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Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system
Physics Letters A 374 (2009) 208–213
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system Yongxiang Zhang a,∗ , Guiqin Kong b a b
College of Science, Shenyang Agricultural University, Shenyang 110161, China Jiujiang Precision Measuring Technology Research Institute, Jiujiang 332000, China
a r t i c l e
i n f o
Article history: Received 26 August 2009 Received in revised form 10 October 2009 Accepted 19 October 2009 Available online 24 October 2009 Communicated by A.R. Bishop PACS: 05.45.Ac 05.45.Df 05.45.Pq
a b s t r a c t A variety of different dynamical regimes involving strange nonchaotic attractors (SNAs) can be observed in a quasiperiodically forced delayed system. We describe some numerical experiments giving evidences of intertwined basin boundaries (smooth, non-Wada fractal and Wada property) for SNAs. In particular, we show that Wada property, fractality and smoothness can be intertwined on arbitrarily fine scales. This suggests that SNAs can exhibit the final state sensitivity and unpredictable behaviors. An interesting dynamical transition of SNAs together with associated mechanisms from non-Wada fractal to Wada intertwined basin boundaries is examined. A scaling exponent is used to characterize the intertwined basin boundaries. © 2009 Elsevier B.V. All rights reserved.
Keywords: Strange nonchaotic attractors Unpredictability Intertwined basin boundaries Wada property
1. Introduction Prediction is one of the fundamental goals of science. When prediction becomes impossible, it may be thought that one of the foundations of science will be shattered. Many physical, chemical, biological, and engineering problems are known to possess multiple coexisting final states. Multiple coexisting final states are determined by both initial conditions and system parameters. For a given parameter, different choices of the initial condition can lead to distinctly different asymptotic attractors, each with its own basin of attraction. The boundaries that separate different basins of attraction are basin boundaries, which can be either smooth or fractal [1–3]. A basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins, is called a Wada basin, and its boundary is called a Wada basin boundary. In particular, fractal basin boundaries of more than two basins of attraction, the Wada basin boundaries, have been studied [4–7]. Very recently, different types of fractal basins, such as Wada and riddled basins, have been reviewed [8]. It is impossible to predict, with certainty, the asymp-
*
Corresponding author. Tel.: +86 024 88493201. E-mail address: [email protected] (Y. Zhang).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.10.053
totic attractor for initial conditions in the neighborhood of fractal and Wada basin boundaries. Strange nonchaotic attractors (SNAs) typically appear in quasiperiodically forced nonlinear dynamical systems. They are geometrically strange just like chaotic attractors, while all their Lyapunov exponents are not positive, which ensures that the underlying dynamics is nonchaotic. Since the first description of SNAs by Grebogi et al. [9], dynamical behaviors of the quasiperiodically forced systems have been extensively investigated both numerically and experimentally. More work was devoted to the investigation of irregular dynamical transitions and mechanisms, most of which have been reviewed previously [10–12]. Recently, SNAs have been observed in wider dynamical systems, such as LCR circuits [13], chemical systems [14] and random dynamical systems [15]. By applying unstable sets in SNAs, mechanisms of dynamical transitions can be examined, e.g. intermittent transition [16], fractalization [17], and crises [18,19]. While the existence of SNAs was firmly established, a question that remains interesting is whether these SNAs can be predicted or not. However, in the previous work, very few topics dealing with this problem have been reported. The study of fractal and Wada basin boundaries has been restricted to regular attractors (periodic and quasiperiodic) and chaotic attractors. Specifically, our attention will be focused on the analysis of basin boundaries of SNAs. In particular, an interesting and new
Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
basin boundary can be observed, where Wada property, non-Wada fractality and smoothness can be also intertwined on arbitrarily fine scales. We have attempted to assess the unpredictable behaviors of SNAs arising from phase space. One of the goals of nonlinear dynamics is to determine the global structures of the system such as basin boundaries [20]. Another goal is to determine how these global structures come about with variation of a system parameter [21]. In our model, two special kinds of intertwined basin boundaries for SNAs are considered. The most prominent subtype is intertwined basin boundaries (smooth and fractal). Another subtype is more interesting (smoothness, non-Wada fractality and Wada property are intertwined). In this Letter, we examined basin boundaries of coexisting SNAs and their transitions between these two subtypes. Associated mechanisms of transitions are also investigated. A blowout bifurcation can lead to a new coexisting SNA. One of the main consequences of the intertwined basin boundaries is related to the difficulty of predicting to which attractor a given initial condition might go. Consequently, our results show that SNAs cannot be predicted reliably for specific initial conditions on intertwined basin boundaries. 2. The forced delayed system and dynamical regimes The quasiperiodically forced logistic map has been studied extensively, which is a representative model for quasiperiodically forced period-doubling systems. Here, we consider a representative model for quasiperiodically forced Hopf-bifurcational systems. In the present work, we investigate the dynamics of (1) with the additional quasiperiodic forcing,
xn+2 = f (xn , xn+1 , zn ) = ε 1 + p cos(2π zn ) xn+1 + xn − xn3 zn+1 = zn + ω(mod 1) (1)
In the absence of quasiperiodic forcing (p = 0), the system (1) is a simple dynamics model with time delays, which is analogous to the typical Duffing oscillator and is quite different from the delayed logistic equation. It can be applied to the field of ecological systems. If we introduce yn and xn+1 = yn , the system (1) can be rewritten as
⎧x ⎨ n+1 = yn yn+1 = ε 1 + p cos(2π zn ) yn + xn − xn3 ⎩ zn+1 = zn + ω(mod 1)
(2)
where ω and p represent the frequency and amplitude of the quasiperiodic forcing respectively. We set √ the frequency to be the reciprocal of the golden mean, ω = ( 5 − 1)/2. Obviously, the invariant subspace S is given by (xn , yn , zn ) = (0, 0, zn ) in the system (1). Since S is invariant, initial conditions in S result in trajectories which remain in S forever. In order to describe strange nonchaotic dynamics, we obtain the phase diagram as a function of the parameter ε and nonzero p, shown in Fig. 1. We have found it useful to characterize SNAs through both the Lyapunov exponent λx in the x-direction, which is given by
λx = lim
N →∞
N 1
N
i =1
∂ f ln ∂ xi
(3)
and the phase sensitivity exponent which can be obtained from phase sensitivity function Γ N
Γ N (ε , p ) = min
x0 , y 0 , z0
∂ xn max 0n N ∂ z
(4)
209
Fig. 1. Phase diagram in the ε –p plane. Regular, chaotic, SNA, and escape regimes are shown in white, gray, light gray, and black, respectively. Another type of attractor without phase sensitivity is shown in light red, which is marked AT and the largest Lyapunov exponent is approximate zero. A blowout bifurcation occurs by the critical curve L1 at which the largest Lyapunov exponent and the largest transverse Lyapunov exponent are zero. Merging crises of the chaotic attractor occur by the critical curve L2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
∂ xn ∂ z can be obtained by differentiating (2),
⎧ ∂ xn+1 ∂ yn ⎪ ⎪ ∂z = ∂z ⎪ ⎪ ⎨ ∂ y n +1 ∂ z = −2πε p sin(2π zn ) yn ⎪ ⎪ + ε 1 + p cos(2π zn ) ∂∂yzn + 1 − 3xn2 ∂∂xzn ⎪ ⎪ ⎩ zn+1 = zn + ω(mod 1)
(5)
On a SNA, the function Γ N grows with the length of the orbit N, as a power, i.e., Γ N ∼ N γ , where γ is the phase sensitivity exponent [10]. The exponent γ measures the sensitivity with respect to the phase of the quasiperiodic forcing and characterizes the strangeness of an attractor in a quasiperiodically driven system. A smooth torus has a negative Lyapunov exponent and no phase sensitivity (γ = 0). On the other hand, SNAs have negative Lyapunov exponents and high phase sensitivity (γ > 0). Chaotic attractors have positive Lyapunov exponents. In some dynamical regimes (Fig. 1), a striking phenomenon that can be observed is bistability of symmetric attractors. We examine the bifurcation structure of two symmetric attractors with separate basins of attraction, where the phase diagram (in the p–ε plane) is quite identical. Fig. 1 shows a phase diagram with single attractors for reasons of simplicity and clarity. Regular, chaotic, SNA and escape regimes are shown in white, gray, light gray and black, respectively. Quasiperiodical regions are denoted by 1T and 2T and shown in white. We can easily observe that there exist two distinct 2T quasiperiodical tongues (T1 and T2 ), where the different mechanisms may be described. In the T1 region, we show that a blowout bifurcation occurs and a symmetric attractor is born as the parameters p and ε pass through the critical curve L1 (the largest Lyapunov exponent and the largest transverse Lyapunov exponent are zero), while in the T2 region the 2T torus attractor appears via torus-doubling bifurcation. The interesting fact is that a SNA tongue (region S1) is interspersed along the boundary of escape regions, but regions where they occur are extremely long and chaotic attractors are very rare. SNAs can be observed through fractalization (route d in Fig. 1) in another region (region S2 in gray). Note that its main interesting feature is the existence of a small tongue of SNA that penetrates into the quasiperiodical regions T2 . Another type of attractor without phase sensitivity can be located in light red regimes, which is called attractor transients
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Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
(AT) and the largest Lyapunov exponent is approximate zero. The term attractor transient refers to the fact that an orbit can spend a long time in the vicinity of the attractor A before it leaves, finally moving off to the special attractor A. Merging crises of the chaotic attractor occur by the critical curve L2. 3. Intertwined basin boundaries of SNAs in phase space While the existence of different attractors was firmly established, a question that remains interesting is the occurrence and transitions of SNAs as a system parameter changes through a critical value. Another question is whether these SNAs can be predicted or not. How do basin boundary structures of SNAs come about with variation of a system parameter? Some typical routes to SNAs can be observed in Eq. (1), such as Heagy–Hammel route, the fractalization, the type-I intermittency and boundary crises [22]. Symmetry-breaking and symmetry-recovering phenomena can also be observed [23]. Here, we focus on a small portion of this parameter space that SNAs are created through fractalization (routes c in Fig. 1). Such a phenomenon is essentially a gradual fractalization of the doubled torus (2T), but the difference is that here a transition from a two-frequency torus (2T) to SNA is realized through a gradual fractalization process instead of the transition from the 1T torus (routes d in Fig. 1). An example has been analyzed by varying ε for the fixed parameter p = 2.287. Fig. 2(a) shows two coexisting SNAs (red and blue) for the fixed parameter ε = 0.699. For the present case, the maximal Lyapunov exponent λmax is approximate −0.0005. The phase sensitivity function Γ N grows unboundedly with the power-law relation Γ N ∼ N γ , γ ≈ 0.79 [see the box in Fig. 2(a)]. At such a value, two coexisting attractors, Fig. 2(a), possess a geometrically strange property but do not exhibit any sensitivity to initial conditions (the maximal Lyapunov exponent is negative) and so they are indeed strange nonchaotic attractors. We now present numerical evidences for the existence of nonWada fractal and partially Wada basin boundaries [8] in Eq. (1). It is well known that some special kinds of fractal basin boundaries have appeared apart from the nominal fractal basin boundaries. The most prominent subtypes are intertwined, Wada, riddled and sporadically fractal basins. Some dynamical systems have three or more basins sharing the same boundary. These basin boundaries were named Wada basin boundary. However, a basin satisfies the Wada property if any initial condition that is on the boundary of one basin is also simultaneously on the boundary of another two or more basins. To possess the Wada property is stronger than to have fractal basin boundaries. Here we pay special attention to two types of intertwined basin boundaries of SNAs. Grebogi and coworkers discovered that it is common to find basin boundaries that show different dimensions in different regions [24]. These peculiar structures were named intertwined basins, describing situations in which every fractal region of the boundary has subregions inside where the boundary is smooth. Here, we observe this type of intertwined basin boundaries of SNAs in system (1). In particular, an interesting and new basin boundary can be observed, which Wada property, non-Wada fractality and smoothness can be also intertwined on arbitrarily small scales. Basin boundaries with fractal and Wada property lead to final state sensitivity. Final state sensitivity can be quantified by the uncertainty exponent α , which was firstly introduced by Grebogi et al. [25]. The uncertainty exponent α is defined as follows. Randomly choose an initial condition x0 . Define x0 = x0 + δ , where δ is a small perturbation. Determine whether the asymptotic dynamics of the system using these two initial conditions are qualitatively different. For a given perturbation δ , a fraction of uncertain initial conditions f (δ) can be computed by randomly choosing many initial conditions and determining if they are uncertain. The uncertainty exponent α is determined by fitting the scaling law f (δ) ∼ δ α . The uncertainty
Fig. 2. (a) Coexisting SNAs (TF1 and TF2) at p = 2.287 and ε = 0.699. The maximal Lyapunov exponent is approximately −0.0005. A phase sensitivity function diagram is given in the box; (b) The basin section of attraction at zn = 0.25 (SNA (TF1), SNA (TF2) and escape regions are shown in red, blue and white, respectively); (c) A magnification of the area enclosed in the green rectangle in (b); (d) Another magnification of area enclosed in green rectangle in (c); (e) A linear fit yields the following uncertainty exponent: α ≈ 0.52 for region (d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
dimension D of the fractal set embedded in the initial conditions is obtained from D = N − α , where N is the dimension of the phase space. Therefore the uncertainty dimension is defined in the range D ∈ [ N − 1, N ]. The fractal dimension is a common way to measure the complexity of a fractal basin boundary. If it is an integer num-
Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
Fig. 3. Coexisting attractors (SNA (BB), TF1 and TF2) at p = 2.287 and ε = 0.703. The largest Lyapunov exponent of SNA (BB) is approximately −0.013. The largest Lyapunov exponents of TF1 (or TF2) are approximately −0.007. A phase sensitivity function of SNA (BB) diagram is given in the box. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)
211
ber, then the boundary is said to be nonfractal (or smooth), while if it is not an integer, the boundary is fractal. Since it is difficult to study basin structures in three dimensions, we fix the variable z and study the reduced basin structure in two-dimensional surfaces. We fix the variable z at z = 0.25 and a typical basin section of SNAs is shown in Fig. 2(b). Fig. 2(b) shows an example of intertwined basin boundaries for two different SNAs [TF1 and TF2 in Fig. 2(a)]. The basins of SNA (TF1) and SNA (TF2) are shown by red and blue, respectively. Escape regimes are shown in white. A magnification of the area enclosed in the green rectangle in Fig. 2(b) is shown in Fig. 2(c). Another magnification of area enclosed in green rectangle in Fig. 2(c) is shown in Fig. 2(d). Obviously, fractality and smoothness are intertwined on basin boundaries. We calculate an uncertainty exponent α ≈ 0.52 [in Fig. 2(e)] and a fractal dimension of 2.48 at y = 1.55, z = 0.25, but we can always find smooth subregions of the integral dimension. A much more interesting question would be to study the transition from non-Wada fractal to partially Wada basin boundaries. Another interesting question is the nature of basin boundaries for SNAs. We have observed that at ε = ε ∗ ≈ 0.7028622, a blowout bifurcation can lead to a new SNA (BB). Whether SNAs attract or repel initial conditions in the vicinity of S is determined by the
Fig. 4. (a) The basin section of attraction at zn = 0.25 (SNA (BB), TF1, TF2 and escape regions are shown in black, red, blue and white, respectively); (b) A magnification of the area enclosed in the green rectangle in (a); (c) A magnification of the area enclosed in the green rectangle in (b); (d) A magnification of the area enclosed in the green rectangle in (c); (e) A magnification of the area enclosed in the green rectangle in (d); (f) A magnification of the area enclosed in the green rectangle in (e); (g) A linear fit f (δ)
yields the following uncertainty exponent for region (c): α ≈ 0.53 [log10 versus logδ10 for the basin boundaries between the red and blue basins]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
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Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
sign of the largest transverse Lyapunov exponent Λ T computed for trajectories in S with respect to perturbations in the subspace which is transverse to S [26]. The crucial curve L1 (in Fig. 1) plays an important role to analyze blowout bifurcation, where the largest Lyapunov exponent and the largest transverse Lyapunov exponent are zero. For values of ε that are slightly smaller than ε ∗ , there are two coexisting attractors (TF1 and TF2), and the basin boundary is non-Wada fractal. This phenomenon is an interesting example of a transition between a fractal and partially Wada basin boundary when a system parameter is varied. More precisely, a blowout bifurcation creates a new coexisting SNA, so the system passes from two to three attractors. Fig. 3 shows three coexisting attractors (SNA (BB), TF1 and TF2) at p = 2.287 and ε = 0.703. Three different attractors (SNA (BB), TF1 and TF2) can be distinguished by black, red and blue, respectively. The largest Lyapunov exponent of SNA (BB) is approximately −0.013. The largest Lyapunov exponents of TF1 (or TF2) are approximately −0.007. A phase sensitivity function Γ N of SNA (BB) is given in the box, which grows unboundedly with the power-law relation Γ N ∼ N γ , γ ≈ 0.65. Two other attractors (TF1 and TF2) have near-zero phase sensitivity exponents, which may be extremely wrinkled 2T attractors. Fig. 4(a) shows their basin section of attraction at z = 0.25. The basins of SNA (BB), TF1 and TF2 are shown in black, red and blue, respectively. Escape regimes are shown in white. Successive magnifications of small areas within the region illustrate clearly the intertwined nature of the basin boundary [Fig. 4(b)–(e)]. From the pictures, it seems that no matter how much we amplify the magnification, there seem to be regions where Wada property, fractality and smoothness are intertwined. This gives a clear (although nonrigorous) indication that the basin boundaries are intertwined, and also that some subregions of the basin boundaries for this system possess the Wada property. We calculate an uncertainty exponent α ≈ 0.53 f (δ)
and a fractal dimension of 2.47 at y = 1.6, z = 0.25 [log10
ver-
δ
sus log10 for the basin boundary between the red and blue basins in Fig. 4(g)]. The dynamics of the unstable invariant set plays a crucial role in the formation of fractal and Wada basin boundaries, especially the stable and unstable manifolds of the invariant set. In order to more rigorously confirm that the basin boundaries are intertwined (non-Wada fractal and Wada property), we numerically approximate the unstable manifold (in green and yellow) of four representatively accessible orbits [Fig. 5(a)]. It may be provided a verification of fractal and partially Wada basin boundaries. From the pictures [Fig. 5(a)], the green unstable manifold of the periodic orbit P that is accessible from the black basin indeed intersects all three basins. However, the yellow unstable manifold of the periodic orbit Q that is accessible from the red (or blue) basin intersects only the red and blue basins, and therefore these basins are not Wada (partially Wada basin boundaries). An interesting question is why partially Wada basin boundaries and non-Wada fractal basin boundaries are intertwined. Following Poon et al. [7] and Aguirre et al. [8], we explain qualitatively why the basin boundaries are intertwined. Fig. 5(b) shows a schematic diagram of two representatively accessible orbits (P and Q) and their manifolds. Suppose that B1, B2 and B3 are three small sets belonging to three different basins (red, blue and black). The unstable manifold of P intersected all basins, and that of Q crosses two basins (red and blue). The images of B1, B2 and B3 will get arbitrarily close to the stable manifolds of P and Q when the time goes backward. In the limit, all the points on the stable manifold of P and Q are boundary points. Therefore, basin boundary points are partially Wada points and non-Wada points (smooth or fractal basin boundary). It seems to explain why the basin boundaries are intertwined on arbitrarily fine scales. In the future work, we will try to find the Wada basin of SNAs and get some more interesting results.
Fig. 5. (a) The unstable manifold (in green and yellow) of four accessible orbits [marked with four arrows] is shown, which green manifold intersects all three basins and yellow manifold intersects two basins; (b) A schematic diagram of two representatively accessible orbits (P and Q) and their manifolds. The unstable manifold of P intersected all basins, and that of Q crosses two basins (red and blue). For other details, see the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
4. Summary In this Letter, typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished in a quasiperiodically forced delayed system. We have given numerical evidences that multifarious intertwined basin boundaries for SNAs can occur in such a system. In particular, we show that Wada property, non-Wada fractality and smoothness can be intertwined on arbitrarily fine scales. Numerical computations reveal that it is fundamentally difficult to predict the eventual fate of SNAs for initial conditions on basin boundaries. We have shown that there is a transition from non-Wada fractal basin boundaries to Wada intertwined basin boundaries when a blowout bifurcation gives rise to a new SNA. A scaling exponent is used to characterize basin boundaries. The sensitivity on parameters of SNAs and Wada basin of SNAs will be also an interesting topic in the future work. Acknowledgements The authors thank the anonymous reviewers for their helpful comments and suggestions.
Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system Yongxiang Zhang a,∗ , Guiqin Kong b a b
College of Science, Shenyang Agricultural University, Shenyang 110161, China Jiujiang Precision Measuring Technology Research Institute, Jiujiang 332000, China
a r t i c l e
i n f o
Article history: Received 26 August 2009 Received in revised form 10 October 2009 Accepted 19 October 2009 Available online 24 October 2009 Communicated by A.R. Bishop PACS: 05.45.Ac 05.45.Df 05.45.Pq
a b s t r a c t A variety of different dynamical regimes involving strange nonchaotic attractors (SNAs) can be observed in a quasiperiodically forced delayed system. We describe some numerical experiments giving evidences of intertwined basin boundaries (smooth, non-Wada fractal and Wada property) for SNAs. In particular, we show that Wada property, fractality and smoothness can be intertwined on arbitrarily fine scales. This suggests that SNAs can exhibit the final state sensitivity and unpredictable behaviors. An interesting dynamical transition of SNAs together with associated mechanisms from non-Wada fractal to Wada intertwined basin boundaries is examined. A scaling exponent is used to characterize the intertwined basin boundaries. © 2009 Elsevier B.V. All rights reserved.
Keywords: Strange nonchaotic attractors Unpredictability Intertwined basin boundaries Wada property
1. Introduction Prediction is one of the fundamental goals of science. When prediction becomes impossible, it may be thought that one of the foundations of science will be shattered. Many physical, chemical, biological, and engineering problems are known to possess multiple coexisting final states. Multiple coexisting final states are determined by both initial conditions and system parameters. For a given parameter, different choices of the initial condition can lead to distinctly different asymptotic attractors, each with its own basin of attraction. The boundaries that separate different basins of attraction are basin boundaries, which can be either smooth or fractal [1–3]. A basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins, is called a Wada basin, and its boundary is called a Wada basin boundary. In particular, fractal basin boundaries of more than two basins of attraction, the Wada basin boundaries, have been studied [4–7]. Very recently, different types of fractal basins, such as Wada and riddled basins, have been reviewed [8]. It is impossible to predict, with certainty, the asymp-
*
Corresponding author. Tel.: +86 024 88493201. E-mail address: [email protected] (Y. Zhang).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.10.053
totic attractor for initial conditions in the neighborhood of fractal and Wada basin boundaries. Strange nonchaotic attractors (SNAs) typically appear in quasiperiodically forced nonlinear dynamical systems. They are geometrically strange just like chaotic attractors, while all their Lyapunov exponents are not positive, which ensures that the underlying dynamics is nonchaotic. Since the first description of SNAs by Grebogi et al. [9], dynamical behaviors of the quasiperiodically forced systems have been extensively investigated both numerically and experimentally. More work was devoted to the investigation of irregular dynamical transitions and mechanisms, most of which have been reviewed previously [10–12]. Recently, SNAs have been observed in wider dynamical systems, such as LCR circuits [13], chemical systems [14] and random dynamical systems [15]. By applying unstable sets in SNAs, mechanisms of dynamical transitions can be examined, e.g. intermittent transition [16], fractalization [17], and crises [18,19]. While the existence of SNAs was firmly established, a question that remains interesting is whether these SNAs can be predicted or not. However, in the previous work, very few topics dealing with this problem have been reported. The study of fractal and Wada basin boundaries has been restricted to regular attractors (periodic and quasiperiodic) and chaotic attractors. Specifically, our attention will be focused on the analysis of basin boundaries of SNAs. In particular, an interesting and new
Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
basin boundary can be observed, where Wada property, non-Wada fractality and smoothness can be also intertwined on arbitrarily fine scales. We have attempted to assess the unpredictable behaviors of SNAs arising from phase space. One of the goals of nonlinear dynamics is to determine the global structures of the system such as basin boundaries [20]. Another goal is to determine how these global structures come about with variation of a system parameter [21]. In our model, two special kinds of intertwined basin boundaries for SNAs are considered. The most prominent subtype is intertwined basin boundaries (smooth and fractal). Another subtype is more interesting (smoothness, non-Wada fractality and Wada property are intertwined). In this Letter, we examined basin boundaries of coexisting SNAs and their transitions between these two subtypes. Associated mechanisms of transitions are also investigated. A blowout bifurcation can lead to a new coexisting SNA. One of the main consequences of the intertwined basin boundaries is related to the difficulty of predicting to which attractor a given initial condition might go. Consequently, our results show that SNAs cannot be predicted reliably for specific initial conditions on intertwined basin boundaries. 2. The forced delayed system and dynamical regimes The quasiperiodically forced logistic map has been studied extensively, which is a representative model for quasiperiodically forced period-doubling systems. Here, we consider a representative model for quasiperiodically forced Hopf-bifurcational systems. In the present work, we investigate the dynamics of (1) with the additional quasiperiodic forcing,
xn+2 = f (xn , xn+1 , zn ) = ε 1 + p cos(2π zn ) xn+1 + xn − xn3 zn+1 = zn + ω(mod 1) (1)
In the absence of quasiperiodic forcing (p = 0), the system (1) is a simple dynamics model with time delays, which is analogous to the typical Duffing oscillator and is quite different from the delayed logistic equation. It can be applied to the field of ecological systems. If we introduce yn and xn+1 = yn , the system (1) can be rewritten as
⎧x ⎨ n+1 = yn yn+1 = ε 1 + p cos(2π zn ) yn + xn − xn3 ⎩ zn+1 = zn + ω(mod 1)
(2)
where ω and p represent the frequency and amplitude of the quasiperiodic forcing respectively. We set √ the frequency to be the reciprocal of the golden mean, ω = ( 5 − 1)/2. Obviously, the invariant subspace S is given by (xn , yn , zn ) = (0, 0, zn ) in the system (1). Since S is invariant, initial conditions in S result in trajectories which remain in S forever. In order to describe strange nonchaotic dynamics, we obtain the phase diagram as a function of the parameter ε and nonzero p, shown in Fig. 1. We have found it useful to characterize SNAs through both the Lyapunov exponent λx in the x-direction, which is given by
λx = lim
N →∞
N 1
N
i =1
∂ f ln ∂ xi
(3)
and the phase sensitivity exponent which can be obtained from phase sensitivity function Γ N
Γ N (ε , p ) = min
x0 , y 0 , z0
∂ xn max 0n N ∂ z
(4)
209
Fig. 1. Phase diagram in the ε –p plane. Regular, chaotic, SNA, and escape regimes are shown in white, gray, light gray, and black, respectively. Another type of attractor without phase sensitivity is shown in light red, which is marked AT and the largest Lyapunov exponent is approximate zero. A blowout bifurcation occurs by the critical curve L1 at which the largest Lyapunov exponent and the largest transverse Lyapunov exponent are zero. Merging crises of the chaotic attractor occur by the critical curve L2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
∂ xn ∂ z can be obtained by differentiating (2),
⎧ ∂ xn+1 ∂ yn ⎪ ⎪ ∂z = ∂z ⎪ ⎪ ⎨ ∂ y n +1 ∂ z = −2πε p sin(2π zn ) yn ⎪ ⎪ + ε 1 + p cos(2π zn ) ∂∂yzn + 1 − 3xn2 ∂∂xzn ⎪ ⎪ ⎩ zn+1 = zn + ω(mod 1)
(5)
On a SNA, the function Γ N grows with the length of the orbit N, as a power, i.e., Γ N ∼ N γ , where γ is the phase sensitivity exponent [10]. The exponent γ measures the sensitivity with respect to the phase of the quasiperiodic forcing and characterizes the strangeness of an attractor in a quasiperiodically driven system. A smooth torus has a negative Lyapunov exponent and no phase sensitivity (γ = 0). On the other hand, SNAs have negative Lyapunov exponents and high phase sensitivity (γ > 0). Chaotic attractors have positive Lyapunov exponents. In some dynamical regimes (Fig. 1), a striking phenomenon that can be observed is bistability of symmetric attractors. We examine the bifurcation structure of two symmetric attractors with separate basins of attraction, where the phase diagram (in the p–ε plane) is quite identical. Fig. 1 shows a phase diagram with single attractors for reasons of simplicity and clarity. Regular, chaotic, SNA and escape regimes are shown in white, gray, light gray and black, respectively. Quasiperiodical regions are denoted by 1T and 2T and shown in white. We can easily observe that there exist two distinct 2T quasiperiodical tongues (T1 and T2 ), where the different mechanisms may be described. In the T1 region, we show that a blowout bifurcation occurs and a symmetric attractor is born as the parameters p and ε pass through the critical curve L1 (the largest Lyapunov exponent and the largest transverse Lyapunov exponent are zero), while in the T2 region the 2T torus attractor appears via torus-doubling bifurcation. The interesting fact is that a SNA tongue (region S1) is interspersed along the boundary of escape regions, but regions where they occur are extremely long and chaotic attractors are very rare. SNAs can be observed through fractalization (route d in Fig. 1) in another region (region S2 in gray). Note that its main interesting feature is the existence of a small tongue of SNA that penetrates into the quasiperiodical regions T2 . Another type of attractor without phase sensitivity can be located in light red regimes, which is called attractor transients
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(AT) and the largest Lyapunov exponent is approximate zero. The term attractor transient refers to the fact that an orbit can spend a long time in the vicinity of the attractor A before it leaves, finally moving off to the special attractor A. Merging crises of the chaotic attractor occur by the critical curve L2. 3. Intertwined basin boundaries of SNAs in phase space While the existence of different attractors was firmly established, a question that remains interesting is the occurrence and transitions of SNAs as a system parameter changes through a critical value. Another question is whether these SNAs can be predicted or not. How do basin boundary structures of SNAs come about with variation of a system parameter? Some typical routes to SNAs can be observed in Eq. (1), such as Heagy–Hammel route, the fractalization, the type-I intermittency and boundary crises [22]. Symmetry-breaking and symmetry-recovering phenomena can also be observed [23]. Here, we focus on a small portion of this parameter space that SNAs are created through fractalization (routes c in Fig. 1). Such a phenomenon is essentially a gradual fractalization of the doubled torus (2T), but the difference is that here a transition from a two-frequency torus (2T) to SNA is realized through a gradual fractalization process instead of the transition from the 1T torus (routes d in Fig. 1). An example has been analyzed by varying ε for the fixed parameter p = 2.287. Fig. 2(a) shows two coexisting SNAs (red and blue) for the fixed parameter ε = 0.699. For the present case, the maximal Lyapunov exponent λmax is approximate −0.0005. The phase sensitivity function Γ N grows unboundedly with the power-law relation Γ N ∼ N γ , γ ≈ 0.79 [see the box in Fig. 2(a)]. At such a value, two coexisting attractors, Fig. 2(a), possess a geometrically strange property but do not exhibit any sensitivity to initial conditions (the maximal Lyapunov exponent is negative) and so they are indeed strange nonchaotic attractors. We now present numerical evidences for the existence of nonWada fractal and partially Wada basin boundaries [8] in Eq. (1). It is well known that some special kinds of fractal basin boundaries have appeared apart from the nominal fractal basin boundaries. The most prominent subtypes are intertwined, Wada, riddled and sporadically fractal basins. Some dynamical systems have three or more basins sharing the same boundary. These basin boundaries were named Wada basin boundary. However, a basin satisfies the Wada property if any initial condition that is on the boundary of one basin is also simultaneously on the boundary of another two or more basins. To possess the Wada property is stronger than to have fractal basin boundaries. Here we pay special attention to two types of intertwined basin boundaries of SNAs. Grebogi and coworkers discovered that it is common to find basin boundaries that show different dimensions in different regions [24]. These peculiar structures were named intertwined basins, describing situations in which every fractal region of the boundary has subregions inside where the boundary is smooth. Here, we observe this type of intertwined basin boundaries of SNAs in system (1). In particular, an interesting and new basin boundary can be observed, which Wada property, non-Wada fractality and smoothness can be also intertwined on arbitrarily small scales. Basin boundaries with fractal and Wada property lead to final state sensitivity. Final state sensitivity can be quantified by the uncertainty exponent α , which was firstly introduced by Grebogi et al. [25]. The uncertainty exponent α is defined as follows. Randomly choose an initial condition x0 . Define x0 = x0 + δ , where δ is a small perturbation. Determine whether the asymptotic dynamics of the system using these two initial conditions are qualitatively different. For a given perturbation δ , a fraction of uncertain initial conditions f (δ) can be computed by randomly choosing many initial conditions and determining if they are uncertain. The uncertainty exponent α is determined by fitting the scaling law f (δ) ∼ δ α . The uncertainty
Fig. 2. (a) Coexisting SNAs (TF1 and TF2) at p = 2.287 and ε = 0.699. The maximal Lyapunov exponent is approximately −0.0005. A phase sensitivity function diagram is given in the box; (b) The basin section of attraction at zn = 0.25 (SNA (TF1), SNA (TF2) and escape regions are shown in red, blue and white, respectively); (c) A magnification of the area enclosed in the green rectangle in (b); (d) Another magnification of area enclosed in green rectangle in (c); (e) A linear fit yields the following uncertainty exponent: α ≈ 0.52 for region (d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
dimension D of the fractal set embedded in the initial conditions is obtained from D = N − α , where N is the dimension of the phase space. Therefore the uncertainty dimension is defined in the range D ∈ [ N − 1, N ]. The fractal dimension is a common way to measure the complexity of a fractal basin boundary. If it is an integer num-
Y. Zhang, G. Kong / Physics Letters A 374 (2009) 208–213
Fig. 3. Coexisting attractors (SNA (BB), TF1 and TF2) at p = 2.287 and ε = 0.703. The largest Lyapunov exponent of SNA (BB) is approximately −0.013. The largest Lyapunov exponents of TF1 (or TF2) are approximately −0.007. A phase sensitivity function of SNA (BB) diagram is given in the box. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)
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ber, then the boundary is said to be nonfractal (or smooth), while if it is not an integer, the boundary is fractal. Since it is difficult to study basin structures in three dimensions, we fix the variable z and study the reduced basin structure in two-dimensional surfaces. We fix the variable z at z = 0.25 and a typical basin section of SNAs is shown in Fig. 2(b). Fig. 2(b) shows an example of intertwined basin boundaries for two different SNAs [TF1 and TF2 in Fig. 2(a)]. The basins of SNA (TF1) and SNA (TF2) are shown by red and blue, respectively. Escape regimes are shown in white. A magnification of the area enclosed in the green rectangle in Fig. 2(b) is shown in Fig. 2(c). Another magnification of area enclosed in green rectangle in Fig. 2(c) is shown in Fig. 2(d). Obviously, fractality and smoothness are intertwined on basin boundaries. We calculate an uncertainty exponent α ≈ 0.52 [in Fig. 2(e)] and a fractal dimension of 2.48 at y = 1.55, z = 0.25, but we can always find smooth subregions of the integral dimension. A much more interesting question would be to study the transition from non-Wada fractal to partially Wada basin boundaries. Another interesting question is the nature of basin boundaries for SNAs. We have observed that at ε = ε ∗ ≈ 0.7028622, a blowout bifurcation can lead to a new SNA (BB). Whether SNAs attract or repel initial conditions in the vicinity of S is determined by the
Fig. 4. (a) The basin section of attraction at zn = 0.25 (SNA (BB), TF1, TF2 and escape regions are shown in black, red, blue and white, respectively); (b) A magnification of the area enclosed in the green rectangle in (a); (c) A magnification of the area enclosed in the green rectangle in (b); (d) A magnification of the area enclosed in the green rectangle in (c); (e) A magnification of the area enclosed in the green rectangle in (d); (f) A magnification of the area enclosed in the green rectangle in (e); (g) A linear fit f (δ)
yields the following uncertainty exponent for region (c): α ≈ 0.53 [log10 versus logδ10 for the basin boundaries between the red and blue basins]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
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sign of the largest transverse Lyapunov exponent Λ T computed for trajectories in S with respect to perturbations in the subspace which is transverse to S [26]. The crucial curve L1 (in Fig. 1) plays an important role to analyze blowout bifurcation, where the largest Lyapunov exponent and the largest transverse Lyapunov exponent are zero. For values of ε that are slightly smaller than ε ∗ , there are two coexisting attractors (TF1 and TF2), and the basin boundary is non-Wada fractal. This phenomenon is an interesting example of a transition between a fractal and partially Wada basin boundary when a system parameter is varied. More precisely, a blowout bifurcation creates a new coexisting SNA, so the system passes from two to three attractors. Fig. 3 shows three coexisting attractors (SNA (BB), TF1 and TF2) at p = 2.287 and ε = 0.703. Three different attractors (SNA (BB), TF1 and TF2) can be distinguished by black, red and blue, respectively. The largest Lyapunov exponent of SNA (BB) is approximately −0.013. The largest Lyapunov exponents of TF1 (or TF2) are approximately −0.007. A phase sensitivity function Γ N of SNA (BB) is given in the box, which grows unboundedly with the power-law relation Γ N ∼ N γ , γ ≈ 0.65. Two other attractors (TF1 and TF2) have near-zero phase sensitivity exponents, which may be extremely wrinkled 2T attractors. Fig. 4(a) shows their basin section of attraction at z = 0.25. The basins of SNA (BB), TF1 and TF2 are shown in black, red and blue, respectively. Escape regimes are shown in white. Successive magnifications of small areas within the region illustrate clearly the intertwined nature of the basin boundary [Fig. 4(b)–(e)]. From the pictures, it seems that no matter how much we amplify the magnification, there seem to be regions where Wada property, fractality and smoothness are intertwined. This gives a clear (although nonrigorous) indication that the basin boundaries are intertwined, and also that some subregions of the basin boundaries for this system possess the Wada property. We calculate an uncertainty exponent α ≈ 0.53 f (δ)
and a fractal dimension of 2.47 at y = 1.6, z = 0.25 [log10
ver-
δ
sus log10 for the basin boundary between the red and blue basins in Fig. 4(g)]. The dynamics of the unstable invariant set plays a crucial role in the formation of fractal and Wada basin boundaries, especially the stable and unstable manifolds of the invariant set. In order to more rigorously confirm that the basin boundaries are intertwined (non-Wada fractal and Wada property), we numerically approximate the unstable manifold (in green and yellow) of four representatively accessible orbits [Fig. 5(a)]. It may be provided a verification of fractal and partially Wada basin boundaries. From the pictures [Fig. 5(a)], the green unstable manifold of the periodic orbit P that is accessible from the black basin indeed intersects all three basins. However, the yellow unstable manifold of the periodic orbit Q that is accessible from the red (or blue) basin intersects only the red and blue basins, and therefore these basins are not Wada (partially Wada basin boundaries). An interesting question is why partially Wada basin boundaries and non-Wada fractal basin boundaries are intertwined. Following Poon et al. [7] and Aguirre et al. [8], we explain qualitatively why the basin boundaries are intertwined. Fig. 5(b) shows a schematic diagram of two representatively accessible orbits (P and Q) and their manifolds. Suppose that B1, B2 and B3 are three small sets belonging to three different basins (red, blue and black). The unstable manifold of P intersected all basins, and that of Q crosses two basins (red and blue). The images of B1, B2 and B3 will get arbitrarily close to the stable manifolds of P and Q when the time goes backward. In the limit, all the points on the stable manifold of P and Q are boundary points. Therefore, basin boundary points are partially Wada points and non-Wada points (smooth or fractal basin boundary). It seems to explain why the basin boundaries are intertwined on arbitrarily fine scales. In the future work, we will try to find the Wada basin of SNAs and get some more interesting results.
Fig. 5. (a) The unstable manifold (in green and yellow) of four accessible orbits [marked with four arrows] is shown, which green manifold intersects all three basins and yellow manifold intersects two basins; (b) A schematic diagram of two representatively accessible orbits (P and Q) and their manifolds. The unstable manifold of P intersected all basins, and that of Q crosses two basins (red and blue). For other details, see the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
4. Summary In this Letter, typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished in a quasiperiodically forced delayed system. We have given numerical evidences that multifarious intertwined basin boundaries for SNAs can occur in such a system. In particular, we show that Wada property, non-Wada fractality and smoothness can be intertwined on arbitrarily fine scales. Numerical computations reveal that it is fundamentally difficult to predict the eventual fate of SNAs for initial conditions on basin boundaries. We have shown that there is a transition from non-Wada fractal basin boundaries to Wada intertwined basin boundaries when a blowout bifurcation gives rise to a new SNA. A scaling exponent is used to characterize basin boundaries. The sensitivity on parameters of SNAs and Wada basin of SNAs will be also an interesting topic in the future work. Acknowledgements The authors thank the anonymous reviewers for their helpful comments and suggestions.
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