embraces z-axis and a 1D unstable manifold W u loc , which is expressed as Proof. When c − d > 0 and b > 0, it is obvious that λ 2 < 0, λ 3 < 0, and λ 1 > 0. erefore, E 0 is a saddle, which consists of a 2D stable manifold that includes z-axis, and a 1D unstable manifold W u loc , which is written as where . Similar to Lemma 1, we may derive that us, we complete the proof of Lemma 2.

e Case c − d � 0
Lemma 3. Suppose c − d � 0, then we have the following true statements: Figure 3. (ii) For b < 0, E 0 is a nonhyperbolic point, and it is of saddle-center type.

is a nonhyperbolic point and asymptotically stable. e corresponding illustrative phase portrait is depicted in
Proof.
(1) When c − d � 0, d < a, and b > 0, E 0 has three eigenvalues λ 1 < 0, λ 3 < 0, and λ 2 � 0. us, it is nonhyperbolic. In the following, we will use the center manifold theory to further determine its stability. Let en, system (5) turns into Based on the center manifold theory, we can determine the stability of the equilibrium point E 0 by investigating a first-order ODE restrict to its center manifold, and the center manifold can be expressed as with δ sufficiently small. Assume By some computations, we obtain Mathematical Problems in Engineering Hence, system (5) restricted to the center manifold may be written as It then follows that the equilibrium point E 0 is asymptotically stable.

2.2.3.
e Case c − d < 0. When c − d < 0, the dynamical properties of the equilibrium point E 0 are formulated in Table 1.
Especially, when c − d < 0, b > 0, and c − a � 0, we have the lemma as follows.
Lemma 4. Suppose c − d < 0, c − a � 0, and b > 0, then the Hopf bifurcation will happen to system (5) at the equilibrium point E 0 , and there exists an orbit with the period ). e corresponding illustrative phase portrait is shown in Figure 4.
By Hopf bifurcation theory [31], a Hopf bifurcation will happen to system (5) at E 0 . is means a limit cycle will occur at c � a, which has initial period T � (π/ ������� a(d − c) ) and increases with c. So, the proof of Lemma 4 is then completed.
Summing up the above discussions, we get the main theorem as follows. (5) has the local dynamical properties which are given in Table 2.

Local Dynamic Behavior of the Nonisolated Equilibrium
e characteristic equation at the nonisolated points E z of system (5) is By analyzing the characteristic roots of the above equation, we can get the results as follows.
and g + f > 0, then the equilibrium points E z have the local dynamical properties which are summed in Table 3.

Local Dynamical Behavior of the Equilibrium Points E ± .
In what follows, we deal with the dynamical behavior of E ± , which means b(c − d) > 0. By symmetry of (5), we know that the stability of E − can be achieved from that of the equilibrium point E + . us, we just consider the local dynamical behavior of E + .
For convenience in our later discussion, we first give some results of the polynomial with real coefficient: Set three roots of equation p(λ) � 0 be λ 1 , λ 2 , and λ 3 . en, it is easy to obtain Mathematical Problems in Engineering 7 We can obtain the principal minors for H which are H 1 � a 1 , H 2 � a 1 a 2 − a 3 , and H 3 � a 3 H 2 . us, we have the following results. Proof.
(1) It follows from a 3 � − λ 1 λ 2 λ 3 > 0 that λ i ≠ 0(i � 1, 2, 3). So, we finish the proof of (1). (2) When a 3 > 0, suppose that p (λ) � 0 has a real root λ 1 and a pair of pure imaginary roots λ 2,3 � ± ωi(ω > 0). It follows from (1) that λ 1 ≠ 0; then, from (11), we can derive H 2 � a 1 a 2 − a 3 � 0, completing the proof of (2). (3) Assume a 3 > 0, when p (λ) � 0 has three real roots or a pair of conjugate roots with nonzero real parts and a real root; then, it follows from the Routh-Hurwitz criterion that p (λ) � 0 has at least a root λ i with Re (λ i ) > 0 (i � 1, 2, 3). And, when p(λ) � 0 has a real root λ 1 and a pair of pure imaginary roots λ 2,3 , if H 2 < 0, then according to (2), we know H 2 � 0, a contradiction; if H 3 < 0, then from H 3 � a 3 H 2 and    e Jacobian matrix A at E + of system (5) is Denote three characteristic roots of (31) by λ 1 , λ 2 , and λ 3 . Combining equations (27) and (31), we obtain en, and the principal minors for H are For p(c), we have the following properties: (1) If f � 0 and d ≤ a, then p(c) � 0 has a root c 0 , which can be expressed as (2) If f > 0, it follows that (i) For d < a, p(c) � 0 has two real roots c ± , which can be expressed in this form: (ii) For d � a and 2a(g + f) + bf ≠ 0, p(c) � 0 has two real roots c ± , which can be expressed as (iii) For d � a and 2a(f + g) + bf � 0, p(c) � 0 has a unique real root c � a. Based on the above analysis, it is not difficult to deduce the following lemmas about c 0 and c ± . Lemma 6. When b > 0 and d < a, it follows

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Lemma 7. When b < 0 and d < a, then we have the following true statements: (1) For f � 0, (2) For f > 0, en, we can conclude the results in the following. (1) For f � 0,

then system (5) will undergo a Hopf bifurcation at E + and a periodic orbit with period
) will occur. e illustrative phase portraits are depicted in Figure 5.
Next, we prove (2) (iii). For f > 0 and c � c − , we know that k(λ) has a pair of pure imaginary roots According to the Hopf bifurcation theory [31], one knows that a limit cycle occurs at c � c − , which increases with respect to c and has the initial period . e above arguments imply that a Hopf bifurcation will happen at E + of system (5). Using the same method as (2) (iii), we can prove the results of (1) (iii). □ Theorem 5. Suppose b < 0 and d < a, then E + is unstable. e corresponding illustrative phase portraits are depicted in Figure 7.
Proof. When b < 0, in order to guarantee the existence of E + , we require c < d. If f � 0, based on the statements of Lemma 7 (1), we know that there exists at least a H i < 0 (i � 1, 2, 3); then, according to Lemma 5 (3), we get that k(λ) � 0 has at least one root with positive real part; thus, E + is unstable. If f > 0, with similar argument to the proof of f � 0, combining Lemma 7 (2) and Lemma 5 (3), we achieve the unstability of E + . Consequently, we finish the proof of eorem 5.

Theorem 6.
Assume d � a, then E + is unstable. e corresponding illustrative phase portraits are depicted in Figure 8.
Because E + and E − are symmetrical, the following results can be achieved. a > 0, b, c, d ∈ R, d ≤ a, g ≥ 0, f ≥ 0, and g + f > 0, the local dynamical properties of the equilibrium points E ± are summed up in Table 4. Remark 2. For the case d > a, we cannot find an effective method to classify the parameters to investigate the dynamical behavior of E ± , but we present numerical simulation for parameters (a, b, c, d, f, g) � (2, 1, 4, 3, 0, 1) and (a, b, c, d, f, g) � (2, − 1, 3, 4, 0, 1) which are shown in Figure 9.

Dynamics at Infinity on System (5)
In this section, the dynamics at infinity on system (5) will be discussed with the aid of the way to Poincaré compactification in R 3 [32,33]. (5). To study the dynamics at infinity, we should investigate the Poincaré compactification for system (5) in the local charts U i and V i (i � 1, 2, 3), respectively.

e Infinite Equilibrium Points of System
3.1.1. In the Local Charts U 1 and V 1 . Based on [32,33], the Poincaré compactification p(X) for system (5) in the chart U 1 may be displayed as As we know, to study the points of the sphere S 2 at infinity, one needs to set z 3 � 0; it then follows that One may get that system (42) has no equilibrium point for g � 0, while, for g > 0, system (42) has a unique equilibrium point (− (f/g), 0). at is to say, system (41) has no equilibrium point for g � 0 and has a single equilibrium point P 1 (− (f/g), 0, 0) for g > 0. Moreover, a first integral,

Mathematical Problems in Engineering
has been found in system (42), which shows the equilibrium point P 1 of system (41) is a center. And, the phase portrait for (43) has been drawn in Figure 10, which helps us to get the corresponding phase portrait for our system (5) on the sphere at infinity. In order to find out the dynamics of the local property of P 1 , we can analyze the flow on the center manifold related to its eigenvalue 0 and achieve the statement listed as follows.

Lemma 8.
When g > 0, the equilibrium point P 1 of system (40) has a 1D center manifold W c loc , and it is unstable along W c loc .
Proof. Since the equilibrium point P 1 has eigenvalues λ 1,2 � ± � � g √ i and λ 3 � 0, which implies P 1 has a 1D W c loc , set k 1 � z 1 + (f/g), k 2 � z 2 , and k 3 � z 3 , which translates the singular point P 1 into the origin O(0, 0, 0). en, system (41) is converted into us, the stability of P 1 restricted on the one-dimensional center manifold can be established according to that of the origin on a center manifold. Assume that the graph of a function h: Expand h at k 3 � 0 with Taylor series, which gives Substituting them into (44) and comparing the coefficients of k 3 on both sides, we obtain , which implies that when z 3 > 0, P 1 is local unstable along its center manifold.
Notice that multiplying -1 in the compactified vector field p(X) in U 1 can give the compactified vector field p(X) in V 1 , so the flow in V 1 is the same as the flow in U 1 if we reverse the time. We have to point out that z 3 < 0 should be considered in the neighborhood of infinity in V 1 , see Figure 11.

In the Local Charts
e Poincaré compactification p(X) for system (5) in U 2 has the following form: By setting z 3 � 0, system (48) becomes By some simple computations, one knows that, for g � 0 and f > 0 or g > 0 and f � 0, system (49) has a line of equilibrium points (0, z 2 ), while, for g > 0 and f > 0, system (48) has an equilibrium point (− (g/f), 0) except the line of equilibrium points (0, z 2 ), which means system (48) has a

Theorem 8.
Suppose g ≥ 0, f ≥ 0, and g + f > 0, the infinite dynamical behavior of system (5) on the sphere is summarized in Figure 14.

e Existence of Infinite Heteroclinic
Orbits. Now, we study how the orbits in the vicinity of the Poincaré ball reach to the equilibrium point of the sphere along z-axis. In fact, we only need to study how the orbits in a neighborhood of (0, 0, z 3 )(0 < z 3 ≪ 1) in system (26) tend to the origin on the chart U 3 .

Lemma 9.
e equilibrium points at the endpoints of z-axis on the sphere at infinity are asymptotically stable (resp. unstable) along z-axis for b < 0 (resp. b > 0).
Proof. Since the equilibrium point is at the end point of positive (resp. negative) z-axis on the sphere at infinity coincide with the origin of the Poincaré compactification p(X) for system (5) in the local chart U 3 (resp. V 3 ), we deal  proof to the existence of the chaos, chaos control, and applications of our studied system.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.