Multiple Chaotic Attractors in Coupled Lorenz Systems
aa r X i v : . [ n li n . C D ] J un Multiple Chaotic Attractors in Coupled Lorenz Systems
Mehmet Onur Fen a, ∗ a Department of Mathematics, TED University, 06420 Ankara, Turkey
Abstract
Unidirectionally coupled Lorenz systems in which the drive possesses a chaotic attractor and the responseadmits two stable equilibria in the absence of the driving is under investigation. It is found that doublechaotic attractors coexist in the dynamics. The approach is applicable for chains of coupled Lorenzsystems. The existence of four chaotic attractors in three coupled Lorenz systems is also demonstrated.
Keywords:
Multiple chaotic attractors; Lorenz system; Unidirectional coupling
The system of differential equations ˙ x = − σx + σx ˙ x = − x x + rx − x ˙ x = x x − bx , (1.1)where σ , r , and b are constants, was presented by Lorenz [1] to investigate the dynamics of the atmo-sphere. System (1.1) is capable of exhibiting stable equilibria, periodic orbits, homoclinic explosions,period-doubling bifurcations, and chaos with different values of σ , r , and b [2].Extension of chaos in dynamics of unidirectionally coupled Lorenz systems in which the responseadmits either a stable equilibrium point or a stable limit cycle in the absence of driving was demonstratedin study [3]. Moreover, it was shown in paper [4] that under certain conditions chaos is present in thedynamics of the response system even if generalized synchronization is not present. Motivated by theresults of [3] and [4], in this paper we also investigate unidirectionally coupled Lorenz systems, but thistime the response is system with two stable equilibrium points in the absence of driving. The mainpurpose of the present study is the demonstration of the coexistence of multiple chaotic attractors in thedynamics of Lorenz systems when they are coupled in a unidirectional way. The coexistence of two andfour chaotic attractors are shown.The paper is structured as follows. In Section 2 the model of coupled Lorenz systems is introduced andthe presence of sensitivity, which is the main ingredient of chaos [1, 5], is discussed from the theoretical ∗ Corresponding Author Tel.: +90 312 585 0217, E-mail: [email protected]
We consider the drive system in the form of (1.1) and the response system is ˙ y = − σy + σy + µ f ( x ( t ))˙ y = − y y + ry − y + µ f ( x ( t ))˙ y = y y − by + µ f ( x ( t )) , (2.2)where σ, r , b , µ , µ , µ are constants, x ( t ) = ( x ( t ) , x ( t ) , x ( t )) is a solution of (1.1), and the functions f , f , and f are continuous. We mainly assume that the drive system (1.1) is chaotic, i.e., it admitssensitivity and infinitely many unstable periodic orbits embedded in the chaotic attractor, and that theconstants σ , r , b in the response system (2.2) are such that the Lorenz system ˙ u = − σu + σu ˙ u = − u u + ru − u ˙ u = u u − bu , (2.3)admits two stable equilibrium points.It is worth noting that the coupled system (1.1)-(2.2) has a skew product structure. Since the drive(1.1) is chaotic, it possesses a compact invariant set Λ ⊂ R . Another assumption on system (2.2) is theexistence of a positive number L satisfying k f ( x ) − f ( x ) k ≥ L k x − x k for all x, x ∈ Λ , where the function f : Λ → R is defined by f ( x ) = ( µ f ( x ) , µ f ( x ) , µ f ( x )) and k . k denotes the usual Euclidean norm in R .We call a solution x ( t ) of the drive (1.1) satisfying x (0) = x , where x is a point which belongs tothe chaotic attractor of the system, a chaotic solution since it is used as a perturbation in (2.2). Chaoticsolutions may be irregular as well as regular, i.e., periodic and unstable [1, 2, 5].System (1.1) is called sensitive if there exist positive numbers ǫ and ∆ such that for an arbitrarypositive number δ and for each chaotic solution x ( t ) of (1.1), there exist a chaotic solution x ( t ) of thesystem and an interval J ⊂ [0 , ∞ ) with a length no less than ∆ such that k x (0) − x (0) k < δ and2 x ( t ) − x ( t ) k > ǫ for every t in J [3, 4].For the discussion of sensitivity in the response system (2.2), we require that the system possesses acompact invariant set U ⊂ R for each chaotic solution x ( t ) of (1.1). The existence of such an invariantset can be shown, for instance, by means of Lyapunov functions [3, 6].Let us denote by φ x ( t ) ( t, y ) the solution of (2.2) satisfying φ x ( t ) (0 , y ) = y , where x ( t ) is a solutionof the drive (1.1) and y is a point in U . We say that system (2.2) is sensitive if there exist positivenumbers ǫ and e ∆ such that for an arbitrary positive number δ , each y ∈ U , and a chaotic solution x ( t ) of (1.1), there exist y ∈ U , a chaotic solution x ( t ) of (1.1), and an interval e J ⊂ [0 , ∞ ) with a lengthno less than e ∆ such that k y − y k < δ and (cid:13)(cid:13) φ x ( t ) ( t, y ) − φ x ( t ) ( t, y ) (cid:13)(cid:13) > ǫ for all t in e J [3, 4].It can be proved in a very similar way to Theorem 3.1 presented in paper [4] that the response system(2.2) is sensitive.The coexistence of two chaotic attractors in the dynamics of the coupled system (1.1)-(2.2) is demon-strated in the next section, provided that the constants µ , µ , and µ are sufficiently small in absolutevalue. In order to demonstrate the coexistence of two chaotic attractors in the dynamics of the -dimensionalsystem (1.1)-(2.2), we set σ = 10 , r = 28 , and b = 8 / such that the drive system (1.1) admits achaotic attractor [1, 2]. Additionally, we consider the response system (2.2) with σ = 10 , r = 12 , b = 8 / , µ = µ = µ = 0 . , and f ( x , x , x ) = 7 . x + cos x , f ( x , x , x ) = 1 . x + 0 . x , f ( x , x , x ) = 3 . x + 0 . e − x . The points ( − p / , − p / , and (2 p / , p / , arethe stable equilibria of system (2.3) with the aforementioned choices of the parameters σ , r , and b [2].Figure 1 depicts the projections of two chaotic trajectories of the coupled system (1.1)-(2.2) onthe y y y -space. The trajectory in blue corresponds to the initial data x (0) = − . , x (0) = − . , x (0) = 26 . , y (0) = − . , y (0) = − . , y (0) = 10 . , whereas the trajectory inred corresponds to the initial data x (0) = 13 . , x (0) = 10 . , x (0) = 36 . , y (0) = 3 . , y (0) = 4 . , y (0) = 11 . . Figure 2, on the other hand, shows the time series of the y -coordinatesof these trajectories. Figure 2 (a) and (b) respectively represent the time series of the trajectory in blueand the trajectory in red shown in Figure 1. Both Figure 1 and Figure 2 manifest that double chaoticattractors coexist in the dynamics of the coupled system (1.1)-(2.2).3 y y -5 51416 0 05 -510 -10 Figure 1: Chaotic trajectories of the -dimensional system (1.1)-(2.2). The figure reveals the coexistenceof two chaotic attractors. t -8-6-4-2 y (a) t y (b) Figure 2: Time series of the y coordinates of two chaotic solutions of the coupled system (1.1)-(2.2).(a) The time series corresponding to the initial data x (0) = − . , x (0) = − . , x (0) = 26 . , y (0) = − . , y (0) = − . , y (0) = 10 . ; (b) The time series corresponding to the initial data x (0) = 13 . , x (0) = 10 . , x (0) = 36 . , y (0) = 3 . , y (0) = 4 . , y (0) = 11 . . In this section we will show the coexistence of four chaotic attractors in three coupled Lorenz systems.For that purpose, in addition to the coupled system (1.1)-(2.2), we set up the system ˙ z = − z + 10 z + 0 . y ( t )˙ z = − z z + 3 . z − z + 0 . y ( t )˙ z = z z − z + 0 . y ( t ) , (4.4)4here y ( t ) = ( y ( t ) , y ( t ) , y ( t )) is a solution of (2.2). Considering the coupling between the systems(2.2) and (4.4), system (2.2) is the drive and system (4.4) is the response. The parameters of system(4.4) are such that the system possesses two stable equilibrium points in the absence of the driving, thatis, the Lorenz system ˙ v = − v + 10 v ˙ v = − v v + 3 . v − v ˙ v = v v − v (4.5)admits the stable equilibrium points ( − p / , − p / , / and ( − p / , − p / , / [2]. We again set σ = 10 , r = 28 , b = 8 / in system (1.1), and σ = 10 , r = 12 , b = 8 / , µ = µ = µ =0 . , f ( x , x , x ) = 7 . x + cos x , f ( x , x , x ) = 1 . x + 0 . x , f ( x , x , x ) = 3 . x + 0 . e − x in system (2.2) as in Section 3. Figure 3 shows the projections of four chaotic trajectories of the -dimensional system (1.1)-(2.2)-(4.4) on the z z z -space. Moreover, we represent in Figure 4 theprojections of the same chaotic trajectories on the z z -plane. The color of each of the trajectoriesdepicted in Figure 3 is the same with the color of its counterpart shown in Figure 4. The initial data ofthese trajectories are provided in Table 1. The simulation results shown in Figures 3 and 4 reveal thatfour chaotic attractors coexist in the dynamics of the -dimensional system (1.1)-(2.2)-(4.4). z z z
23 03.5 02 -24 -4
Figure 3: Projections of four chaotic trajectories of the -dimensional coupled Lorenz systems (1.1)-(2.2)-(4.4) on the z z z -space. The initial data of the trajectories are provided in Table 1. The figureconfirms the coexistence of four chaotic attractors in the dynamics.5 z z (a) z z (b) -2.2 -2.1 -2 -1.9 z z (c) z z (d) Figure 4: Projections of four chaotic trajectories of the -dimensional system (1.1)-(2.2)-(4.4) on the z z -plane. Table 1: Initial data of the trajectories represented in Figures 3 and 4. x (0) x (0) x (0) y (0) y (0) y (0) z (0) z (0) z (0) Trajectory in blue − . − .
221 29 . − . − .
892 11 . − . − .
461 3 . Trajectory in red .
436 2 .
501 16 .
269 4 .
278 4 .
565 10 .
058 2 .
579 2 .
484 3 . Trajectory in green .
969 6 .
963 18 .
602 4 .
838 3 .
498 12 . − . − .
961 2 . Trajectory in black .
537 5 .
098 21 . − . − .
884 10 .
654 1 .
949 2 .
034 2 . In this study we demonstrate under certain conditions that two chaotic attractors coexist in the dynamicsof unidirectionally coupled Lorenz systems. High dimensional systems with multiple chaotic attractorscan be obtained by applying the same type of coupling provided in Section 2 to chains of Lorenz systems,and an example of a -dimensional system possessing four chaotic attractors is revealed in Section 4.This is the first time in the literature that the coexistence of four chaotic attractors is obtained.Global unpredictable behavior of the weather dynamics is one of the subjects associated with ourresults. An effort was made in study [3] to answer the question why the weather is unpredictable at eachpoint of the atmosphere on the basis of Lorenz systems. The whole atmosphere of the Earth was assumedto be partitioned in a finite number of subregions such that the dynamics of each of them is governed bya Lorenz system with certain coefficients. Considering sensitivity as unpredictability of weather in themeteorological sense, it was further assumed in [3] that there are subregions whose corresponding Lorenzsystems admit chaos with the main ingredient as sensitivity and subregions whose corresponding Lorenzsystems are non-chaotic. The cases of one stable equilibrium point and stable limit cycle were taken intoaccount for the non-chaotic ones. It was deduced that if a subregion with a chaotic dynamics influences6nother one with a non-chaotic dynamics, then the latter also becomes unpredictable.One can confirm that the results of this paper are complementary to the discussions mentioned in[3] such that if the corresponding Lorenz system of a subregion with a non-chaotic dynamics possessestwo stable equilibrium points, then that subregion will become unpredictable under the influence of achaotic subregion and two chaotic attractors may take place in the dynamics. Moreover, consideringfurther interactions between chaotic and non-chaotic subregions, multiple chaotic attractors may occurdepending on the number of stable equilibrium points of the Lorenz systems corresponding to non-chaoticsubregions. These inferences may be helpful for analyzes on the complex dynamics of the atmosphere.Our results may also be used as tools in secure communication [7] and for designing coupled Lorenzlasers [8, 9] possessing multiple chaotic attractors. References [1] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130–141.[2] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer-Verlag,New York, 1982.[3] M. Akhmet, M.O. Fen, Extension of Lorenz unpredictability, Int. J. Bifurcat. Chaos 25 (2015)1550126.[4] M.O. Fen, Persistence of chaos in coupled Lorenz systems, Chaos Solit. Fract. 95 (2017) 200–205.[5] S. Wiggins, Global Bifurcations and Chaos, Springer, New York, 1988.[6] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solu-tions, Springer-Verlag, New-York, Heidelberg, Berlin, 1975.[7] K. M. Cuomo, A. V. Oppenheim, S. H. Strogatz, Synchronization of Lorenz-based chaotic circuitswith applications to communications, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process40 (1993) 626–633.[8] H. Haken, Analogy between higher instabilities in fluids and lasers, Phys. Lett. A 53 (1975) 77–78.[9] N.M. Lawandy, K. Lee, Stability analysis of two coupled Lorenz lasers and the coupling-inducedperiodic →→