- Email: [email protected]
On a four-dimensional chaotic system
Chaos, Solitons and Fractals 23 (2005) 1671–1682 www.elsevier.com/locate/chaos
On a four-dimensional chaotic system Guoyuan Qi a,*, Shengzhi Du b, Guanrong Chen c, Zengqiang Chen b, Zhuzhi yuan b a
Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, PR China b Department of Automation, Nankai University, Tianjin 300071, PR China c Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, PR China Accepted 22 June 2004 Communicated by Prof. T. Kapitaniak
Abstract This paper reports a new four-dimensional continuous autonomous chaotic system, in which each equation in the system contains a 3-term cross product. Basic properties of the system are analyzed by means of Lyapunov exponents and bifurcation diagrams. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction In 1963, Lorenz found the first chaotic attractor in a three-dimensional (3D) autonomous system when he studied atmospheric convection [1]. As the first chaotic model, the Lorenz system has become a paradigm of chaos research. Later, Ro¨sslor [2] constructed an even simpler 3D chaotic system. Notably, over the last two decades, chaos in engineering systems, such as nonlinear circuits, has gradually moved from simply being a scientific curiosity to a promising subject with practical significance and applications. It has been noticed that purposefully creating chaos can be a key issue in many technological applications. In this pursuit, Chen constructed a 3D autonomous chaotic system from an engineering feedback control approach [4,5], followed by a closely related Lu¨ system [7], and a unified system that combines the three systems as its special cases [8]. According to the canonical-form classification of Vanecek and Celikovsky [9], in the linear part A = [aij] of 3D autonomous systems with quadratic nonlinearities, the Lorenz system satisfies a12a21 > 0, the Chen system satisfies a12a21 < 0, while the Lu¨ system satisfies a12a21 = 0 and they are not topologically equivalent. In this sense, they together constitute a complete family of generalized Lorenz dynamical systems. In a different perspective, the Ro¨ssler system contains a quadratic cross-product term and does not belong to the family of generalized Lorenz systems mentioned above. In the recent study of Ro¨ssler-like 3D autonomous chaotic systems with quadratic cross-product terms, Liu and Chen [6] found a chaotic system with a quadratic cross-product term
*
Corresponding author. E-mail addresses: [email protected] (G. Qi), [email protected] (S. Du), [email protected] (G. Chen), [email protected] (Z. Chen). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.054
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in each equation, which can produces two co-existing double-scroll attractors. This paper further extends the same idea of using only cross-product nonlinearities to generate chaos from an autonomous system, and constructs a new 4D autonomous chaotic system with only one cubic cross-product term in each equation that can generate complex dynamics including chaos. Basic properties of the new chaotic system will be analyzed by means of Lyapunov exponents and bifurcation diagrams. Simulation results show that the system can generate various complex chaotic attractors when the system parameters are chosen appropriately.
2. The new 4D chaotic system and its basic properties The 4D autonomous system is described by x_ 1 ¼ aðx2 x1 Þ þ x2 x3 x4 x_ 2 ¼ bðx1 þ x2 Þ x1 x3 x4
ð1Þ
x_ 3 ¼ cx3 þ x1 x2 x4 x_ 4 ¼ dx4 þ x1 x2 x3
where xi (i = 1, 2, 3, 4) are the state variables of the system, and a, b, c, d are all positive real constant parameters. The finding of this system structure and the determination of its parameter values such that the system can become chaotic follow some basic ideas of chaotification [3,10], namely, to construct an autonomous chaotic system or to chaotify a non-chaotic autonomous system, the following general conditions should be considered: 1) The system is dissipative, namely, the energy of the system is decreasing (unless Hamiltonian type of systems are considered). 2) The system has unstable equilibria, namely, the Jacobian evaluated at the equilibria have unstable eigen values. 3) The system has at least one cross-product term, namely, dynamical influence between different variables can be accounted for. 4) The system orbits are all bounded, which means the reducing and increasing system energy maintains a dynamical balance. Although there are some systematic procedures for chaotifying a discrete system [3], there are no general guidelines for the continuous case to date. For the latter, one usually needs to combine the above-described analytic conditions with trial-and-error computer simulations to achieve an intended chaotification task. System (1) was also coined in this way. Now, some basic properties of system (1) are analyzed. For this system, one has rV ¼
o_x1 o_x2 o_x3 o_x4 þ þ þ ¼ b ða þ c þ dÞ: ox1 ox2 ox3 ox4
ð2Þ
Therefore, to ensure system (1) being dissipative, it is required that b ða þ c þ dÞ < 0:
ð3Þ
The equilibria of system (1) can be found by solving the following equations simultaneously: aðx2 x1 Þ þ x2 x3 x4 ¼ 0;
bðx1 þ x2 Þ x1 x3 x4 ¼ 0;
cx3 þ x1 x2 x4 ¼ 0;
dx4 þ x1 x2 x3 ¼ 0:
Obviously, S0 = [0, 0, 0, 0] is one equilibrium. Furthermore, from (4), one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 c bx41 ða þ bÞqx21 aq2 ¼ 0; x2 ¼ ; x3 ¼ adx21 adq; x4 ¼ x3 x1 q q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi where q ¼ cd . Setting p ¼ a2 þ 6ab þ b2 , one obtains 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1;2 2aða þ b pÞq; aða þ b þ pÞq; x3;4 1 ¼ pffiffiffi 1 ¼ 2a 2a 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aða þ b pÞq; x7;8 2aða þ b þ pÞq: x5;6 1 ¼ pffiffiffi 1 ¼ 2a 2a
ð4Þ
ð5Þ
ð6Þ
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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10
Lyapunov exponents
0 -10 -20 -30 -40 -50 -60 -1
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
d (a) Lyaponov-exponent spectrum: d ∈[0 35] 9 8 7 6
x4
5 4 3 2 1 0
0
5
10
15
20
d
25
30
35
(b) Bifurcation diagram of x4 (d ∈[0, 33])
6
6
5
5
4
4
x4 axis
x4 axis
Fig. 1. Lyaponov-exponents spectrum and bifurcation diagram w.r.t. d.
3
3
2
2
1
1
0 -6
-4
-2
0
2
x1 axis
(a) Projection on the x1−x4 plane.
4
6
0 -6
-4
-2
0
2
4
x2 axis
(b) Projection on the x2−x4 plane.
Fig. 2. Chaotic attractor of system (1): a = 35, b = 10, c = 1, d = 10.
Thus, one can easily calculate all equilibria by using other three expressions in (5). Notice that x4 has 64 roots, which leads to 64 equilibria in addition to S0. This is a very complex system. Only the stability of the zero equilibrium S0 is discussed here. By linearizing system (1) at S0, one obtains the Jaccobian
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682 4
4
3
3.5
2
3
1
2.5
x4 axis
x2 axis
1674
0 -1
2 1.5
-2
1
-3
0.5
-4 -5
0
0 -4
5
-2
0
x1 axis
2
4
x2 axis
(a) Projection on the x1−x2 plane.
(b) Projection on the x3−x4 plane.
4
x4 axis
3 2 1 0 10 5
9 0
8 7
xis
x2 a
-5
x3
is
ax
(c) 3D view on the x2−x3−x4 plane.
Fig. 3. Periodic orbit of system (1): a = 35, b = 10, c = 1, d = 25.
-0.5
3
-1
2.5
x 4 axis
x 2 axis
-1.5 -2
2
1.5
-2.5
1
-3
0.5
-3.5 -4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
9
9.2
9.4
x1 axis
(a) Projection on the x1 – x2 plane.
9.6 x3 axis
9.8
10
10.2
(b) Projection on the x3 – x4 plane.
Fig. 4. Cyclic orbit of system (1): a = 35, b = 10, c = 1, d = 34.
2
a 6 b 6 J ¼6 4 0 0
a 0 b 0 0 c 0 0
3 0 0 7 7 7: c 5 d
ð7Þ
The eigenvalues of matrix J are k1;2 ¼
ab 1 p; 2 2
k3 ¼ c; k3 ¼ d:
ð8Þ
Because a, b, c, d are all positive real numbers, one can easily found k2 > 0, implying that the equilibrium S0 is unstable.
2.5
18
2
16 x3 axis
x2 axis
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
1.5
1675
14
12 1
4 2
10 2.5 0.5 -1
0
1
x1 axis
2
3
0
2
1.5
4
1
0.5
x2 axis
(a) Projection on x1 − x 2 plane.
(b) 3D view on
x1 − x 2 − x3
-2
x1 axis
plane.
Fig. 5. Fixed-point solution of system (1): a = 35, b = 10, c = 1, d = 40.
For those nonzero equilibria, numerically evaluating their stabilities is still possible. Since one unstable equilibrium has been found at zero, this tedious numerical evaluation becomes less important therefore is not further discussed here.
3. Observation of new chaotic attractors Subject to condition (3), several simulations have been carried out, with some discoveries summarized as follows: 3.1. Fix a = 35, b = 10, c = 1, and vary d As can be seen from the Lyapunov-exponents spectrum and the bifurcation diagram with respect to the varying parameter d, shown in Fig. 1, when d 2 (0, 21.88], system (1) is chaotic with a positive Lyapunov exponent (for example, with d = 10, the phase portrait is shown in Fig. 2); while for d 2 (21.88, 26.8], the maximum Lyapunov exponent equals 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90
16
26
36
46
56
66
76
86
96
(a) Lyaponov-exponents spectrum: a ∈ [17, 90] 7 6 5
x4
4 3 2 1 0 10
20
30
40
a
50
60
70
80
(b) Bifurcation diagram of x 4 ( a ∈ [17, 90] ) Fig. 6. Lyaponov-exponents spectrum and bifurcation diagram w.r.t. a.
90
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G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682 10.2
50
10
x3axis
x3 axis
40 30
9.8 9.6
20 9.4
10 9.2 4
0 20
2
10
20
5 0
10
0
x2 axis
-20
-4
x2 axis
-10 -20
0
-2
0
-10
-5
x1 axis
x1 axis
a = 16
(a) 3D view (with the transient part):
a = 16
(b) 3D view (without the transient part):
6
6
5
5
4
4
x4 axis
x4 axis
Fig. 7. Periodic orbit of system (1): b = 10, c = 1, d = 10.
3
3
2
2
1
1
0 -5
-4
-3
-2
-1
0
1
2
3
4
0 -5
5
-4
-3
-2
-1
0
1
2
3
4
5
x2 axis
x2 axis
(a) Chaotic attractor: a = 30
(b) Chaotic attractor: a = 70
4.5 4 3.5
x4 axis
3 2.5 2 1.5 1 0.5 0 -2
x2 axis
-4
8.5
8
7.5
7
6.5
6
x3 axis
(c) Fix point: a = 83
Fig. 8. Different evolving modes of system (1): b = 10, c = 1, d = 10.
zero, implying that the system has a periodic orbit (Fig. 3 shows the periodic orbit when d = 25). When d 2 (26.8, 33], the system goes back to be chaotic again, but at this time the shape of its chaotic attractor is quite deferent from the
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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0
Lyapunov Exponents
-50
-100
-150
-200
-250
-300
-350
0
50
100
150
200
250
300
350
c
Fig. 9. Lyaponov-exponents spectrum once c 2 [0, 350].
previous one, as can be clearly seen from the bifurcations of the two different regions. The phase portrait evolves to another cyclic orbit when d is varying in (33, 36.5], as shown in Fig. 4 (where d = 34). Finally, when d > 36.5 (not shown in the Lyapunov-exponent spectrum), the system orbit is a fixed point (see Fig. 5, where d = 40). 3.2. Fix b = 10, c = 1, d = 10, and vary a
8
8
6
6
4
4
2
2
x2 axis
x2 axis
When 0 < a < 15, system (1) wanders around and then escapes to infinity. The corresponding Lyaponov-exponents spectrum and bifurcation diagram with a 2 [17, 90] are shown in Figs. 6(a) and (b), respectively. When 17 6 a < 23.98, the system has some typical periodic orbits, as shown in Figs. 6(a) and (b). In this region, the system eventually tends a cyclic orbit, which the progress shown in the state space of (x1, x2, x3). The initial state is located on the outside surface; the orbit moves up spirally till reaching the top, then moves to the inside surface spirally, and finally settles at a periodic orbit, as shown in Fig. 7. When 23.98 6 a 6 80.65, there appears a chaotic attractor, and the chaotic orbit evolves as a is increasing. As shown in the bifurcation diagram of x4 (Fig. 6), the attractor shapes a hole after a > 35, which is demonstrated in Figs. 8(a) and (b). When a > 80.7, the orbit converges to a fixed (shown in Fig. 8(c)).
0
0
-2
-2
-4
-4
-6
-6
-8 -10
-8
-6
-4
-2
0
x1 axis
(a) c = 14
2
4
6
8
10
-8 -10
-8
-6
-4
-2
0
x1 axis (b) c = 31
Fig. 10. Inverse bifurcation from chaos to periodic orbit: a = 30, b = 10, d = 10.
2
4
6
8
10
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3.3. Fix a = 30, b = 10, d = 10, and vary c Now, the dynamics of system (1) in the region of 0 < c < 350 is studied. Fig. 9 shows system (1) has very rich dynamical behaviors, which are summarized as follows. When 0 < c 6 15.3, the system is chaotic, as shown in Fig. 9 (the maximum Lyapunov exponent is positive). When c is around the point c0 = 33, the system orbit is divergent (the Lyapunov exponents are all infinite, which show broken points in Fig. 9). On both sides of this divergent point c0, there are two areas of periodic orbits as shown in Fig. 9, where two kinds of period solutions of system (1) exist. After crossing c0, as shown in Figs. 10(a) and (b), the periodic orbit of the system changes to another one, both in shape and in location. This implies that the system may go into another chaotic mode. This hypothesis is confirmed by Figs. 10(a) and 11(d), in which the chaotic orbits of system (1) in the two chaos regions have very different shapes. Next, the system evolution process is investigated in more detail. Starting with the first chaotic region (0 < c 6 15.3), Fig. 10(a) shows the chaotic orbit of system (1). With c increasing, the orbit evolves to a loop-like periodic one through inverse bifurcation as shown in Fig. 10(b). Although these two orbits are different in dynamical mode, they have some similarity in shape and location. Then, on the right side of c0, with c increasing, beginning with a Hopf periodic orbit (for example, result of c = 35 is shown in Fig. 11(a)), the orbit continuously evolves to a complex one through a com-
15
15
10
10
5
5
x2 axis
20
x2 axis
20
0 -5
-5
-10
-10
-15
-15
-20 -20
-10
0
10
20
-20 -20
20
15
15
10
10
5
5
0
-5
-10
-10
-15
-15
-20
-20 -5
-10
-8
0
x1 axis (c) c = 36
5
10
15
20
25
-6
-4
-2
0
10
15
20
25
0
-5
-10
-12
(b) c = 35.3
20
-15
-14
(a) c = 35 25
-20
-16
x2 axis
25
-25 -25
-18
x1 axis
x2 axis
x2 axis
0
-25 -25
-20
-15
-10
-5
0
5
x1 axis (d) c = 37
Fig. 11. Complex bifurcation: from Hopf periodic solution through period-doubling and tangent bifurcations to chaos: a = 30, b = 10, d = 10.
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bination of period-doubling and tangential bifurcations, as demonstrated in Figs. 11(b)–(c). At last, it goes into chaos when c = 37, as shown in Fig. 11(d). In this evolution process, the shapes of different states orbits change continuously, showing some similarities between adjacent periodic orbits. The interesting discontinuous feature of chaotic dynamics of system (1) deserves some discussion. After c > 37, the system orbit suddenly trends to a wide cyclic area without any obvious evidence of going through a inverse bifurcation, and then comes back to chaotic motion without passing through any obvious bifurcation (91 < c < 126.7). This demonstrates the discontinuous feature of chaos evolution––the system transit from chaotic to periodic motion or the reverse process can occur suddenly just at a critical point. The chaos attracters or periodic orbits can also collapse instantaneously. 3.4. Fix a = 30, c = 1, d = 10, and vary b As shown in Figs. 12(a) and (b), it can be clearly seen that the convergent area, chaotic area, and periodic area appear one by one when b is increased gradually. The transition points are b = 4.25 and 14.3. 3.5. Fix b = 10 and d = 10, and vary 2 parameters a and c Now, fix b = 10 and d = 10 and let parameters a and c be varied. Fig. 13 shows a local parameters division (7 6 a 6 60 and 90 6 c 6 130) according to the changes of the chaos-related properties of system (1). In Fig. 13, there are mainly three regions: deep blue (F), light blue (P), and red (C) divisions, representing areas of convergent, periodic, and chaotic orbits respectively. There is also a range of half tints between these areas representing some other dynamics including transitions. Firstly, the orbit evolution is demonstrated when parameters are varied following the trace: a ! d (a 2 [7, 28] and c = 110) shown in Fig. 13. The line a ! d begins with the convergent (F) area outside the colored region, the crosses the periodic strip (P) and transition (T) areas, and finally reaches the chaotic (C) area. Fig. 14(a) shows how the orbit bifurcates to chaos. It is clearly that the system evolves to chaos through a series period-doubling bifurcations. 5
Lyapunov exponents
0 -5 - 10 - 15 - 20 - 25 - 30 - 35
0
2
4
6
8
10
12
b (a) Lyapunov-exponents spectrum:
14
16
18
20
a = 30, c = 1, d = 10
3.5 3
x4
2.5 2 1.5 1 0.5 0 0
2
4
6
8
10
12
14
16
18
b
(b) Bifurcation diagram : a = 30, c = 1, d = 10 Fig. 12. Lyaponov-exponents spectrum and bifurcation diagram w.r.t. b.
20
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G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
Fig. 13. Division of parameters: F––convergent region; P––periodic orbit; T––transition; C––chaos region.
20
14 12
15
x4
8
10
6
5 4 2 5
10
15
a
20
25
0 30
30
32
34
36
a (b) e → g ( a ∈ [30, 38] , c = 108 )
(a) a → d ( a ∈ [7, 28] and c = 110 )
13 12 11 10 9
x4
x4
10
8 7 6 5 4 3 99
100
101
102
103
104
c
(c) h → i ( a = 22 and c ∈ [99, 105] ) Fig. 14. Transiting processes of system (1).
105
38
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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Fig. 15. Chaotic attractors observed from system (1) with a time-varying parameter.
Secondly, the trace e ! g (a 2 [30, 38], c = 108) is examined, which shows the chaos–non-chaos and non-chaos– chaos processes of the system with a periodic segment splitting a chaotic area into parts. Perfect inverse-period-doubling and period-doubling bifurcations can be clearly seen in Fig. 14(b). Finally, the transition from chaos to periodic orbits is investigated by following the trace: h ! i (a = 22 and c 2 [99, 105]). Fig. 14(c) shows that in the half tint areas of Fig. 13 the system is chaotic but there exists a broad hollow band, showing some holes (different dynamics) around the system equilibria in the phase space. 3.6. Time-varying parameters The above studies are restricted to constant parameters. Now, consider system (1) with some time-varying parameters. In doing so, system (1) becomes non-autonomous, and is equivalent to a 5D autonomous system as is well known. Therefore, very complex dynamics can be expected. Simulations show that as long as the time-varying parameters are within the chaotic parameters regions discussed above, system (1) remains to be chaotic. For example, take a = 30, b = 10, d = 10, c = 95 + 10 sin(t). The system produces a chaotic attractor in a spindle shape, as shown in Fig. 15(a). Take a = 30, b = 10, c = 1, d = 16 + 12 sin(t), as another example. The chaotic attractor looks similar to a fishnet, as shown in Fig. 15(b). 4. Conclusions This paper has reported and described a new four-dimensional continuous autonomous chaotic system, in which each equation in the system has a 3-term cross product. Basic properties of the system have been analyzed by means of Lyapunov exponents and bifurcation diagrams. This seemingly simple 4D autonomous system has very complex dynamics, such as having 65 equilibria and various chaotic and bifurcation behaviors. Apparently there are much more left out for in-depth study about this new system in terms of dynamics and complexity, for example the possibility of hyperchaos in the system with a time-varying parameter.
Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant 60374037) and Nankai University Innovation Foundation.
References [1] Sparrow C. The lorenz equations: bifurcations, chaos, and strange attractors. NY: Springer-Verlag; 1982. [2] Ro¨sslor OE. An equation for continuous chaos. Phys Lett A 1976;57:397–8. [3] Chen G. Chaotification via feedback: the discrete case. In: Chen G, Yu X, editors. Chaos control: theory and applications. Berlin: Springer; 2003. p. 159–77.
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Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:1465–6. Ueta T, Chen G. Bifurcation analysis of ChenÕs equation. Int J Bifurcat Chaos 2000;10:1917–31. Liu WB, Chen G. A new chaotic system and its generation. Int J Bifurcat Chaos 2003;13:261–7. Lu¨ JH, Chen G. A new chaotic attractor coined. Int J Bifurcat Chaos 2002;12:659–61. Lu¨ JH, Chen G, Celikovsky´ S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917–26. [9] Vanecek A, Celikovsky´ S. Control systems: from linear analysis to synthesis of chaos. London: Prentice-Hall; 1996. [10] Wang XF. Generating chaos in continuous-time systems via feedback control. In: Chen G, Yu X, editors. Chaos control: theory and applications. Berlin: Springer; 2003. p. 179–204.
On a four-dimensional chaotic system Guoyuan Qi a,*, Shengzhi Du b, Guanrong Chen c, Zengqiang Chen b, Zhuzhi yuan b a
Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, PR China b Department of Automation, Nankai University, Tianjin 300071, PR China c Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, PR China Accepted 22 June 2004 Communicated by Prof. T. Kapitaniak
Abstract This paper reports a new four-dimensional continuous autonomous chaotic system, in which each equation in the system contains a 3-term cross product. Basic properties of the system are analyzed by means of Lyapunov exponents and bifurcation diagrams. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction In 1963, Lorenz found the first chaotic attractor in a three-dimensional (3D) autonomous system when he studied atmospheric convection [1]. As the first chaotic model, the Lorenz system has become a paradigm of chaos research. Later, Ro¨sslor [2] constructed an even simpler 3D chaotic system. Notably, over the last two decades, chaos in engineering systems, such as nonlinear circuits, has gradually moved from simply being a scientific curiosity to a promising subject with practical significance and applications. It has been noticed that purposefully creating chaos can be a key issue in many technological applications. In this pursuit, Chen constructed a 3D autonomous chaotic system from an engineering feedback control approach [4,5], followed by a closely related Lu¨ system [7], and a unified system that combines the three systems as its special cases [8]. According to the canonical-form classification of Vanecek and Celikovsky [9], in the linear part A = [aij] of 3D autonomous systems with quadratic nonlinearities, the Lorenz system satisfies a12a21 > 0, the Chen system satisfies a12a21 < 0, while the Lu¨ system satisfies a12a21 = 0 and they are not topologically equivalent. In this sense, they together constitute a complete family of generalized Lorenz dynamical systems. In a different perspective, the Ro¨ssler system contains a quadratic cross-product term and does not belong to the family of generalized Lorenz systems mentioned above. In the recent study of Ro¨ssler-like 3D autonomous chaotic systems with quadratic cross-product terms, Liu and Chen [6] found a chaotic system with a quadratic cross-product term
*
Corresponding author. E-mail addresses: [email protected] (G. Qi), [email protected] (S. Du), [email protected] (G. Chen), [email protected] (Z. Chen). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.054
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G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
in each equation, which can produces two co-existing double-scroll attractors. This paper further extends the same idea of using only cross-product nonlinearities to generate chaos from an autonomous system, and constructs a new 4D autonomous chaotic system with only one cubic cross-product term in each equation that can generate complex dynamics including chaos. Basic properties of the new chaotic system will be analyzed by means of Lyapunov exponents and bifurcation diagrams. Simulation results show that the system can generate various complex chaotic attractors when the system parameters are chosen appropriately.
2. The new 4D chaotic system and its basic properties The 4D autonomous system is described by x_ 1 ¼ aðx2 x1 Þ þ x2 x3 x4 x_ 2 ¼ bðx1 þ x2 Þ x1 x3 x4
ð1Þ
x_ 3 ¼ cx3 þ x1 x2 x4 x_ 4 ¼ dx4 þ x1 x2 x3
where xi (i = 1, 2, 3, 4) are the state variables of the system, and a, b, c, d are all positive real constant parameters. The finding of this system structure and the determination of its parameter values such that the system can become chaotic follow some basic ideas of chaotification [3,10], namely, to construct an autonomous chaotic system or to chaotify a non-chaotic autonomous system, the following general conditions should be considered: 1) The system is dissipative, namely, the energy of the system is decreasing (unless Hamiltonian type of systems are considered). 2) The system has unstable equilibria, namely, the Jacobian evaluated at the equilibria have unstable eigen values. 3) The system has at least one cross-product term, namely, dynamical influence between different variables can be accounted for. 4) The system orbits are all bounded, which means the reducing and increasing system energy maintains a dynamical balance. Although there are some systematic procedures for chaotifying a discrete system [3], there are no general guidelines for the continuous case to date. For the latter, one usually needs to combine the above-described analytic conditions with trial-and-error computer simulations to achieve an intended chaotification task. System (1) was also coined in this way. Now, some basic properties of system (1) are analyzed. For this system, one has rV ¼
o_x1 o_x2 o_x3 o_x4 þ þ þ ¼ b ða þ c þ dÞ: ox1 ox2 ox3 ox4
ð2Þ
Therefore, to ensure system (1) being dissipative, it is required that b ða þ c þ dÞ < 0:
ð3Þ
The equilibria of system (1) can be found by solving the following equations simultaneously: aðx2 x1 Þ þ x2 x3 x4 ¼ 0;
bðx1 þ x2 Þ x1 x3 x4 ¼ 0;
cx3 þ x1 x2 x4 ¼ 0;
dx4 þ x1 x2 x3 ¼ 0:
Obviously, S0 = [0, 0, 0, 0] is one equilibrium. Furthermore, from (4), one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 c bx41 ða þ bÞqx21 aq2 ¼ 0; x2 ¼ ; x3 ¼ adx21 adq; x4 ¼ x3 x1 q q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi where q ¼ cd . Setting p ¼ a2 þ 6ab þ b2 , one obtains 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1;2 2aða þ b pÞq; aða þ b þ pÞq; x3;4 1 ¼ pffiffiffi 1 ¼ 2a 2a 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aða þ b pÞq; x7;8 2aða þ b þ pÞq: x5;6 1 ¼ pffiffiffi 1 ¼ 2a 2a
ð4Þ
ð5Þ
ð6Þ
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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10
Lyapunov exponents
0 -10 -20 -30 -40 -50 -60 -1
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
d (a) Lyaponov-exponent spectrum: d ∈[0 35] 9 8 7 6
x4
5 4 3 2 1 0
0
5
10
15
20
d
25
30
35
(b) Bifurcation diagram of x4 (d ∈[0, 33])
6
6
5
5
4
4
x4 axis
x4 axis
Fig. 1. Lyaponov-exponents spectrum and bifurcation diagram w.r.t. d.
3
3
2
2
1
1
0 -6
-4
-2
0
2
x1 axis
(a) Projection on the x1−x4 plane.
4
6
0 -6
-4
-2
0
2
4
x2 axis
(b) Projection on the x2−x4 plane.
Fig. 2. Chaotic attractor of system (1): a = 35, b = 10, c = 1, d = 10.
Thus, one can easily calculate all equilibria by using other three expressions in (5). Notice that x4 has 64 roots, which leads to 64 equilibria in addition to S0. This is a very complex system. Only the stability of the zero equilibrium S0 is discussed here. By linearizing system (1) at S0, one obtains the Jaccobian
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682 4
4
3
3.5
2
3
1
2.5
x4 axis
x2 axis
1674
0 -1
2 1.5
-2
1
-3
0.5
-4 -5
0
0 -4
5
-2
0
x1 axis
2
4
x2 axis
(a) Projection on the x1−x2 plane.
(b) Projection on the x3−x4 plane.
4
x4 axis
3 2 1 0 10 5
9 0
8 7
xis
x2 a
-5
x3
is
ax
(c) 3D view on the x2−x3−x4 plane.
Fig. 3. Periodic orbit of system (1): a = 35, b = 10, c = 1, d = 25.
-0.5
3
-1
2.5
x 4 axis
x 2 axis
-1.5 -2
2
1.5
-2.5
1
-3
0.5
-3.5 -4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
9
9.2
9.4
x1 axis
(a) Projection on the x1 – x2 plane.
9.6 x3 axis
9.8
10
10.2
(b) Projection on the x3 – x4 plane.
Fig. 4. Cyclic orbit of system (1): a = 35, b = 10, c = 1, d = 34.
2
a 6 b 6 J ¼6 4 0 0
a 0 b 0 0 c 0 0
3 0 0 7 7 7: c 5 d
ð7Þ
The eigenvalues of matrix J are k1;2 ¼
ab 1 p; 2 2
k3 ¼ c; k3 ¼ d:
ð8Þ
Because a, b, c, d are all positive real numbers, one can easily found k2 > 0, implying that the equilibrium S0 is unstable.
2.5
18
2
16 x3 axis
x2 axis
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
1.5
1675
14
12 1
4 2
10 2.5 0.5 -1
0
1
x1 axis
2
3
0
2
1.5
4
1
0.5
x2 axis
(a) Projection on x1 − x 2 plane.
(b) 3D view on
x1 − x 2 − x3
-2
x1 axis
plane.
Fig. 5. Fixed-point solution of system (1): a = 35, b = 10, c = 1, d = 40.
For those nonzero equilibria, numerically evaluating their stabilities is still possible. Since one unstable equilibrium has been found at zero, this tedious numerical evaluation becomes less important therefore is not further discussed here.
3. Observation of new chaotic attractors Subject to condition (3), several simulations have been carried out, with some discoveries summarized as follows: 3.1. Fix a = 35, b = 10, c = 1, and vary d As can be seen from the Lyapunov-exponents spectrum and the bifurcation diagram with respect to the varying parameter d, shown in Fig. 1, when d 2 (0, 21.88], system (1) is chaotic with a positive Lyapunov exponent (for example, with d = 10, the phase portrait is shown in Fig. 2); while for d 2 (21.88, 26.8], the maximum Lyapunov exponent equals 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90
16
26
36
46
56
66
76
86
96
(a) Lyaponov-exponents spectrum: a ∈ [17, 90] 7 6 5
x4
4 3 2 1 0 10
20
30
40
a
50
60
70
80
(b) Bifurcation diagram of x 4 ( a ∈ [17, 90] ) Fig. 6. Lyaponov-exponents spectrum and bifurcation diagram w.r.t. a.
90
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G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682 10.2
50
10
x3axis
x3 axis
40 30
9.8 9.6
20 9.4
10 9.2 4
0 20
2
10
20
5 0
10
0
x2 axis
-20
-4
x2 axis
-10 -20
0
-2
0
-10
-5
x1 axis
x1 axis
a = 16
(a) 3D view (with the transient part):
a = 16
(b) 3D view (without the transient part):
6
6
5
5
4
4
x4 axis
x4 axis
Fig. 7. Periodic orbit of system (1): b = 10, c = 1, d = 10.
3
3
2
2
1
1
0 -5
-4
-3
-2
-1
0
1
2
3
4
0 -5
5
-4
-3
-2
-1
0
1
2
3
4
5
x2 axis
x2 axis
(a) Chaotic attractor: a = 30
(b) Chaotic attractor: a = 70
4.5 4 3.5
x4 axis
3 2.5 2 1.5 1 0.5 0 -2
x2 axis
-4
8.5
8
7.5
7
6.5
6
x3 axis
(c) Fix point: a = 83
Fig. 8. Different evolving modes of system (1): b = 10, c = 1, d = 10.
zero, implying that the system has a periodic orbit (Fig. 3 shows the periodic orbit when d = 25). When d 2 (26.8, 33], the system goes back to be chaotic again, but at this time the shape of its chaotic attractor is quite deferent from the
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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0
Lyapunov Exponents
-50
-100
-150
-200
-250
-300
-350
0
50
100
150
200
250
300
350
c
Fig. 9. Lyaponov-exponents spectrum once c 2 [0, 350].
previous one, as can be clearly seen from the bifurcations of the two different regions. The phase portrait evolves to another cyclic orbit when d is varying in (33, 36.5], as shown in Fig. 4 (where d = 34). Finally, when d > 36.5 (not shown in the Lyapunov-exponent spectrum), the system orbit is a fixed point (see Fig. 5, where d = 40). 3.2. Fix b = 10, c = 1, d = 10, and vary a
8
8
6
6
4
4
2
2
x2 axis
x2 axis
When 0 < a < 15, system (1) wanders around and then escapes to infinity. The corresponding Lyaponov-exponents spectrum and bifurcation diagram with a 2 [17, 90] are shown in Figs. 6(a) and (b), respectively. When 17 6 a < 23.98, the system has some typical periodic orbits, as shown in Figs. 6(a) and (b). In this region, the system eventually tends a cyclic orbit, which the progress shown in the state space of (x1, x2, x3). The initial state is located on the outside surface; the orbit moves up spirally till reaching the top, then moves to the inside surface spirally, and finally settles at a periodic orbit, as shown in Fig. 7. When 23.98 6 a 6 80.65, there appears a chaotic attractor, and the chaotic orbit evolves as a is increasing. As shown in the bifurcation diagram of x4 (Fig. 6), the attractor shapes a hole after a > 35, which is demonstrated in Figs. 8(a) and (b). When a > 80.7, the orbit converges to a fixed (shown in Fig. 8(c)).
0
0
-2
-2
-4
-4
-6
-6
-8 -10
-8
-6
-4
-2
0
x1 axis
(a) c = 14
2
4
6
8
10
-8 -10
-8
-6
-4
-2
0
x1 axis (b) c = 31
Fig. 10. Inverse bifurcation from chaos to periodic orbit: a = 30, b = 10, d = 10.
2
4
6
8
10
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3.3. Fix a = 30, b = 10, d = 10, and vary c Now, the dynamics of system (1) in the region of 0 < c < 350 is studied. Fig. 9 shows system (1) has very rich dynamical behaviors, which are summarized as follows. When 0 < c 6 15.3, the system is chaotic, as shown in Fig. 9 (the maximum Lyapunov exponent is positive). When c is around the point c0 = 33, the system orbit is divergent (the Lyapunov exponents are all infinite, which show broken points in Fig. 9). On both sides of this divergent point c0, there are two areas of periodic orbits as shown in Fig. 9, where two kinds of period solutions of system (1) exist. After crossing c0, as shown in Figs. 10(a) and (b), the periodic orbit of the system changes to another one, both in shape and in location. This implies that the system may go into another chaotic mode. This hypothesis is confirmed by Figs. 10(a) and 11(d), in which the chaotic orbits of system (1) in the two chaos regions have very different shapes. Next, the system evolution process is investigated in more detail. Starting with the first chaotic region (0 < c 6 15.3), Fig. 10(a) shows the chaotic orbit of system (1). With c increasing, the orbit evolves to a loop-like periodic one through inverse bifurcation as shown in Fig. 10(b). Although these two orbits are different in dynamical mode, they have some similarity in shape and location. Then, on the right side of c0, with c increasing, beginning with a Hopf periodic orbit (for example, result of c = 35 is shown in Fig. 11(a)), the orbit continuously evolves to a complex one through a com-
15
15
10
10
5
5
x2 axis
20
x2 axis
20
0 -5
-5
-10
-10
-15
-15
-20 -20
-10
0
10
20
-20 -20
20
15
15
10
10
5
5
0
-5
-10
-10
-15
-15
-20
-20 -5
-10
-8
0
x1 axis (c) c = 36
5
10
15
20
25
-6
-4
-2
0
10
15
20
25
0
-5
-10
-12
(b) c = 35.3
20
-15
-14
(a) c = 35 25
-20
-16
x2 axis
25
-25 -25
-18
x1 axis
x2 axis
x2 axis
0
-25 -25
-20
-15
-10
-5
0
5
x1 axis (d) c = 37
Fig. 11. Complex bifurcation: from Hopf periodic solution through period-doubling and tangent bifurcations to chaos: a = 30, b = 10, d = 10.
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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bination of period-doubling and tangential bifurcations, as demonstrated in Figs. 11(b)–(c). At last, it goes into chaos when c = 37, as shown in Fig. 11(d). In this evolution process, the shapes of different states orbits change continuously, showing some similarities between adjacent periodic orbits. The interesting discontinuous feature of chaotic dynamics of system (1) deserves some discussion. After c > 37, the system orbit suddenly trends to a wide cyclic area without any obvious evidence of going through a inverse bifurcation, and then comes back to chaotic motion without passing through any obvious bifurcation (91 < c < 126.7). This demonstrates the discontinuous feature of chaos evolution––the system transit from chaotic to periodic motion or the reverse process can occur suddenly just at a critical point. The chaos attracters or periodic orbits can also collapse instantaneously. 3.4. Fix a = 30, c = 1, d = 10, and vary b As shown in Figs. 12(a) and (b), it can be clearly seen that the convergent area, chaotic area, and periodic area appear one by one when b is increased gradually. The transition points are b = 4.25 and 14.3. 3.5. Fix b = 10 and d = 10, and vary 2 parameters a and c Now, fix b = 10 and d = 10 and let parameters a and c be varied. Fig. 13 shows a local parameters division (7 6 a 6 60 and 90 6 c 6 130) according to the changes of the chaos-related properties of system (1). In Fig. 13, there are mainly three regions: deep blue (F), light blue (P), and red (C) divisions, representing areas of convergent, periodic, and chaotic orbits respectively. There is also a range of half tints between these areas representing some other dynamics including transitions. Firstly, the orbit evolution is demonstrated when parameters are varied following the trace: a ! d (a 2 [7, 28] and c = 110) shown in Fig. 13. The line a ! d begins with the convergent (F) area outside the colored region, the crosses the periodic strip (P) and transition (T) areas, and finally reaches the chaotic (C) area. Fig. 14(a) shows how the orbit bifurcates to chaos. It is clearly that the system evolves to chaos through a series period-doubling bifurcations. 5
Lyapunov exponents
0 -5 - 10 - 15 - 20 - 25 - 30 - 35
0
2
4
6
8
10
12
b (a) Lyapunov-exponents spectrum:
14
16
18
20
a = 30, c = 1, d = 10
3.5 3
x4
2.5 2 1.5 1 0.5 0 0
2
4
6
8
10
12
14
16
18
b
(b) Bifurcation diagram : a = 30, c = 1, d = 10 Fig. 12. Lyaponov-exponents spectrum and bifurcation diagram w.r.t. b.
20
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Fig. 13. Division of parameters: F––convergent region; P––periodic orbit; T––transition; C––chaos region.
20
14 12
15
x4
8
10
6
5 4 2 5
10
15
a
20
25
0 30
30
32
34
36
a (b) e → g ( a ∈ [30, 38] , c = 108 )
(a) a → d ( a ∈ [7, 28] and c = 110 )
13 12 11 10 9
x4
x4
10
8 7 6 5 4 3 99
100
101
102
103
104
c
(c) h → i ( a = 22 and c ∈ [99, 105] ) Fig. 14. Transiting processes of system (1).
105
38
G. Qi et al. / Chaos, Solitons and Fractals 23 (2005) 1671–1682
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Fig. 15. Chaotic attractors observed from system (1) with a time-varying parameter.
Secondly, the trace e ! g (a 2 [30, 38], c = 108) is examined, which shows the chaos–non-chaos and non-chaos– chaos processes of the system with a periodic segment splitting a chaotic area into parts. Perfect inverse-period-doubling and period-doubling bifurcations can be clearly seen in Fig. 14(b). Finally, the transition from chaos to periodic orbits is investigated by following the trace: h ! i (a = 22 and c 2 [99, 105]). Fig. 14(c) shows that in the half tint areas of Fig. 13 the system is chaotic but there exists a broad hollow band, showing some holes (different dynamics) around the system equilibria in the phase space. 3.6. Time-varying parameters The above studies are restricted to constant parameters. Now, consider system (1) with some time-varying parameters. In doing so, system (1) becomes non-autonomous, and is equivalent to a 5D autonomous system as is well known. Therefore, very complex dynamics can be expected. Simulations show that as long as the time-varying parameters are within the chaotic parameters regions discussed above, system (1) remains to be chaotic. For example, take a = 30, b = 10, d = 10, c = 95 + 10 sin(t). The system produces a chaotic attractor in a spindle shape, as shown in Fig. 15(a). Take a = 30, b = 10, c = 1, d = 16 + 12 sin(t), as another example. The chaotic attractor looks similar to a fishnet, as shown in Fig. 15(b). 4. Conclusions This paper has reported and described a new four-dimensional continuous autonomous chaotic system, in which each equation in the system has a 3-term cross product. Basic properties of the system have been analyzed by means of Lyapunov exponents and bifurcation diagrams. This seemingly simple 4D autonomous system has very complex dynamics, such as having 65 equilibria and various chaotic and bifurcation behaviors. Apparently there are much more left out for in-depth study about this new system in terms of dynamics and complexity, for example the possibility of hyperchaos in the system with a time-varying parameter.
Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant 60374037) and Nankai University Innovation Foundation.
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