In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system x·1=-ax1-fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22 by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.

References

References
1.
Edward
,
O.
,
2002
,
Chaos in Dynamical Systems
,
Second ed.
,
Cambridge University Press
,
Cambridge, UK
.
2.
Alvarez
,
G.
,
Li
,
S.
,
Montoya
,
F.
,
Pastor
,
G.
, and
Romera
,
M.
,
2005
, “
Breaking Projective Chaos Sychronization Secure Communication Using Filtering and Generalized Synchronization
,”
Chaos Solitons Fractals
,
24
(
3
), pp.
775
783
.10.1016/j.chaos.2004.09.038
3.
Ottino
,
J. M.
,
Leong
,
C. W.
,
Rising
,
H.
, and
Swanson
,
P. D.
,
1988
, “
Morphological Structures Produced by Mixing in Chaotic Flows
,”
Nature
,
333
(
2
), pp.
419
425
.10.1038/333419a0
4.
Wang
,
Z.
,
Chi
,
Z.
,
Wu
,
J.
, and
Lu
,
H.
,
2011
, “
Chaotic Time Series Method Combined With Particle Swarm Optimization and Trend Adjustment for Electricity Demand Forecasting
,”
Expert Syst. Appl.
,
38
(
7
), pp.
8419
8429
.10.1016/j.eswa.2011.01.037
5.
Chen
,
G.
,
1999
,
Controlling Chaos and Bifurcations in Engineering Systems
,
CRC Press
,
Boca Raton, FL
.
6.
Lorenz
,
E. N.
,
1963
, “
Deterministic Non-Periodic Flow
,”
J. Atmos. Sci.
,
20
(
2
), pp.
130
141
.10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
7.
Rössler
,
O. E.
,
1976
, “
An Equation for Continuous Chaos
,”
Phys. Lett. A
,
57
(
5
), pp.
397
398
.10.1016/0375-9601(76)90101-8
8.
Chen
,
G.
, and
Ueta
,
T.
,
1999
, “
Yet Another Chaotic Attractor
,”
Int. J. Bifurcation Chaos
,
9
(
8
), pp.
1465
1466
.10.1142/S0218127499001024
9.
Li
,
X.
,
Liu
,
T.
,
Liu
,
L.
, and
Liu
,
K.
,
2004
, “
A New Chaotic Attractor
,”
Chaos Solitons Fractals
,
22
(
5
), pp.
1031
1048
.10.1016/j.chaos.2004.02.060
10.
Li
,
X.
, and
Wang
,
H.
,
2011
, “
Homoclinic and Heteroclinic Orbits and Bifurcations of a New Lorenz-type System
,”
Int. J. Bifurcation Chaos
,
21
(
9
), pp.
2695
2712
.10.1142/S0218127411030039
11.
Liu
,
Y.
, and
Yang
,
Q.
,
2010
, “
Dynamics of a New Lorenz-Like Chaotic System
,”
Nonlinear Anal.: Real World Appl.
,
11
(
4
), pp.
2563
2572
.10.1016/j.nonrwa.2009.09.001
12.
Li
,
X.
, and
Ou
,
Q.
,
2011
, “
Dynamical Properties and Simulation of a New Lorenz-Like Chaotic System
,”
Nonlinear Dyn.
,
65
(
3
), pp.
255
270
.10.1007/s11071-010-9887-z
13.
Dias
,
F. S.
,
Mello
,
L. F.
, and
Zhang
,
J.
,
2010
, “
Nonlinear Analysis in a Lorenz-Like System
,”
Nonlinear Anal.: Real World Appl.
,
11
(
5
), pp.
3491
3500
.10.1016/j.nonrwa.2009.12.010
14.
Yang
,
Q.
, and
Wei
,
Z.
,
2010
, “
An Unusual 3D Autonomous Quadratic Chaotic System With Two Stable Node-Foci
,”
Int. J. Bifurcation Chaos
,
20
(
4
), pp.
1061
1083
.10.1142/S0218127410026320
15.
Wei
,
Z.
, and
Yang
,
Q.
,
2010
, “
Dynamics Analysis of a New Autonomous 3-D Chaotic System Only With Stable Equilibria
,”
Nonlinear Anal.: Real World Appl.
,
12
(
1
), pp.
106
118
10.1016/j.nonrwa.2010.05.038.
16.
Liu
,
Y.
,
Pang
,
S.
, and
Chen
,
D.
,
2013
, “
An Unusual Chaotic System and Its Control
,”
Math. Comput. Model
,
57
(
9–10
), pp.
2473
2493
.10.1016/j.mcm.2012.12.006
17.
Liu
,
C.
,
Liu
,
L.
, and
Liu
,
T.
,
2009
, “
A Novel Three-Dimensional Autonomous Chaos System
,”
Chaos Solitons Fractals
,
39
(
4
), pp.
1950
1958
.10.1016/j.chaos.2007.06.079
18.
Li
,
C.
,
Li
,
H.
, and
Tong
,
Y.
,
2013
, “
Analysis of a Novel Three-Dimensional Chaotic System
,”
Optik
,
124
(
13
), pp.
1516
1522
.10.1016/j.ijleo.2012.04.005
19.
Mees
,
A. I.
, and
Chapman
,
P. B.
,
1987
, “
Homoclinic and Heteroclinic Orbits in the Double Scroll Attractor
,”
IEEE Trans. Circuits Syst.
,
34
(
9
), pp.
1115
1120
.10.1109/TCS.1987.1086251
20.
Silva
,
C. P.
,
1993
, “
Shil'nikov's Theorem—A Tutorial
,”
IEEE Trans. Circuits Syst.
,
40
(
10
), pp.
675
682
.10.1109/81.246142
21.
Kuzenetsov
,
Y. A.
,
2004
,
Elements of Applied Bifurcation Theory
,
Third ed.
,
Springer-Verlag
,
New York
.
22.
Kokubu
,
H.
, and
Roussarie
,
R.
,
2004
, “
Existence of a Singularly Degenerate Heteroclinic Cycle in the Lorenz System and Its Dynamical Consequences: Part I
,”
J. Dyn. Differ. Equations
,
16
(
2
), pp.
513
557
10.1007/s10884-004-4290-4.
23.
Messias
,
M.
,
2009
, “
Dynamics at Infinity and the Existence of Singularly Degenerate Heteroclinic Cycles in the Lorenz System
,”
J. Phys. A: Math. Theor.
,
42
(11), p.
115101
.10.1088/1751-8113/42/11/115101
24.
Liu
,
Y.
,
2012
, “
Dynamics at Infinity and the Existence of Singularly Degenerate Heteroclinic Cycles in the Conjugate Lorenz-type System
,”
Nonlinear Anal.: Real World Appl.
,
13
(
6
), pp.
2466
2475
.10.1016/j.nonrwa.2012.02.011
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