On singular orbits and a given conjecture for a 3D Lorenz-like system

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

“…The proof of Proposition 6 follows from the Routh-Hurwitz criterion and is similar to the one [ [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 .…”

“…Existence of herteroclinic orbit. Utilizing two different Lyapunov functions, concepts of both α-limit set and ω-limit set [5,16,17,19,20,21,24,34,35,39,40,41,44,45], this section devotes to investigating the existence of heteroclinic orbits of the system (1), aiming at complementing and extending the obtained results in [5,Theorem 5,p.579]. The fundamental work of the proof is how to construct suitable Lyapunov functions for the system (1).…”

“…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.…”

Some new insights into a known Chen‐like system

“…Anyway, these are only simulation results, which are not theoretically verified. To the best of our knowledge, the classic method-using Lyapunov functional together with the !limit set and˛-limit set [17,18,20,[24][25][26][27][28][29][30]-is not applicable to the system (1). Although there exist some other powerful analytical methods for the study of the existence of homoclinic and heteroclinic orbits for polynomial systems of ODEs, for example, the so-called Fishing principle [67][68][69][70][71][72][73][74] and the one in [75], we have not yet known whether or not these methods are effective for the system (1).…”

“…The pioneer work for the study of chaos is the report of the Lorenz system [6,7] given by E. N. Lorenz in 1963. Since then, many researchers have been motivated into this aspect; on the one hand, they constructed some new and interesting chaotic systems, for example, the Chen system [8], the Lü system [9] (which should belong to the Lorenz system family [10,11] according to [12]), and others [13][14][15][16][17][18]; on the other hand, they studied rich dynamics of such systems [7,[17][18][19][20][21][22][23][24][25][26][27][28][29][30], mainly the stability of equilibrium point, various bifurcations, homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and some other complex dynamical behaviors. All of these work should be helpful to understand the real essence of chaos.…”