Phase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos

Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 682 688 c International Academic Publishers Vol. 35, No. 6, June 15, 2001 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos ZHENG Zhi-Gang and HU Gang Department of Physics, Beijing Normal University, Beijing 100875, China (Received August 15, 2000; Revised October 19, 2000) Abstract Phase is an important degree of freedom in studies of chaotic oscillations. Phase coherence and localization in coupled chaotic elements are studied. It is shown that phase desynchronization is a key mechanism responsible for the transitions from low- to high-dimensional chaos. The route from low-dimensional chaos to high-dimensional toroidal chaos is accompanied by a cascade of phase desynchronizations. Phase synchronization tree is adopted to exhibit the entrainment process. This bifurcation tree implies an intrinsic cascade of order embedded in irregular motions. PACS numbers: 05.45.Xt Key words: phase synchronization, phase localization, Lyapunov exponents 1 Introduction Studies of the route to chaos, i.e., via what a way a nonlinear dynamical system becomes chaotic as a system parameter varies, have been a fundamental and central problem for a long time. In low-dimensional chaotic systems, or specifically say, systems with only one positive Lyapunov exponent, this transition often occurs via the following four well-known routes: (i) the period-doubling cascade route; [1] (ii) the intermittency transition route from the laminar phase to a turbulent phase; [2] (iii) the crisis route; [3] (iv) the route to chaos via quasiperiodicity instability [4] and strange non-chaotic attractors. [5] For general systems, the transition to chaos may be a combination of these routes. On the other hand, there has been growing tide of interest in high-dimensional chaotic systems. These systems usually possess more than one positive Lyapunov exponent for typical trajectories in the phase space, while the dimensionality of the phase space could be arbitrarily high. Though many studies have been done for high-dimensional chaotic systems, [6] the route to highdimensional chaos, i.e., how high-dimensional chaos arises as the system parameter changes, is still a less explored area. Recently, this problem was attempted by Harrison and Lai. [7] It was found that the system passes from chaos (one positive Lyapunov exponent) to hyperchaos (more than one positive Lyapunov exponent) via a continuous way. Till now no further investigations were made due to the difficulty in studies of high-dimensional phase space. The system composed of interacting chaotic oscillators should belong to a very good candidate in studies of transitions from low-dimensional chaos to high-dimensional chaos. Synchronizations of coupled chaotic oscillators, which were recently shown for different degrees (identical synchronization or complete synchronization, [8] generalized synchronization, [9] and phase synchronization [10] ), should be important and novel for coupled chaotic systems. Phase synchronization has been numerically and experimentally observed in a variety of systems [10,11] which touches upon the rotational dynamics of chaotic oscillators. The rotational features of chaotic attractors, which were recently emphasized for its significance, have long been ignored due to the widespread applications of Poincaré surface of sections. [12] Phase synchronization of interacting chaotic elements is a type of collective behavior much similar to interacting periodic oscillators. Rosenblum et al. [10] demonstrated that, for coupled Rossler oscillators with mismatched natural frequencies, in the synchronous regime the appropriately defined phases are locked, while the amplitudes vary chaotically and are practically noncorrelated. They also revealed that the synchronization of phases is accompanied by a topological transition of the chaotic attractor. [10] In this paper, we will present a mechanism of the transition from lowto high-dimensional chaotic motions. We demonstrate that, for more than two coupled chaotic oscillators, desynchronizations of phases are responsible for the topological transition of the chaotic attractor. This cascade of transitions manifests for the chaotic attractor through transitions from high-dimensional chaotic torus (e.g., T n toroidal chaos) to low-dimensional chaotic torus (e.g., T n 1 ), which is similar to the transitions we proposed for coupled limit cycles. [13] 2 Phase Coherence and Localization Let us choose N Rossler oscillators with the nearestneighbor interactions as our paradigmatic model. [14] For The project supported by National Natural Science Foundation of China (19805002), the Special Funds for Major State Basic Research Projects (G2000077304), and the Foundation for University Key Teacher by the Ministry of Education of China

No. 6 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos 683 the x-coupling, in dimensionless form, the equation of motion can be written as ẋ i = ω i x i z i + K(x i+1 2x i + x i 1 ), ẏ i = ω i x i + ay i, ż i = f + z i (x i c), (1) where we adopt a = 0.165, f = 0.2 and c = 10 in the present work, i = 1,2,...,N. The parameters ω i = ω 0 + i are different for different oscillators, where the misfit i [, ]. K is the coupling strength among the nearest neighbors. For general chaotic systems, the definition of a phase is difficult due to the multiple centers of rotation. A typical definition of a time series s(t) is to introduce a principal integral [12] s = 1 π P + s(τ) t τ dτ, where P means that the integral is taken in the sense of the Cauchy principal value. A phase θ(t) of this signal s(t) thus can be defined as ψ(t) = s(t)+ı s(t) = A(t)exp[ıθ(t)]. For a Rossler oscillator, s(t) x(t), s(t) y(t). Therefore x(t) and y(t) can be considered as ideal variables to define the phase of the i-th oscillator by θ i (t) = tan 1 [y i (t)/x i (t)]. (2) The evolution of the phases of all oscillators is usually chaotic due to the intrinsic stochasticity. However, due to interactions among oscillators, phases of oscillators will tend to organize themselves to coherent states, leading to collective order in chaos. To study this self-organized order, we may define the average winding number of the i-th oscillator as the temporal average of the phase velocity, 1 Ω i = lim T T T 0 θ i (t)dt. (3) Due to the introduction of phase variables, one is able to rewrite Eq. (1) into coupled phase and amplitude equations by inserting x i (t) = A i (t)cos[θ i (t)], y i (t) = A i (t)sin[θ i (t)] into Eq. (1), θ i = ω i + asin θ i cos θ i + z i A i sin θ i K sin θ i (A i+1 cos θ i+1 2A i cos θ i A i + A i 1 cos θ i 1 ), Ȧ i = aa i sin 2 θ i z i cos θ i + K cos θ i (A i+1 cos θ i+1 2A i cos θ i + A i 1 cos θ i 1 ), ż i = f cz i + A i z i cos θ i. (4) Under some circumstances, one may neglect the effect of the amplitudes A i (t) and z i (t) on the phase equation, thus a phase model can be constructed θ i = ω i + F({θ j, j = 1,2,...,N},K), (5) where F is a nonlinear function of phase variables with K being the coupling strength. A particular case has been studied by us, [13] where the phase equations satisfy θ i = ω i + K 3 [sin(θ i+1 θ i ) + sin(θ i 1 θ i )]. (6) This system exhibits abundant synchronization dynamics. In the following we shall see that system (1) behaves much similar to this phase model. In order to study the phase relations of coupled oscillators as the coupling strength varies, we observe the profile of the phase of all other oscillators as the phase of a chosen oscillator, e.g., the first oscillator θ 1, passes 2nπ, where n denotes integers. One may make statistics of the distribution of each phase profile. In Fig. 1, we give the statistics of θ 2 and θ 3 with 2π modulus when θ 1 passes 2nπ for three interacting Rossler oscillators with ω i = 1.006,1.0,0.988, and K = 0.001, 0.0065, 0.008 and 0.011 in Figs 1a 1d, respectively. In Fig. 1a, one may find that as θ 1 = 2nπ, P(θ 2 ) and P(θ 3 ) distribute over the whole range from 0 to 2π, i.e., θ 1 does not form a relation with the other two phase variables. As the coupling strength increases, one may find that the distribution of θ 2, P(θ 2 ), becomes localized [Though P(θ 3 ) has a low peak at θ 3 2.8, there is still probability for all other values, which implies noncoherent relation between θ 1 and θ 3 ]. This interesting phase localization behavior indicates that a coherent relation builds between θ 1 and θ 2. Phase localizations for both the (θ 1,θ 2 ) and (θ 1,θ 3 ) occur when the coupling is further increased, as shown in Fig. 1c for K = 0.008 and Fig. 1d for K = 0.011. It can be found that as long as a coherence relation is built, the localization distribution is a typical Gaussian form. This can be understood as the influence of noise. In the absence of noises, the relation between phases should be a solid one, i.e., the localization distribution P(θ 2 ) should be a δ-function. In the present system, noise arises from the chaoticity of motion. The chaotic noise makes the phase coherence a fluctuating one around the center. It is interesting to investigate phase dynamics before the onset of phase localization. We can study phase difference ϕ i,j (t) = θ j (t) θ i (t) before coherence of θ i and θ j. In Fig. 2a the evolution of the phase difference ϕ 1,2 (t) is shown for two different coupling strengths for K = 0.0054 and 0.0056. It is shown that the evolution exhibits a stickslip motion. This is a typical intermittent motion between saddle points along the heteroclinic orbit. However, the motion is an irregular one. We record time interval τ between penetration of ϕ 1,2 from 2nπ to 2(n + 1)π, called stick time and do statistics. In Fig. 2b, we give statistics of the stick time for K = 0.0054 and 0.0056. Different from

684 ZHENG Zhi-Gang and HU Gang Vol. 35 Gaussian phase distribution, the stick time distribution possesses an exponential long-time tail, i.e., a Poisson distribution is obeyed for long-time sticks. The mechanism for this exponential long tail is still an open issue. Fig. 1 Statistics of θ 2 and θ 3 with 2π modulus when θ 1 passes 2nπ for N = 3 coupled Rossler oscillators with ω i = 1.006, 1.0, 0.988. K = 0.001, 0.0065, 0.008 and 0.011 correspond to (a) (d), respectively. Fig. 2 (a) The evolution of the phase difference ϕ 1,2(t) for K = 0.0054 and 0.0056. The evolution exhibits a stick-slip motion. N = 3 Rossler oscillators with the same parameter as Fig. 1. (b) Statistics of the stick time for K = 0.0054 and 0.0056. An exponential long-time tail can be found.

No. 6 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos 685 3 Phase Synchronization Tree and Topology Transitions The above finding of phase localization implies a macroscopic order immersed in the chaotic background. In fact, it is instructive to investigate the relation between averaged winding numbers and the coupling in order to reveal this intrinsic order. The localization indicates a temporal localization of the phase difference ϕ ij (t) = θ i (t) θ j (t). The 1:1 phase synchronization between the elements i and j just satisfies this temporal localization, implying the formation of the relation Ω i = Ω j for i j. The relation Ω i K thus forms a synchronization tree. In Figs 3a 3c, we plot the bifurcation tree of the average frequencies for N = 3,5,15 as a function of the coupling strength for the coupled Rossler systems with different natural frequency misfits. We found very interesting phase synchronization tree structures similar to those for coupled limit cycles, [13] as an example shown in Fig. 3d. Although the coupled system is chaotic (usually hyperchaotic, in fact), oscillators form phase synchronized clusters as the coupling strength is increased. In Fig. 3d, we plot a case of phase synchronization of system (6) for five coupled limit cycles, where the phase for each oscillator is naturally defined, and the motion for individual oscillator is periodic in the absence of interactions among them. The comparison shows a clear similarity, whereas the coupled Rossler system is chaotic. Therefore, this phase synchronization tree gives a vivid picture of macroscopic order in chaotic motions. Phase dynamics plays a crucial role in constructing spatiotemporal patterns, where the synchronization of phases is necessary. The macroscopic order in chaos is very interesting and instructive in understanding the collective dynamics and spatiotemporal patterns of coupled chaotic systems. Fig. 3 In (a) (c), we show the synchronization bifurcation trees of the average frequencies for N = 3, 5, 15 respectively as functions of the coupling strength for the coupled Rossler systems with different natural frequency misfits. (d) The tree of phase synchronization for five coupled limit cycles. How oscillators organize themselves to phase entrainment, i.e., how the topology of attractor changes during synchronization bifurcations, is an intriguing point. To characterize this feature, let us study Lyapunov exponent spectrum {λ 1 λ 2 } of the coupled system. It is known that if there is more than one positive exponent, the attractor should be hyperchaotic. If λ 1 = 0, and there are totally M zero exponents, we say the motion is on an M-torus T M, i.e., a quasiperiodic motion with M incommensurate winding numbers. Moreover, if λ 1 > 0,

686 ZHENG Zhi-Gang and HU Gang Vol. 35 and there are totally M zero exponents, the motion is still on M-torus while chaotic (called chaotic M-torus). As a system parameter varies, Lyapunov exponent spectrum changes too, leading to topological transitions of the attractor. As one zero Lyapunov exponent becomes negative, the dimensionality of attractor should change by one. Fig. 4 Lyapunov exponent spectrum varies against the coupling strength. (a) N = 3 coupled Rossler oscillators; (b) N = 5 coupled Rossler oscillators; (c) N = 5 coupled limit cycles. It was found by Rosenblum et al. [10] using two coupled Rossler oscillators that phase synchronization is accompanied by the transition of the topology of the attractor, i.e., one of the zero Lyapunov exponents becomes negative. This indicates that the degree of freedom of the system decreases by one. For N(> 2) coupled oscillators with small parameter mismatches, phase synchronization, or inversely say, phase desynchronization, should be responsible for topology transitions of the attractor. This is shown in Fig. 4a for N = 3 and Fig. 4b for N = 5, where the first 2N Lyapunov exponents against the coupling strength are plotted (the other N exponents are all negative and thus not the focus here). One may clearly find that for strong couplings, the first N exponents are all positive with only one zero exponent, implying a hyperchaotic motion. One zero exponent denotes the direction along the chaotic manifold. As one decreases the coupling, a striking transition occurs at K = 0.0115 for N = 3 and K = 0.032 for N = 5, where one zero exponent touches and keeps zero. This indicates a transition from chaotic T 1 to chaotic T 2, where the dimensionality of the attractor increases by one. By comparing the Lyapunov exponents and the synchronization bifurcation tree shown in Figs 3a and 3b, we find that the first phase desynchronization takes place at K = 0.011 for N = 3 and K = 0.033 for N = 5, in good agreement with the values for the topology transition. Furthermore, the second and high-order phase desynchronizations all correspond to positions for one exponent passing from negative to zero values. This strongly verifies our proposition that phase desynchronization is a key mechanism responsible for transitions from low- to high-dimensional chaos. The presence of phase synchronization signifies the coherence of phases of some oscillators, i.e., these phases build certain relations and therefore are localized. In Eq. (6), due to the formation of relations for some phases, the number of independent variables decreases (in fact, these phase relations correspond to slow variables, whereas fluctuations around them are fast variables and thus can be adiabatically eliminated). This effectively reduces the dimensionality of phase space. Therefore, through phase desynchronization the attractor of motion may change from low-dimensional chaotic torus

No. 6 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos 687 to high-dimensional torus. Moreover, we find that in a large coupling regime, the chaotic (hyperchaotic) tori are stable. This is similar to the case of coupled limit cycles. Figure 4c gives the Lyapunov exponents varying against the coupling for coupled limit cycles of Eq. (6). One may find strong similarity to chaotic cases as shown in Figs 4a and 4b, where the topological transition takes place via a way from periodicity to low-dimensional torus and highdimensional quasiperiodicity with the decrease of coupling strength. For coupled chaotic cases, the motion is chaotic without breaking the torus. A chaotic torus becomes unstable through the phase desynchronization, specifically say, through intermittency. In Figs 5a 5c, we exhibit the Poincaré map θ 2 (n) θ 1 (n) as θ 3 (t) passes 2nπ (n being integer) for N = 3. In Fig. 5a for K = 0.011, where a full phase synchronization is achieved, the section is a small dotted region [15] (For coupled limit cycles the section is only composed of some dots, implying periodic motion T 1 ). The motion therefore is chaotic on T 1. By decreasing the coupling, phase desynchronization can be observed with only one synchronized cluster (1,2). In Fig. 5b for K = 0.008, a chaotic 2-torus can be found, corresponding to a strip on the phase plane. In Fig. 5c, where all three phases are desynchronized, we observe a global scattering on the phase plane, indicating the 3-toroidal chaos. Fig. 5 The Poincaré map θ 2(n) θ 1(n) as θ 3(t) passes 2nπ (n being integer) for N = 3. (a) K = 0.011, the motion is chaotic on T 1 ; (b) K = 0.008, a chaotic 2-torus can be found; (c) K = 0.001, the 3-toroidal chaos is found. 4 Concluding Remarks To summarize, we proposed in this paper a mechanism for the route to high-dimensional chaos, i.e., the route through phase desynchronization. As examples we chose the coupled chaotic system to study the role of phase entrainment in the transitions from low-dimensional to high-dimensional chaos. We found the phase coherence and localization behavior, which reflect the relation between phases in chaos. Moreover, we adopted the synchronization bifurcation tree to show this intrinsic order embedded in highly chaotic motions. This tree shows a structure much similar to that of coupled limit cycles. We also studied the Lyapunov exponent spectrum to study the topology of the attractor. We found that the phase desynchronization is accompanied by the transition from lowdimensional to high-dimensional chaos. These chaotic motions are identified as hyperchaotic tori. We also showed the stability of these chaotic tori in a large regime of cou-

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