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Blowout bifurcation and chaos–hyperchaos transition in five-dimensional continuous autonomous systems
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Chaos, Solitons and Fractals 40 (2009) 1012–1020 www.elsevier.com/locate/chaos
Blowout bifurcation and chaos–hyperchaos transition in five-dimensional continuous autonomous systems Qian Zhou *, Zeng-qiang Chen, Zhu-zhi Yuan Department of Automation, Nankai University, Tianjin 300071, PR China Accepted 29 August 2007
Abstract Blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric subspace, becomes transversely unstable. There has been previous reports of chaos–hyperchaos transition via blowout bifurcation in synchronization of identical chaotic systems. In this paper, two five-dimensional continuous autonomous systems are considered, in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcations occur in these systems and bring on changes of the systems’ dynamics. It is observed that one system undergoes a symmetric hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the dynamical behaviours before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole systems, which provides a possible route to hyperchaos. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been theoretical and practical interest in the study of deterministic higher-dimensional dynamical systems, and in particular some coupled systems [1–19]. When varying a system parameter, such systems can produce some interesting dynamical behaviours such as synchronization [1–4], symmetry-breaking or symmetryincreasing [5,6], on–off intermittency [7–11] shown to be related to blowout bifurcation [10–13], crisis induced intermittency [14], and transition from chaos to hyperchaos, i.e., the second largest Lyapunov exponent becomes positive [3,4,6,15–18]. Blowout bifurcation occurs in systems with some type of symmetry, when a chaotic attractor lying in some symmetric subspace becomes transversely unstable [10]. The existence of symmetry in the system often leads to a low-dimensional invariant subspace in the phase space. Depending on the global dynamics, there are two kinds of blowout bifurcation, nonhysteretic blowout bifurcation and hysteretic blowout bifurcation. A good example of the former case
*
Corresponding author. E-mail addresses: [email protected] (Q. Zhou), [email protected] (Z.-q. Chen).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.091
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is the synchronization of coupled chaotic systems [3,18,19]. When identical chaotic systems synchronize, the synchronization state is an invariant subspace. As the coupling parameter passes through a critical value, desynchronization occurs via a blowout bifurcation, and an asynchronous attractor is born, which exhibits intermittent bursting. For the case of hysteretic blowout bifurcation, when a system parameter passes through a critical value, the attractor in the invariant subspace disappears abruptly, and typical trajectories starting near the invariant manifold are attracted to another attractor off the manifold. In this paper the bifurcation discussed is nonhysteretic blowout bifurcation. Hyperchaos was first presented by Ro¨ssler [20] for a model of a chemical reaction. He used this term to name chaotic phenomenon with more than one positive Lyapunov exponent. With more complex dynamical behaviour and application value than chaos, hyperchaos has been a hot research topic and recently the transition from chaos to hyperchaos has attracted much attention. Its study can help us to have a better understanding of the route to high-dimensional chaos. An interesting phenomenon is that in some coupled identical chaotic systems chaos–hyperchaos transition occurs through blowout bifurcation [3,4,18,19]. Kim et al. [19] investigated the dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcation in coupled chaotic systems. They found that whether the asynchronous attractor appearing through a blowout bifurcation is chaotic or hyperchaotic, is determined by the competition between its laminar and bursting components. The theory of unstable periodic orbits (UPO’s) has been used to explain higher-dimensional dynamical phenomenon of blowout bifurcation [21] and chaos–hyperchaos transition [22]. It has been shown that chaos–hyperchaos transition is caused by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. And blowout bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. In this sense, the mechanisms of the two phenomenon have the same characteristic features. Up to now, the reports of chaos–hyperchaos transition via blowout bifurcation are generally in the synchronization of chaotic system. In this paper we consider two five-dimensional nonlinear systems in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcations occur in these systems and bring on changes on the systems’ dynamics. We observe that one system undergoes a hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the two systems’ dynamical behaviours before and after the blowout bifurcation and make an analysis of the transition process. It is shown that in such a coupled chaotic continuous system, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole system. The transition is soft and continuous as described in [23] and marked by blowout bifurcation. Our work provides a possible route to hyperchaos and indicates applications to the design of high-dimensional hyperchaos. 2. Blowout bifurcation and chaos–chaos transition The models considered in this paper are two drive-response systems. The basic ingredients for a system to exhibit a blowout bifurcation are the following [21]: (i) the phase space contains an invariant subspace, (ii) there is a chaotic attractor in the invariant subspace, and (iii) the chaotic dynamics in the invariant subspace is coupled to the dynamics in the transverse subspace. Drive-response systems are easy to satisfy these features. We take a three-dimensional autonomous chaotic system as the drive subsystem, and the response subsystem has the following form: y_ 1 ¼ y 2 ; ð1Þ y_ 2 ¼ y 31 þ mSy 1 þ ny 2 ; where S is a signal from a drive system, m is a control parameter and n is a constant. In this section, we consider blowout bifurcation and chaos–chaos transition in a chaotic system, i.e., as a system parameter varies, blowout bifurcation happens and after the bifurcation the second largest Lyapunov exponent of the whole system does not become positive. However, the dynamic of the chaotic attractor of the system changes before and after the blowout bifurcation, which is called chaos–chaos transition [24]. As an example, consider the following five-dimensional continuous dynamical system [9]: 8 x_ 1 ¼ rðx2 x1 Þ; > > > > > x > < _ 2 ¼ x1 x3 þ rx1 x2 ; x_ 3 ¼ x1 x2 bx3 ; ð2Þ > > > _ y ¼ y ; > 1 2 > > : y_ 2 ¼ y 31 þ mx1 y 1 þ ny 2 :
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In this model, the drive system A : fx1 ; x2 ; x3 g is Lorenz chaotic system [25], which has three Lyapunov exponents: (0.9056, 0, 14.5723) under typical parameters r ¼ 10; r ¼ 28; b ¼ 8=3. The response system B : fy 1 ; y 2 g is unidirectionally coupled by A whose dynamics is independent of that of B. Moreover, system (2) is symmetrical with respect to ðy 1 ; y 2 Þ ! ðy 1 ; y 2 Þ and y 1 ¼ y 2 ¼ 0 is a three-dimensional invariant manifold in the phase space. In this invariant subspace, the dynamics evolves on the Lorenz chaotic attractor. As the coupling coefficient m varies, the transverse stability of the invariant manifold y 1 ¼ y 2 ¼ 0 changes. It can be measured by the transverse Lyapunov exponent k? of system (1), which is computed via k? ¼ limn!1 1t ln½dðtÞ=dð0Þ where dðtÞ ¼ f½dy 1 ðtÞ2 þ ½dy 2 ðtÞ2 g1=2 and ðdy 1 ; dy 2 Þ is an infinitesimal perturbation transverse to the invariant subspace. In our simulations we take r ¼ 10; r ¼ 28; b ¼ 8=3 and n ¼ 0:65. It is found that system (2) has two blowout bifurcation points at mc1 0:11769 and mc2 0:11825 as shown in Fig. 1. At these two points the transverse Lyapunov exponent k? of system (2) passes through zero. Near each blowout bifurcation point, while the control parameter m takes values corresponding to slightly positive k? , we observe typical on–off intermittency in the time series of y 1 ðtÞ and y 2 ðtÞ as shown in Fig. 2. The Lyapunov exponents of system (2) are calculated. We find that in the parameter range m 2 ½0:15; 0:15, the second largest Lyapunov exponent k2 does not change and k2 0:0004. However, the 3D projections of the chaotic attractor in Fig. 3 shows that the attractor’s dynamics change before and after the blowout bifurcation. After the blowout bifurcation, because of the on–off intermittent bursting in the time series of y 1 ðtÞ and y 2 ðtÞ, the chaotic attractor of system (2) becomes a bursting attractor. This chaotic dynamics change is called chaos–chaos transition.
Fig. 1. The transverse Lyapunov exponent k? versus the parameter m for m 2 ½0:15; 0:15. The exponent k? passes through zero at mc1 0:11769 and k2 0:0004.
Fig. 2. On–off intermittent time series of y 1 ðtÞ at m = 0.13 from an arbitrary initial condition.
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Fig. 3. Three-dimensional projections of chaotic attractors of system (2), (a) before the blowout bifurcation at m = 0.10; (b) after the bifurcation at m = 0.13.
3. Blowout bifurcation and chaos–hyperchaos transition In this section, we give an example of system in which chaos–hyperchaos transition occurs via blowout bifurcation, i.e., as a system parameter varies, blowout bifurcation happens and at the same time the second largest Lyapunov exponent of the system passes through zero and becomes positive at the blowout bifurcation points. Consider the following system: 8 x_ 1 ¼ rðx2 x1 Þ þ x2 x3 ; > > > > > > < x_ 2 ¼ x1 x3 þ rx1 x2 ; x_ 3 ¼ x1 x2 bx3 ; ð3Þ > > > _ y ¼ y ; > 1 2 > > : y_ 2 ¼ y 31 þ mx1 y 1 þ ny 2 : Here, the drive system A : fx1 ; x2 ; x3 g is a modified Lorenz chaotic attractor. (A nonlinear term x2 x3 is added to the first equation’s right hand.). Still the response system B : fy 1 ; y 2 g is uni-directional coupled by A, and y 1 ¼ y 2 ¼ 0 is a threedimensional invariant manifold in the phase space. In this invariant subspace, the dynamics is determined by the drive system A. In our numerical studies we take the following parameter values r ¼ 10; r ¼ 28; b ¼ 8=3; n ¼ 0:65 and consider the coupling coefficient m as a control parameter. Since the divergence of system (3) is negative, the system is a dissipative system. A two-dimensional projection of the chaotic attractor on the invariant manifold are shown in Fig. 4. It is characterized by the Lyapunov exponents k1 1:9381; k2 0 and k3 14:9384.
Fig. 4. Two-dimensional projection of the chaotic attractor on the invariant manifold y 1 ¼ y 2 ¼ 0 of system (3).
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Fig. 5. The transverse Lyapunov exponent k? and the second largest Lyapunov exponent k2 versus the control parameter m for m 2 ½0:5; 0:5.
As the coupling coefficient m varies, the transverse stability of the invariant manifold y 1 ¼ y 2 ¼ 0 changes. A plot of the transverse Lyapunov exponent k? versus m is shown in Fig. 5 (the dotted line). We see k? passes through zero at mc1 0:2625 and mc2 0:2605, corresponding to two blowout bifurcation points of system (3) in the parameter range. After the blowout bifurcation, typical on–off intermittent behaviour is observed in the time series of y 1 ðtÞ and y 2 ðtÞ as shown in Fig. 6. It is interesting to observe that with the increase of the coupling parameter m, system (3) undergoes a symmetric transition of hyperchaos–chaos–hyperchaos. A variation of the five Lyapunov exponents of system (3) versus m is shown in Fig. 7. The transition process can be seen clearly from the variation of the second largest Lyapunov exponent k2 as shown in Fig. 5. In Fig. 5 we see that k2 passes through zero at mc1 and mc2 . Thus the two blowout bifurcation points coincide with the two transition points of chaos–hyperchaos, i.e., when blowout bifurcation occurs, the sign of the second largest Lyapunov exponent changes from negative to positive. When m < mc1 or m > mc2 , the system (3) has two positive Lyapunov exponents and the attractor is a hyperchaotic one. When mc1 < m < mc2 the attractor is chaotic. The statistical distribution of laminar phase length T of the intermittency in system (3) is investigated. We collect about 18,000 laminar phases from time series of y 1 ðtÞ for m ¼ 0:27. As we can see in Fig. 8, the distribution agrees with a 3/2 power-law scaling in [26], which is the statistical characteristic of on–off intermittency. Fig. 9 shows the time series of y 1 ðtÞ and y 2 ðtÞ at m ¼ 0:2 before the blowout bifurcation, the subsystem B : fy 1 ; y 2 g stabilizes asymptotically at its fixed point ðy 1 ¼ 0; y 2 ¼ 0Þ. And after the blowout bifurcation the fixed point loses its stability, the time series of y 1 and y 2 exhibit on–off intermittent bursting (Fig. 6), and the chaotic attractor becomes a bursting hyperchaotic attractor as shown in Fig. 10.
Fig. 6. (a) Typical on–off intermittency in the time series of y 1 ; (b) typical on–off intermittency in the time series of y 2 for m = 0.27.
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Fig. 7. The Lyapunov exponents of system (3) versus the parameter m for m 2 ½1; 1.
Fig. 8. Distribution of the laminar phase duration of the intermittency in system (3).
Fig. 9. (a) Time series of y 1 ; (b) time series of y 2 for m = 0.2.
4. Analysis In this section, we compare the dynamical behaviours of the two blowout bifurcation systems and analyze the processes of chaos–chaos transition and chaos–hyperchaos transition.
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Fig. 10. Projections of the hyperchaotic attractor for m = 0.27.
From Figs. 1, 5 and 7, we see that the dynamics of system (2) and system (3) have a mparameter symmetry, i.e., the solution of the two systems for the constant m is also a solution of the two systems for the parameter m. The symmetry can be explained as follows. It is shown that in the two systems, the time series of x1 and y 1 is always approximately symmetrical about x1 ¼ 0 and y 1 ¼ 0, respectively. since y_ 2 ¼ y 31 þ mx1 y 1 þ ny 2 ¼ y 31 þ ðmÞðx1 Þy 1 þ ny 2 ¼ y 31 þ ðmÞx1 ðy 1 Þ þ ny 2 , for a fixed value of m the dynamics of the two systems are approximately the same as those of them at m. This also explains the symmetry of the transition in system (3). From Figs. 3a and 9, we see that before the blowout bifurcation, the subsystem B : fy 1 ; y 2 g of the two systems stabilizes asymptotically at its fixed point ðy 1 ¼ 0; y 2 ¼ 0Þ. And after the blowout bifurcation the fixed point loses its
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stability, the time series of y 1 and y 2 exhibit on–off intermittent bursting in both systems as shown in Figs. 3b and 10. However, because the dynamics of the invariant manifolds y 1 ¼ y 2 ¼ 0 are different in the two systems, after losing the transverse stability the chaotic attractor in system (3) becomes hyperchaotic and the attractor in system (2) remains chaotic. From the Lyapunov exponents spectrum of system (3) (Fig. 7), we observe that the largest Lyapunov exponent k1 and the smallest Lyapunov exponent k5 almost remain unchanged, while the other three exponents change continuously as the control parameter m varies. And these two exponents are corresponding to those of the drive subsystem A. In fact, the same phenomenon is observed in system (2). Therefore, we conclude that in system (3) as the coupling parameter m varies, the transverse stability of the invariant subspace y 1 ¼ y 2 ¼ 0 changes. On the loss of the transverse stability, the subsystem B : fy 1 ; y 2 g which is chaotically driven by x1 becomes chaotic, leading to an additional positive Lyapunov exponent for the whole system. Therefore, the whole system transits to hyperchaos from chaos in system (3). The transition confirms the general route to high-dimensional chaos presented by Harrison and Lai [23]. A characteristic feature of this route is that the second largest Lyapunov exponent passes through zero continuously as shown in Fig. 5. This gives evidence that the transition from chaos to hyperchaos is continuous and smooth. Here we try to explain why this chaos–hyperchaos transition does not happen in the blowout bifurcation system (2). According to the theory of unstable periodic orbits, chaos–hyperchaos transition is caused by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor [22]. And blowout bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor [21]. It is likely that in a dynamical system, the loss of local stability results the change of the stability of the whole system, like the route towards chaos through local bifurcation. We conclude that the change in the transverse stability of unstable periodic orbits embedded in the chaotic attractor may induce the loss of the global stability of the unstable periodic orbits like the chaos–hyperchaos transition in system (3). When it happens, the chaotic system becomes hyperchaotic, and when it does not happen, the system remains chaotic like in system (2). Thus in a coupled chaotic system, the occurrence of blowout bifurcation does not ensure but may indicate a chaos–hyperchaos transition of the whole system.
5. Conclusions Previous reports of chaos–hyperchaos transition via blowout bifurcation are generally in the synchronization of chaotic system. In this paper we consider two five-dimensional nonlinear systems in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcation occurs in these systems and brings on a change on the systems’ dynamics. We observe that one system undergoes a hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the systems’ dynamical behaviours before and after the blowout bifurcation and make an analysis of the transition process. It is shown that in such a coupled chaotic continuous system, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole system, which provide a possible route to hyperchaos.
Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60774088, 60574036), the Program for New Century Excellent Talents in University of China (NCET), The Science and Technology Key Project of Education Ministry of China (No. 107024) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20050055013).
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Chaos, Solitons and Fractals 40 (2009) 1012–1020 www.elsevier.com/locate/chaos
Blowout bifurcation and chaos–hyperchaos transition in five-dimensional continuous autonomous systems Qian Zhou *, Zeng-qiang Chen, Zhu-zhi Yuan Department of Automation, Nankai University, Tianjin 300071, PR China Accepted 29 August 2007
Abstract Blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric subspace, becomes transversely unstable. There has been previous reports of chaos–hyperchaos transition via blowout bifurcation in synchronization of identical chaotic systems. In this paper, two five-dimensional continuous autonomous systems are considered, in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcations occur in these systems and bring on changes of the systems’ dynamics. It is observed that one system undergoes a symmetric hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the dynamical behaviours before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole systems, which provides a possible route to hyperchaos. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been theoretical and practical interest in the study of deterministic higher-dimensional dynamical systems, and in particular some coupled systems [1–19]. When varying a system parameter, such systems can produce some interesting dynamical behaviours such as synchronization [1–4], symmetry-breaking or symmetryincreasing [5,6], on–off intermittency [7–11] shown to be related to blowout bifurcation [10–13], crisis induced intermittency [14], and transition from chaos to hyperchaos, i.e., the second largest Lyapunov exponent becomes positive [3,4,6,15–18]. Blowout bifurcation occurs in systems with some type of symmetry, when a chaotic attractor lying in some symmetric subspace becomes transversely unstable [10]. The existence of symmetry in the system often leads to a low-dimensional invariant subspace in the phase space. Depending on the global dynamics, there are two kinds of blowout bifurcation, nonhysteretic blowout bifurcation and hysteretic blowout bifurcation. A good example of the former case
*
Corresponding author. E-mail addresses: [email protected] (Q. Zhou), [email protected] (Z.-q. Chen).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.091
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is the synchronization of coupled chaotic systems [3,18,19]. When identical chaotic systems synchronize, the synchronization state is an invariant subspace. As the coupling parameter passes through a critical value, desynchronization occurs via a blowout bifurcation, and an asynchronous attractor is born, which exhibits intermittent bursting. For the case of hysteretic blowout bifurcation, when a system parameter passes through a critical value, the attractor in the invariant subspace disappears abruptly, and typical trajectories starting near the invariant manifold are attracted to another attractor off the manifold. In this paper the bifurcation discussed is nonhysteretic blowout bifurcation. Hyperchaos was first presented by Ro¨ssler [20] for a model of a chemical reaction. He used this term to name chaotic phenomenon with more than one positive Lyapunov exponent. With more complex dynamical behaviour and application value than chaos, hyperchaos has been a hot research topic and recently the transition from chaos to hyperchaos has attracted much attention. Its study can help us to have a better understanding of the route to high-dimensional chaos. An interesting phenomenon is that in some coupled identical chaotic systems chaos–hyperchaos transition occurs through blowout bifurcation [3,4,18,19]. Kim et al. [19] investigated the dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcation in coupled chaotic systems. They found that whether the asynchronous attractor appearing through a blowout bifurcation is chaotic or hyperchaotic, is determined by the competition between its laminar and bursting components. The theory of unstable periodic orbits (UPO’s) has been used to explain higher-dimensional dynamical phenomenon of blowout bifurcation [21] and chaos–hyperchaos transition [22]. It has been shown that chaos–hyperchaos transition is caused by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. And blowout bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. In this sense, the mechanisms of the two phenomenon have the same characteristic features. Up to now, the reports of chaos–hyperchaos transition via blowout bifurcation are generally in the synchronization of chaotic system. In this paper we consider two five-dimensional nonlinear systems in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcations occur in these systems and bring on changes on the systems’ dynamics. We observe that one system undergoes a hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the two systems’ dynamical behaviours before and after the blowout bifurcation and make an analysis of the transition process. It is shown that in such a coupled chaotic continuous system, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole system. The transition is soft and continuous as described in [23] and marked by blowout bifurcation. Our work provides a possible route to hyperchaos and indicates applications to the design of high-dimensional hyperchaos. 2. Blowout bifurcation and chaos–chaos transition The models considered in this paper are two drive-response systems. The basic ingredients for a system to exhibit a blowout bifurcation are the following [21]: (i) the phase space contains an invariant subspace, (ii) there is a chaotic attractor in the invariant subspace, and (iii) the chaotic dynamics in the invariant subspace is coupled to the dynamics in the transverse subspace. Drive-response systems are easy to satisfy these features. We take a three-dimensional autonomous chaotic system as the drive subsystem, and the response subsystem has the following form: y_ 1 ¼ y 2 ; ð1Þ y_ 2 ¼ y 31 þ mSy 1 þ ny 2 ; where S is a signal from a drive system, m is a control parameter and n is a constant. In this section, we consider blowout bifurcation and chaos–chaos transition in a chaotic system, i.e., as a system parameter varies, blowout bifurcation happens and after the bifurcation the second largest Lyapunov exponent of the whole system does not become positive. However, the dynamic of the chaotic attractor of the system changes before and after the blowout bifurcation, which is called chaos–chaos transition [24]. As an example, consider the following five-dimensional continuous dynamical system [9]: 8 x_ 1 ¼ rðx2 x1 Þ; > > > > > x > < _ 2 ¼ x1 x3 þ rx1 x2 ; x_ 3 ¼ x1 x2 bx3 ; ð2Þ > > > _ y ¼ y ; > 1 2 > > : y_ 2 ¼ y 31 þ mx1 y 1 þ ny 2 :
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In this model, the drive system A : fx1 ; x2 ; x3 g is Lorenz chaotic system [25], which has three Lyapunov exponents: (0.9056, 0, 14.5723) under typical parameters r ¼ 10; r ¼ 28; b ¼ 8=3. The response system B : fy 1 ; y 2 g is unidirectionally coupled by A whose dynamics is independent of that of B. Moreover, system (2) is symmetrical with respect to ðy 1 ; y 2 Þ ! ðy 1 ; y 2 Þ and y 1 ¼ y 2 ¼ 0 is a three-dimensional invariant manifold in the phase space. In this invariant subspace, the dynamics evolves on the Lorenz chaotic attractor. As the coupling coefficient m varies, the transverse stability of the invariant manifold y 1 ¼ y 2 ¼ 0 changes. It can be measured by the transverse Lyapunov exponent k? of system (1), which is computed via k? ¼ limn!1 1t ln½dðtÞ=dð0Þ where dðtÞ ¼ f½dy 1 ðtÞ2 þ ½dy 2 ðtÞ2 g1=2 and ðdy 1 ; dy 2 Þ is an infinitesimal perturbation transverse to the invariant subspace. In our simulations we take r ¼ 10; r ¼ 28; b ¼ 8=3 and n ¼ 0:65. It is found that system (2) has two blowout bifurcation points at mc1 0:11769 and mc2 0:11825 as shown in Fig. 1. At these two points the transverse Lyapunov exponent k? of system (2) passes through zero. Near each blowout bifurcation point, while the control parameter m takes values corresponding to slightly positive k? , we observe typical on–off intermittency in the time series of y 1 ðtÞ and y 2 ðtÞ as shown in Fig. 2. The Lyapunov exponents of system (2) are calculated. We find that in the parameter range m 2 ½0:15; 0:15, the second largest Lyapunov exponent k2 does not change and k2 0:0004. However, the 3D projections of the chaotic attractor in Fig. 3 shows that the attractor’s dynamics change before and after the blowout bifurcation. After the blowout bifurcation, because of the on–off intermittent bursting in the time series of y 1 ðtÞ and y 2 ðtÞ, the chaotic attractor of system (2) becomes a bursting attractor. This chaotic dynamics change is called chaos–chaos transition.
Fig. 1. The transverse Lyapunov exponent k? versus the parameter m for m 2 ½0:15; 0:15. The exponent k? passes through zero at mc1 0:11769 and k2 0:0004.
Fig. 2. On–off intermittent time series of y 1 ðtÞ at m = 0.13 from an arbitrary initial condition.
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Fig. 3. Three-dimensional projections of chaotic attractors of system (2), (a) before the blowout bifurcation at m = 0.10; (b) after the bifurcation at m = 0.13.
3. Blowout bifurcation and chaos–hyperchaos transition In this section, we give an example of system in which chaos–hyperchaos transition occurs via blowout bifurcation, i.e., as a system parameter varies, blowout bifurcation happens and at the same time the second largest Lyapunov exponent of the system passes through zero and becomes positive at the blowout bifurcation points. Consider the following system: 8 x_ 1 ¼ rðx2 x1 Þ þ x2 x3 ; > > > > > > < x_ 2 ¼ x1 x3 þ rx1 x2 ; x_ 3 ¼ x1 x2 bx3 ; ð3Þ > > > _ y ¼ y ; > 1 2 > > : y_ 2 ¼ y 31 þ mx1 y 1 þ ny 2 : Here, the drive system A : fx1 ; x2 ; x3 g is a modified Lorenz chaotic attractor. (A nonlinear term x2 x3 is added to the first equation’s right hand.). Still the response system B : fy 1 ; y 2 g is uni-directional coupled by A, and y 1 ¼ y 2 ¼ 0 is a threedimensional invariant manifold in the phase space. In this invariant subspace, the dynamics is determined by the drive system A. In our numerical studies we take the following parameter values r ¼ 10; r ¼ 28; b ¼ 8=3; n ¼ 0:65 and consider the coupling coefficient m as a control parameter. Since the divergence of system (3) is negative, the system is a dissipative system. A two-dimensional projection of the chaotic attractor on the invariant manifold are shown in Fig. 4. It is characterized by the Lyapunov exponents k1 1:9381; k2 0 and k3 14:9384.
Fig. 4. Two-dimensional projection of the chaotic attractor on the invariant manifold y 1 ¼ y 2 ¼ 0 of system (3).
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Fig. 5. The transverse Lyapunov exponent k? and the second largest Lyapunov exponent k2 versus the control parameter m for m 2 ½0:5; 0:5.
As the coupling coefficient m varies, the transverse stability of the invariant manifold y 1 ¼ y 2 ¼ 0 changes. A plot of the transverse Lyapunov exponent k? versus m is shown in Fig. 5 (the dotted line). We see k? passes through zero at mc1 0:2625 and mc2 0:2605, corresponding to two blowout bifurcation points of system (3) in the parameter range. After the blowout bifurcation, typical on–off intermittent behaviour is observed in the time series of y 1 ðtÞ and y 2 ðtÞ as shown in Fig. 6. It is interesting to observe that with the increase of the coupling parameter m, system (3) undergoes a symmetric transition of hyperchaos–chaos–hyperchaos. A variation of the five Lyapunov exponents of system (3) versus m is shown in Fig. 7. The transition process can be seen clearly from the variation of the second largest Lyapunov exponent k2 as shown in Fig. 5. In Fig. 5 we see that k2 passes through zero at mc1 and mc2 . Thus the two blowout bifurcation points coincide with the two transition points of chaos–hyperchaos, i.e., when blowout bifurcation occurs, the sign of the second largest Lyapunov exponent changes from negative to positive. When m < mc1 or m > mc2 , the system (3) has two positive Lyapunov exponents and the attractor is a hyperchaotic one. When mc1 < m < mc2 the attractor is chaotic. The statistical distribution of laminar phase length T of the intermittency in system (3) is investigated. We collect about 18,000 laminar phases from time series of y 1 ðtÞ for m ¼ 0:27. As we can see in Fig. 8, the distribution agrees with a 3/2 power-law scaling in [26], which is the statistical characteristic of on–off intermittency. Fig. 9 shows the time series of y 1 ðtÞ and y 2 ðtÞ at m ¼ 0:2 before the blowout bifurcation, the subsystem B : fy 1 ; y 2 g stabilizes asymptotically at its fixed point ðy 1 ¼ 0; y 2 ¼ 0Þ. And after the blowout bifurcation the fixed point loses its stability, the time series of y 1 and y 2 exhibit on–off intermittent bursting (Fig. 6), and the chaotic attractor becomes a bursting hyperchaotic attractor as shown in Fig. 10.
Fig. 6. (a) Typical on–off intermittency in the time series of y 1 ; (b) typical on–off intermittency in the time series of y 2 for m = 0.27.
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Fig. 7. The Lyapunov exponents of system (3) versus the parameter m for m 2 ½1; 1.
Fig. 8. Distribution of the laminar phase duration of the intermittency in system (3).
Fig. 9. (a) Time series of y 1 ; (b) time series of y 2 for m = 0.2.
4. Analysis In this section, we compare the dynamical behaviours of the two blowout bifurcation systems and analyze the processes of chaos–chaos transition and chaos–hyperchaos transition.
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Fig. 10. Projections of the hyperchaotic attractor for m = 0.27.
From Figs. 1, 5 and 7, we see that the dynamics of system (2) and system (3) have a mparameter symmetry, i.e., the solution of the two systems for the constant m is also a solution of the two systems for the parameter m. The symmetry can be explained as follows. It is shown that in the two systems, the time series of x1 and y 1 is always approximately symmetrical about x1 ¼ 0 and y 1 ¼ 0, respectively. since y_ 2 ¼ y 31 þ mx1 y 1 þ ny 2 ¼ y 31 þ ðmÞðx1 Þy 1 þ ny 2 ¼ y 31 þ ðmÞx1 ðy 1 Þ þ ny 2 , for a fixed value of m the dynamics of the two systems are approximately the same as those of them at m. This also explains the symmetry of the transition in system (3). From Figs. 3a and 9, we see that before the blowout bifurcation, the subsystem B : fy 1 ; y 2 g of the two systems stabilizes asymptotically at its fixed point ðy 1 ¼ 0; y 2 ¼ 0Þ. And after the blowout bifurcation the fixed point loses its
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stability, the time series of y 1 and y 2 exhibit on–off intermittent bursting in both systems as shown in Figs. 3b and 10. However, because the dynamics of the invariant manifolds y 1 ¼ y 2 ¼ 0 are different in the two systems, after losing the transverse stability the chaotic attractor in system (3) becomes hyperchaotic and the attractor in system (2) remains chaotic. From the Lyapunov exponents spectrum of system (3) (Fig. 7), we observe that the largest Lyapunov exponent k1 and the smallest Lyapunov exponent k5 almost remain unchanged, while the other three exponents change continuously as the control parameter m varies. And these two exponents are corresponding to those of the drive subsystem A. In fact, the same phenomenon is observed in system (2). Therefore, we conclude that in system (3) as the coupling parameter m varies, the transverse stability of the invariant subspace y 1 ¼ y 2 ¼ 0 changes. On the loss of the transverse stability, the subsystem B : fy 1 ; y 2 g which is chaotically driven by x1 becomes chaotic, leading to an additional positive Lyapunov exponent for the whole system. Therefore, the whole system transits to hyperchaos from chaos in system (3). The transition confirms the general route to high-dimensional chaos presented by Harrison and Lai [23]. A characteristic feature of this route is that the second largest Lyapunov exponent passes through zero continuously as shown in Fig. 5. This gives evidence that the transition from chaos to hyperchaos is continuous and smooth. Here we try to explain why this chaos–hyperchaos transition does not happen in the blowout bifurcation system (2). According to the theory of unstable periodic orbits, chaos–hyperchaos transition is caused by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor [22]. And blowout bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor [21]. It is likely that in a dynamical system, the loss of local stability results the change of the stability of the whole system, like the route towards chaos through local bifurcation. We conclude that the change in the transverse stability of unstable periodic orbits embedded in the chaotic attractor may induce the loss of the global stability of the unstable periodic orbits like the chaos–hyperchaos transition in system (3). When it happens, the chaotic system becomes hyperchaotic, and when it does not happen, the system remains chaotic like in system (2). Thus in a coupled chaotic system, the occurrence of blowout bifurcation does not ensure but may indicate a chaos–hyperchaos transition of the whole system.
5. Conclusions Previous reports of chaos–hyperchaos transition via blowout bifurcation are generally in the synchronization of chaotic system. In this paper we consider two five-dimensional nonlinear systems in which a two-dimensional subsystem is driven by a chaotic system. As a system parameter changes, blowout bifurcation occurs in these systems and brings on a change on the systems’ dynamics. We observe that one system undergoes a hyperchaos–chaos–hyperchaos transition via blowout bifurcations, while the other system does not transit to hyperchaos after the bifurcations. We investigate the systems’ dynamical behaviours before and after the blowout bifurcation and make an analysis of the transition process. It is shown that in such a coupled chaotic continuous system, blowout bifurcation may indicate a transition from chaos to hyperchaos for the whole system, which provide a possible route to hyperchaos.
Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60774088, 60574036), the Program for New Century Excellent Talents in University of China (NCET), The Science and Technology Key Project of Education Ministry of China (No. 107024) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20050055013).
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