Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small \begin{document}$ b > 0 $\end{document} , i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with \begin{document}$ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-12, 12, 23,-1, -1, 1, 2.1, -6, -0.2) $\end{document} , consisting of occurrence of degenerate pitchfork bifurcation at \begin{document}$ S_{z} $\end{document} , the change in the stability index of the saddle at the origin as \begin{document}$ b $\end{document} crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.

Therefore, the preliminary remarkable achievements provide insight into subsequent studies. In addition, most of hyperchaotic systems are derived from chaotic systems and some dynamics may depend on the original models, especially the forming mechanism and relationships between hyperchaotic attractors and algebraic structures. More importantly, how to further extend these studies to hyperchaotic systems is therefore not only theoretically significant but also practically important, motivating the work to be presented in this paper.
More precisely, in this paper, a unified hyperchaotic Lorenz-type system is introduced, which includes the unified Lorenz-type system [49] and hyperchaotic Lorenztype system [15] as special cases. Combining theoretical analysis and numerical techniques we fix the following issues.
Issue 1.1: Numerical simulations illustrated that not only classic and conjugate chaotic Lorenz-type attractors, but also the hyperchaotic ones were generated from the collapse of the corresponding singularly degenerate heteroclinic cycles, see [23,25,44,45,48,49]. But it is currently unclear whether the conjugate hyperchaotic Lorenz-type attractors (CHCLTA) exist. Issue 1.2: Referring to [5, Theorem 1, p. 573; Theorem 2, p. 576], the authors only considered the pitchfork and Hopf bifurcation at the origin of the hyperchaotic Lorenz-type system in the case of the bifurcation parameter b. How about the other cases? Issue 1.3: Particularly, the authors in [5,39] rigorously proved that there exists a pair of symmetrical heteroclinic orbits in the unified Lorenz-type system and hyperchaotic Lorenz-type system by aid of Lyapunov function, concepts of both αlimit set and ω-limit set. Whether does the system (1) have this kind of dynamics in the region of parameters other than the one given in [5,Theorem 5,p. 578]? Issue 1.4: In addition, the authors in [15,22,46] only investigated the ultimate bound estimation and global exponential attractive set of that hyperchaotic Lorenztype system with all positive parameters. How about the dynamics in the other range of parameters?
To the best of our knowledge, Issue 1.1-1.4 are not discussed in any public literatures. Therefore, it is worthwhile to propose and study the unified hyperchaotic Lorenz-type system.
The main innovations are as follows: (1) Finding classic and conjugate hyperchaotic Lorenz-type attractors through the collapse of the corresponding singularly degenerate heteroclinic cycles near them. (2) Extending the recent results reported in [5,15,22,46] in a wider range of parameters, i.e., pitchfork bifurcation, Hopf bifurcation, ultimate bound estimation, global exponential attractive sets, heteroclinic orbits, etc.
The remainder of this paper is organized as follows. Section 2 introduces the unified hyperchaotic Lorenz-type system. Section 3 discusses dynamics of equilibria of the proposed system. Section 4 numerically illustrates the existence of different shapes of infinitely many singularly degenerate heteroclinic cycles and together with hyperchaotic attractors near them. Section 5 formulates ultimate bound estimation and global exponential attractive sets. In Sect. 6, one rigorously proves that the system has a pair of symmetric heteroclinic orbits under the case a 2 = 0, b > 0, 2a 1 + b ≥ 0, e < 0, a 1 + d < 0, q < 0, f p(−a 1 + q) > 0 and a 2 cq − a 1 (dq − f p) < 0. Finally, some conclusions are drawn in Sect. 7.
(3) Remark 1. It follows from the algebraic structure of the system (1) that it contains the system (2)-(3) as special cases. In particular, the system (1) is reduced to the original Lorenz system when −a 1 = a 2 > 0, c > 0, e = −1, f = 0 and p = q = 0. Therefore, the research being carried out on the system (1) may be helpful for a deep understanding upon the nature of the original Lorenz system. For b = 0, Figure 1(a) displays that solutions of the system (1) ultimately approach the infinity after a short-duration transient of singularly degenerate heteroclinic cycles, which converge to conjugate hyperchaotic Lorenz-type attractors (CHCLTA) while b = 0.07, as depicted in Figure 1 In this case, the system (1) has two positive Lyapunov exponents, L 1 = 0.07190, L 2 = 0.01254, and the other two are L 3 = 0.000002 and L 4 = −0.384442, respectively. Figure 2 clearly shows the distinct geometrical structures of the conjugate hyperchaotic Lorenz-type attractors (CHCLTA), similar to the conjugate chaotic Lorenz/Chen/Lü attractor [48].  (2) When b = 0, S 0 = (0, 0, 0, 0) is the single equilibrium point of the system (1) for ∆ = a 1 beq[a 1 (f p − dq) + a 2 cq] ≤ 0; while ∆ > 0, the system (1) still has a pair of symmetric equilibria In the following, one performs the analysis of qualitative behavior of S 0 , S ± and S z , respectively. First of all, the characteristic equations of the matrix associated with linearized vector field about S 0 , S ± and S z are with where with respectively.

3.1.
Behavior of S 0 . The main dynamics of S 0 is concluded in the following propositions.
Proposition 3. The generic pitchfork bifurcation occurs at S 0 for the critical value Take the case f 0 = q(a1d−a2c) a1p for example. The generic pitchfork bifurcation happens at S 0 when ρ 2 Moreover, the discussions for the other cases are similar and omitted here. .
Take the case for example. When (d + q)[a 2 1 (d + q) − a 2 c(a 1 − q)] < 0, a 1 + d + q < 0 and b > 0, the system (1) undergoes a Hopf bifurcation at S 0 if f passes through the critical value f * . Furthermore, the corresponding first Lyapunov coefficient is given by where Moreover, (i) If p(d + q) < 0 and l 1 > 0(or p(d + q) > 0 and l 1 < 0), then the Hopf bifurcation at S 0 is subcritical, and the unstable periodic orbit exists for f < f * .
(ii) If p(d + q) < 0 and l 1 < 0(or p(d + q) > 0 and l 1 > 0), then the Hopf bifurcation at S 0 is supercritical, and the stable periodic orbit exists for f > f * .
For the other cases, the discussions are similar and omitted here.
The sketches of proofs of Proposition 2-4 are presented as follows.
Proof of Proposition 2. a) The statement directly follows from the Routh-Hurwitz criterion. b) Define (x, y, z, w) to be any one solution of the system (1).
Take the Lyapunov-like function for the subcase 2a 1 + b > 0, and another one for the subcase 2a 1 + b = 0(At this time, it is easy to verify the x 2 = 2a 2 z, i.e. Q = 0). It follows from the Lyapunov fucntions U 2,3 and Eqs. (8)-(9) thaṫ According to LaSalle theorem [10], S 0 is also globally asymptotically stable. The proof is completed.
. The system (1) has the following equivalent form = 0, the eigenvalues and corresponding eigenvectors of the associated linear vector field , and Taking the following homothetic transformation (x, y, z, w) T = (ξ 1 , ξ 2 , ξ 3 , ξ 4 )(u, v, r, s) T where , In order to determine the stability of S 0 nearf = 0 by the center manifold theory, one needs to study the two-parameter family of first-order ordinary differential equations on its center manifold that represents as a graph over u andf = 0,i.e. Substituting the following expanded expressions of V (u,f ), S(u,f ) and R(u,f ) into the system (11), Consequently, the restricted vector field of system (11) on its center manifold, i.e.
is obtained by substituting expressions in (12) into the system (11). Since the system (1) undergoes a pitchfork bifurcation at S 0 as f passes through the critical value f 0 = q(a1d−a2c) a1p according to the pitchfork bifurcation theory [9,47]. Furthermore, S 0 is unstable (resp. stable) on its 1D center manifold for The proof is over.

HAIJUN WANG AND FUMIN ZHANG
Calculating the derivatives on both sides of Eq. (4) with respect to f and substituting λ with ωi yield which verifies the condition of the transversality.

2) Nondegeneracy
In order to validate the nondegeneracy condition, one has to compute the Lyapunov coefficients by using the project method [13].
When f = f * , the matrix associated with linearized vector field about S 0 Performing further computations, one obtains Finally, one gets the first Lyapunov coefficient as given by (7)  Therefore, if p(d + q) < 0 and l 1 > 0(or p(d + q) > 0 and l 1 < 0), then the Hopf bifurcation at S 0 is subcritical. Otherwise, if p(d + q) < 0 and l 1 < 0(or p(d + q) > 0 and l 1 > 0), then the Hopf bifurcation at S 0 is supercritical.
This completes the proof.

3.2.
Behavior of S ± . The dynamics of S ± on stability and Hopf bifurcation is summarised as follows. (1) simultaneously undergoes Hopf bifurcation at S ± .
Proof. i) The stability of S ± easily follows from the Routh-Hurwitz criterion.
ii) If (a 1 , a 2 , b, c, d, e, f, p, q) ∈ W 2 1 , then Eq. (5) has a pair of conjugate purely imaginary roots λ 3,4 = ±ωi with ω = − b0ε α−b0 and two negative real roots Calculating the derivatives on Eq. (5) with respect to b and substituting the λ with ωi result in which validates the transversal condition. Consequently, Hopf bifurcation simultaneously happens at S ± . The proof is over.

3.3.
Behavior of S z . The dynamics of S z associated with stability and degenerate pitchfork bifurcation [19,24,30] is concluded as follows.
Proposition 6. (a) Every one of S z is a stable normally hyperbolic node or node- (b) The system (1) undergoes a degenerate pitchfork bifurcation at S z when b crosses the null value and The proof of Proposition 6 follows from the Routh-Hurwitz criterion and is similar to the one [ 4. Singularly degenerate heteroclinic cycle. Coining singularly degenerate heteroclinic cycles is an inevitable issue when analyzing a chaotic/hyperchaotic system, since the collapse of them is one route to chaos/hyperchaos [7,12,20,21,23,25,40,41,44,45,49]. In this effort, one in this section tries to give some kind of the forming mechanism of the hyperchaotic attractor [15, p. 867] of the system (1) with (a 1 , a 2 , c, d, e, f, b, p, q) = (−12, 12, 23, −1, −1, 1, 2.1, −6, −0.2) combining numerical techniques and the dynamics of S z .
First of all, the following linear scaling with k > 0 HAIJUN WANG AND FUMIN ZHANG converts the system (1) into the resulting equivalent system Therefore, the hyperchaotic attractor with for system (1) corresponds to the solution of system (14) with Here, set k = 40. Then the aforementioned hyperchaotic attractor corresponds the solution of system (14) with Choosing two groups of initial conditions ( ) of each normally hyperbolic saddle-node S 1 z (resp. saddle-focus S 2 z ) tend toward one of the normally hyperbolic stable node-foci S 3 z as t → ∞, forming singularly degenerate heteroclinic cycles with different geometrical structures, which further also collapse into the hyperchaotic attractors when b = 0.0525. See   A detailed numerical study of the solutions of the system (1) with b = 0 has been made, which clearly indicates that the system presents an infinite set of singularly degenerate heteroclinic cycles. Each one of these cycles is formed by one of the twodimensional unstable manifolds of saddle-node S 1 z (resp. saddle-focus S 2 z ), which connects S 1 z (resp. S 2 z ) with the normally hyperbolic stable node-focus S 3 z , as t → ∞. As the system presents an infinite number of normally hyperbolic saddle-nodes S 1 z (resp. saddle-foci S 2 z ) and stable node-foci S 3 z , there exists an infinite set of singularly degenerate heteroclinic cycles. In Figure 3  x Figure 4. the saddle-node S 1 z (resp. saddle-focus S 2 z ) at the z-axis, a singularly degenerate heteroclinic cycle is created.
Take the same parameter value  4)). We firstly compute the x-coordinate of the solutions, that is the curve (t, x(t)), in order to verify the dependence on initial values, which is one of the main properties of strange attractors. The result is shown in Figure 4(resp. Figure 7). The solutions differ in their colors (or gray scale) and actually show this sensitive dependence. The y-, z-and w-coordinates have the same behaviors. Then, in order to ensure that the system (1) presents hyperchaotic behavior for the same parameter value, ) This displays that two positive Lyapunov exponents exist, indicating the hyperchaotic behavior of the system for the parameter value. The result is also illustrated in Figure 5 (resp. Figure 8). Also, Figure 11-12 verify the dependence on initial values and ensure the existence of hyperchaotic behavior for the system (1)    x Figure  11.  Denote where X = (x, y, z, w) T and R max can be found by calculating the maximum optimization question: Then, Ω is the ultimate bound set of the system (1).
When V λ (X) > L λ , V λ (X 0 ) > L λ , we can get the exponential inequality for the system (1), given by By the definition, the set is the global exponential attractive set of the system (1).
In the following, one reveals the global exponential attractive set of the system (1).

2)Proof of Proposition 8. Set the Lyapunov-like function
Computing the derivative of V λ,m (X(t)) along the trajectory of the system (1) leads to In other words, Integrating both sides of formula (26) and applying comparison principle, one arrives at If V λ,m (X(t)) > L λ,m and V λ,m (X(t 0 )) > L λ,m , the exponential estimation for the According to the definition and taking limit on both sides of the above inequality as t → ∞ results in lim t→∞ V λ,m (X(t)) ≤ L λ,m .
Equivalently, the set is the global exponential attractive set of the system (1). The proof is over.

3)Proof of Proposition 9.
Define Then, its derivative along trajectories of the system (1) As b + 2a 1 < 0, one hasV + bV ≥ 0. For any initial value V (t 0 ) = V 0 , according to the comparison theorem, one has and computes its derivative along the trajectory of the system (1) According to Proposition 9, one has Since lim t→∞ [2a 2 z − x 2 ] > 0, so there exists a positive constant T 0 > 0, when t > T 0 , one arrives at When V λ (X) > L λ and V λ (X 0 ) > L λ . By the definition, taking upper limit on both sides of the above inequality (27) as t → +∞ results in Namely, the set is the global exponential attractive set of the system (1). The proof is finished.
Remark 6. Using the method [1], one may handle the boundedness of the Chentype or Lü-type subsystem of the system (1), which are not contained in Proposition 7-10.
In the following Section 6, one studies the global bifurcation on the existence of homoclinic and heteroclinic orbits for the system (1). First of all, one introduces some notations for the convenience of statement in the discussion in the sequel.
(a) The system (1) has neither homoclinic orbits nor heteroclinic orbits to S + and S − . (b) The system (1) has exactly two heteroclinic orbits to S 0 and S ± .
Proof of Proposition 12. a) Firstly, let us show that there is neither homoclinic orbits nor heteroclinic orbits to S + and S − for a 2 = 0, a 1 < 0, 2a 1 + b ≥ 0, e < 0, a 1 + d < 0, q < 0, f p(−a 1 + q) > 0 and a 2 cq − a 1 (dq − f p) < 0. Assume γ(t, q 0 ) is a homoclinic orbit or a heteroclinic orbit to S + and S − of the system (1) through where s − and s + satisfy either s − = s + ∈ {S 0 , S − , S + } or {s − , s + } = {S − , S + }. It follows from (28)-(29) that In either case, we have the relation V 1,2 (s − ) = V 1,2 (s + ), which suggests V 1,2 (γ(t, q 0 )) = V 1,2 (s + ). The assertion (i) of Proposition 11 results that q 0 is one of the equilibria of the system (1). Hence, neither homoclinic orbits nor heteroclinic orbits to S + and S − exist in the system (1). b) Next, we prove that the system (1) has a heteroclinic orbit to S 0 and S − . Since W u − is the negative branch with respect to x of the unstable manifold at S 0 , there exists a t 1 ∈ R such that x(t 1 , q 0 ) < 0 for q 0 ∈ W u − , which by Proposition 11(ii), gives x(t, q 0 ) < 0 for all t ∈ R and q 0 ∈ W u − , i.e. any orbit on W u − never approach S + , which lies in x > 0. From Proposition 12(a), it also does not tend to S 0 but one of the equilibria. Therefore, denoting by γ − (t) an orbit on W u − , it follows that lim t→∞ γ − (t) = S − , i.e. γ − (t) is a heteroclinic orbit to S 0 and S − lying in x < 0. Let us show now that this heteroclinic orbit is unique in x < 0. Assume that φ t (q 0 ) is a solution of the system with q 0 arbitrary, not necessarily on W u − , with s − , s + as above. lim where s − , s + ∈ {S 0 , S − }, i.e. φ t (q 0 ) is another heteroclinic orbit to S 0 and S − . As V 1,2 are decreasing on the orbits, it follows that for all t ∈ R. As V 1,2 (S 0 ) > V 1,2 (S − ), one gets that s − = S 0 and s + = S − , which gives further that q 0 ∈ W u − , by Proposition 11 (ii), i.e. the orbit φ t (q 0 ) coincides with γ − (t). As the orbits are symmetrical with respect to the z-axis, there exits an unique heteroclinic orbit γ + (t) symmetrical to γ − (t) with respect to the z-axis.
Moreover, numerical simulations are presented to show the correctness of the theoretical result, as depicted in Figure 13-14. This completes the proof.
Remark 7. Except for the results formulated in Proposition 12, numerical simulations also demonstrate that there are some other new heteroclinic orbits when (a 1 , a 2 , b, c, d, e, f, p, q) ∈ W 3 1 , see Figure 15-16.
the forming mechanism of conjugate hyperchaotic Lorenz-type attractors, ultimate bound estimation, global exponential attractive sets, homoclinic and heteroclinic orbits, and so on, are investigated by the linear analysis, center manifold theorem, Routh-Hurwitz criterion, bifurcation theory, Lyapunov function, numerical simulation, etc. In particular, the following important but distinct properties are coined: (i) conjugate hyperchaotic Lorenz-type attractors (CHCLTA) which are bifurcated from singularly degenerate heteroclinic cycles or the solutions approaching infinity after a short-duration transient of singularly degenerate heteroclinic cycles; (ii) some kind of forming mechanism of the well-known hyperchaotic attractor that is collapse of singularly degenerate heteroclinic cycles with different geometrical structures; (iii) ultimate bound estimation and a family of mathematical expressions of global exponential attractive sets; (iv) a pair of heteroclinic orbits.
Since the findings and results obtained in this paper extend some published results in some known literatures, we hope them will have a good potential in control and synchronization of hyperchaos and their engineering applications. It is expected that the basic ideas and the self-contained approach presented in this paper can be applied to explore similar hyperchaotic systems.
Owing to the great potential of hyperchaos in such nontraditional engineering and technological applications as lasers and electronics, encryption and secure communications, and biological networks, among others, it is insightful and important to develop the future work that circles around the further inquiry into the subjects of generating, controlling, synchronizing and applying hyperchaos.