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Phys. Scr. 77 (2008) 045005 (7pp)
doi:10.1088/0031-8949/77/04/045005
Synchronization and bifurcation structures in coupled periodically forced non-identical Duffing oscillators U E Vincent1 and A Kenfack2 1
Department of Physics, Olabisi Onabanjo University, PMB 2002, Ago-Iwoye, Nigeria Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany 2
E-mail:
[email protected]
Received 15 August 2007 Accepted for publication 21 February 2008 Published 20 March 2008 Online at stacks.iop.org/PhysScr/77/045005 Abstract We study the bifurcation structure and the synchronization of a double-well Duffing oscillator coupled to a single-well one and subjected to periodic forces. Using the amplitudes and the frequencies of these driving forces as control parameters, we show that our model presents phenomena which were not observed in a similar system but with identical potentials. In the regime of relatively weak coupling, bubbles of bifurcations and chains of symmetry-breaking are identified. For much stronger couplings, Hopf bifurcations born from orbits of higher periodicity, as well as subcritical and supercritical Neimark bifurcations emerge. Varying the coupling strength, we also find a threshold for which the system remains quasiperiodic. Moreover, tori-breakdown route to a strange non-chaotic attractor is another highlight of features found in this model. In two parameter diagrams, regions of chaos and quasiperiodicity are clearly identified. Finally, threshold parameters for which synchronization occurs have been found. PACS numbers: 02.30.oz, 05.45.Pq, 05.45.Xt (Some figures in this article are in colour only in the electronic version.)
to coupled non-linear systems. Of particular interest in this study are coupled Duffing oscillators—strictly dissipative non-linear oscillators which model the motion of various physical systems such as a pendulum, an electrical circuit or a Josephson-junction to mention but a few. Two or more identical or non-identical oscillators may be coupled in different ways. Thus, the type of the oscillators and couplings as well as the external driving have a significant influence on the dynamics of the coupled system. For instance, Kozlowski et al [6] have shown that the global pattern of bifurcation curves in the parameter space of two coupled identical periodically driven single-well Duffing oscillators consists of repeated subpatterns of Hopf bifurcations—a scenario that differs from that of a single Duffing oscillator. Recently, Kenfack [9] studied the model already considered in [6] but using identical double-well potentials. That model presented many striking departures from the behaviour of coupled single-well Duffing oscillators. Scenarios such as multiple
1. Introduction The dynamics of coupled non-linear oscillators has attracted considerable attention in recent years because they arise in many branches of science. Coupled oscillators are used in the modelling of many physical, chemical, biological and physiological systems such as coupled p–n junctions [1], charge–density waves [2], chemical-reaction systems [3], and biological-oscillation systems [4]. They exhibit varieties of bifurcations and chaos [5–12], pattern formation [13], synchronization [7], [14–22] and so on. Bifurcation analysis is a useful and widely studied subfield of dynamical systems. The observation of the bifurcation scenario allows one to draw qualitative and quantitative conclusions about the structure and dynamics of the systems theory. Numerous theoretical and numerical studies in this direction have been carried out for several specific problems [6–12]. Here, we have referred to recent studies relevant 0031-8949/08/045005+07$30.00
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© 2008 The Royal Swedish Academy of Sciences
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Phys. Scr. 77 (2008) 045005
U E Vincent and A Kenfack
period-doubling (pd) of both types, symmetry-breaking (sb), sudden chaos and a great abundance of Hopf bifurcations were found in that model, in addition to the well-known routes to chaos in a one-dimensional (1D) Duffing oscillator [10]. The physics underlining the dynamics of a coupled oscillator could also be understood by investigating the synchronization behaviour. Indeed, synchronization phenomena in coupled non-linear oscillators are in general of fundamental importance and have been extensively investigated both theoretically and experimentally in the context of several other specific problems arising in laser dynamics, electronics circuits, secure communications and time series analyses, to name a few [14]. For a more detailed and comprehensive description of different types of synchronization likely to occur in coupled systems, the reader can refer to the book of Pikovsky et al [15]. The synchronization of chaotic systems and in particular that of different or non-identical chaotic systems is challenging not only because of the extreme sensitivity of a chaotic system to tiny perturbations but also because of the parameter mismatch introduced when non-identical systems are considered. The present study derives its motivation from [6, 9]. Here, we present a model of periodically driven coupled oscillators consisting of a double-well Duffing oscillator coupled to a single-well Duffing oscillator. We numerically analyse varieties of bifurcations with special emphasis on those which are typical of the model. This is explored for different forcing modes and in different coupling regimes as a function of the control parameters of the system, namely, the amplitudes and the angular frequencies of the driving forces. In addition, the relevant parameters of the system for which synchronization could be achieved is examined. The paper is organized as follows: section 2 describes the system under consideration. In sections 3 and 4, we present our results and conclude the paper in section 5.
exhibiting rich and baffling varieties of regular and chaotic motions. When two equation (2) systems interact with each other through a specific coupling, the dynamics is expected to be even richer and more attractive. The coupling employed in the present paper is a linear feedback which can be interpreted as a perturbation of each oscillator by a signal proportional to the difference of their positions. The potential governing such a coupled system reads V (x, y) =
(3)
where k is the coupling parameter. Depending on the values taken by the parameters α, β, α 0 and β 0 , one distinguishes three categories of globally bounded coupled Duffing oscillators; namely the single–single wells, the single–double wells and the double–double wells. From equation (3), the following equations of motion of the driven, coupled double–single wells (α = −α 0 > 0, β = β 0 > 0) Duffing oscillators can be derived 2 d x dx 2 = −b + αx − βx 3 + k(y − x) + f 1 cos(ωt), dt dt (4) 2 dy d y = −b − αy − βy 3 − k(y − x) + f 2 cos(ωt), dt 2 dt where f 1 and f 2 represent the amplitudes of the driving forces acting on oscillators x and y, respectively. The extended phase space of our model is 5D (R 4 × S 1 ), wherein any element of the state space is denoted by (x, dx/dt, y, dy/dt, θ); S 1 being the unit circle containing the phase angle, θ = ωt. Thus, in our numerical study, we visualize the attractors in the subspace along with their bifurcations by exploring the dynamics in the Poincaré cross-section defined by X (5) = (x, dx/dt, y, dy/dt, θ = θ0 ) ∈ R 4 × S 1 ,
2. The model
where θ0 is a constant determining the location of the Poincaré cross-section on which the coordinates (x, dx/dt, y, dy/dt) ≡ (x1 , v1 , x2 , v2 ) of the attractors are expressed. It is worth noticing that time-reversal symmetry is broken. Due to the symmetry of the potential V (x, y) = V (−x, −y), the system is invariant under the following transformation
The Duffing oscillator [23] that we treat here is a well-known model of non-linear oscillator. It is governed by the following dimensionless second-order differential equation d2 x dx dV (x) +b + = f cos(ωt), 2 dt dt dx
α 2 β 4 α0 2 β 0 4 k x + x + y + y + (x − y)2 , 2 4 2 4 2
(1)
S : (x1 , v1 , x2 , v2 , t) → (−x1 , −v1 , −x2 , −v2 , t + T /2),
where x and b stand for the position coordinate of a particle and its damping parameter, respectively. The righthand side of equation (1) represents the driving force at time t, with the amplitude f and the angular frequency ω. The oscillator belongs to the category of 3D dynamical systems (x, dx/dt, t). The system (1) is a generalization of the classic Duffing oscillator equation and can be considered in three main physical situations, wherein the dimensionless potential
where T = 2π/ω is the period of the driving forces. Therefore in addition to saddle nodes (sn), pd and Hopf (H), one may expect sb. This can be evidenced with the eigenvalues spectra obtained from a simple linear stability analysis. Additional features may arise not only because of the driving force but also because the two coupled oscillators are subject to different potentials.
x2 x4 +β , (2) 2 4 can be a (i) single-well (α > 0, β > 0), (ii) double-well (α < 0, β > 0) or (iii) double-hump (α > 0, β < 0) potential. Each of the above three cases constitutes on its own a classic central model describing inherently non-linear phenomena V (x) = α
3. Bifurcation structures In the numerical results that follow, we investigate the dependence of the system behaviour at given driving strengths f 1 and f 2 , with the coupling strength k and the driving frequency ω as control parameters. The bifurcation diagrams 2
Phys. Scr. 77 (2008) 045005
U E Vincent and A Kenfack
bifurcation diagram in this regime, together with its Lyapunov spectrum λmax . In the periodic window 0.3375 6 ω 6 0.34, a sequence of reverse pd bifurcations yields a period-5 attractor which is later destroyed at ω = 0.34 in a crisis event, say sudden chaos. After the large chaotic domain intermingled with windows made up of periodic orbits, another reverse pd cascade occurs around 0.352 < ω < 0.355. Again, the cascade generates a period-4 attractor which is subsequently destroyed in another crisis event. This bifurcation scenario is in fact the common feature of the system as ω is further increased. However, for ω > 0.94, we find periodic orbits dominating the system behaviour. Notice that different bifurcations observed throughout this study are perfectly traced by the Lyapunov spectra. We show in figure 1(b) the Lyapunov corresponding to the bifurcation diagram of figure 1(a) as a prototypical example. As the driving-force amplitude f increases, different transitions can be captured besides the pd sequence already observed. The bifurcation diagram of figure 1(c), plotted for f = 4.2, shows the occurrence of the attractor bubblings (bubbles) sandwiched by sb. The sequence observed, consisting of a sb, a pd, bubbles, a reversed pd and a sb bifurcations, can simply be denoted by -sb-pd-(bubbles)-pdsb-. This means at the first sb point, a symmetrical periodic orbit splits into two coexisting asymmetrical periodic orbits (one being the mirror image of the other); each of the two asymmetrical periodic orbits give rise to another asymmetrical orbit with pd followed by bubbles. The bubbles in return generate asymmetrical periodic orbits which join by a pair (reversed pd) to form a symmetrical orbit at the second sb point. The sequence -sb–pd-(bubbles)-pd–sb- which is also observed for f > 4.2, was not reported in previous literature for coupled Duffing oscillators. However, we found a similar sequence in a recent study for coupled ratchets exhibiting synchronized dynamics [7]. Subsequently, we will refer to the integer n as the period of an orbit.
Figure 1. Bifurcation diagrams for low coupling k = 0.1 with (a) f = 4.0 and (c) f = 4.2. In (a), a sequence of reverse pd route to chaos and periodic windows sandwiched by chaotic domains are also visible. These structures are fully traced by the Lyapunov spectrum (b). In (c), bubbles of bifurcation (at ω ∼ 0.16) are sandwiched by pd born from sb.
show the projection of the attractors in the Poincaré section onto one of the system coordinates versus ω. To gain further insight about the dynamics of the system under investigation, we compute the Poincaré sections and Lyapunov spectra λmax . These results are obtained solving equation (4) with the help of the standard fourth-order Runge–Kutta algorithm. Starting from an initial condition, the system is numerically integrated for 100 periods of the driving force until the transient has died out. To ascertain periodic, quasiperiodic and chaotic trajectories, the system is further integrated for 180 periods. Next, our results are presented for different coupling regimes (weak, moderate and strong), distinguishing the case when only one oscillator is driven and the case when both oscillators are mutually driven.
3.1.2. Moderate coupling (k = 1). Setting k = 1, we find (besides pd and sb) an abundance of Hopf bifurcations. In the earlier work [6, 9], no periodic orbits with periods larger than n = 3 were found to undergo Hopf bifurcations. However, in the present model, this number is much larger and can reach n = 6. In figure 2, we plot two bifurcation diagrams for: (a) f = 1 and (b) f = 15.7 in order to illustrate these observations. Hopf bifurcations with orbits of periods (a) n = 2 and (b) n = 6 can clearly be seen. By means of Poincaré cross-sections, we further zoom structures in this regime, where Hopf bifurcations are predominant. Depicted in figure 3 is another bifurcation scenario found in our model, namely the Neimark bifurcation. Figure 3(a) displays an unstable fixed point spiralling out for ω = 0.38005. Further increasing ω, the system moves outward from that unstable fixed point, passing through states like the one presented in figure 3(b) for ω = 0.38009, towards the attracting invariant closed curve (torus) shown in figure 3(c) for ω = 0.381. It is noteworthy that the transition is termed secondary Hopf bifurcation or Neimark bifurcation [24, 25]. Moreover, it is a supercritical Neimark bifurcation since the system moves outward from near an unstable fixed point towards the attracting invariant
3.1. The case of one driven oscillator We begin by considering the case in which the second oscillator is unforced, i.e. f 2 = 0; while f 1 = f is non-zero. 3.1.1. Weak coupling (0 < k < 1). At weak coupling strength, bifurcation structures found are essentially similar to those commonly observed in 1D Duffing oscillators. For k = 0.1, we computed bifurcation diagrams in a large range of f . In figure 1, for instance, we set f = 4.0 and plot a typical 3
Phys. Scr. 77 (2008) 045005
U E Vincent and A Kenfack
Figure 2. Bifurcation diagram showing varieties of Hopf bifurcations for k = 1.0: (a) n = 2, f = 1 and (b) n = 6, f = 15.7. Here, n represents the period of the periodic orbit.
closed curve. In general, the bifurcated solution can be stable, say supercritical or subcritical otherwise. In figure 2(a), sudden chaos occurs at ω ≈ 0.3826. Above this critical ω value, a steady state chaotic motion emerges and spans a relatively wide range of the driving frequency. By gradually increasing k, with ω = 0.38285, the system which is predominantly chaotic begins to establish correlations between x1 and x2 and reach the quasiperiodic regime as k becomes larger than the threshold kth ≈ 2.95, see figure 4.
Figure 3. Poincaré sections (x1 , v1 ) illustrating the supercritical Neimark bifurcation for k = 1 and f = 1: (a) ω = 0.38005, (b) ω = 0.38009 and (c) ω = 0.3810.
0.08
λmax
0.04
3.1.3. Strong coupling (k = 5). Considering k = 5, as used in [6, 9], and for low ω values, we find that periodic orbits dominate for large values of the driving amplitude ( f > 20). However in figure 5, phenomena such as pd, sb, sn, resonance (R), bubbles and sudden chaos can be found in the frequency range 0.1 6 ω 6 0.4, for moderate values of f 6 20. We have observed throughout, a variety of Hopf bifurcations of higher period n 6 6 (not shown), much like in the case k = 1. Interestingly, another transition found in this model is the passing from low-dimensional to high-dimensional quasiperiodic orbits via tori-breakdown. The sequence of the transformation from tori (low-dimensional quasiperiodic orbit) to a strange non-chaotic attractor (high-dimensional quasiperiodic orbit) are shown in figure 6. In figure 6(a), we plot a Poincaré section for ω = 1.38, showing a smooth quasiperiodic attractor visiting three folded tori in the (x2 , v2 ) sub-space. This folding state of tori is the onset of toribreakdown. Upon increasing the frequency to ω = 1.412, for instance, the smooth folded tori become completely broken, as can be seen in figure 6(b). The corresponding Lyapunov exponent, λmax = −0.0174, shows that the attractor is not chaotic as the eye-test may tend to show; but rather it is a
0
–0.04 0
1
2
3
k
4
5
6
Figure 4. Lyapunov spectrum, for ω = 0.38285 and f = 1, versus the coupling strength showing the threshold (kthr ≈ 2.95) for which the system becomes quasiperiodic.
strange non-chaotic attractor. One would in principle expect chaos from that transition as was reported by Kozlov [26] in a system of coupled Van der Pol oscillators. 3.2. Mutually driven oscillators Here, we examine broadly a situation wherein a common external forcing acts simultaneously on the two oscillators, i.e. f 1 = f 2 = f . In this case, we have also computed several bifurcation diagrams. The weak coupling regime possesses structures similar to what was found when only one of the 4
Phys. Scr. 77 (2008) 045005
U E Vincent and A Kenfack
Figure 7. Poincaré sections (x2 , v2 ) illustrating the subcritical Neimark bifurcation for k = 1 and f = 1: (a) ω = 0.4076, (b) ω = 0.4078 and (c) ω = 0.41. Figure 5. Bifurcation diagrams for k = 5, (a) f = 20 and (b) f = 4. Bubbles at around ω = 0.26 and several regions of R, pd, sn, sb and chaos are clearly seen.
Figure 8. Two-parameter bifurcation diagrams in (a) ( f − ω) plane for k = 1 (b) ( f − ω) plane for k = 5 (c) (k − ω) plane for f = 1 and (d) (k − ω) plane for f = 1, showing regions of chaos (yellow), non-attraction (white) and quasiperiodicity of period-1 (magenta), period-2 (green) and period-3 (red).
closed curve, figure 7(c), via the transition state in figure 7(b). However, the bifurcation in this mutually excited system leads to a chaotic state, figure 7(d), via the tori doubling shown in figure 7(c) as the driving frequency ω increases from 0.395 to 0.43. During this transition, the subcritical Neimark bifurcation takes place in which the system moves away from the attracting 2T invariant closed curve and goes through a similar state like in figure 7(b) and eventually ends in the chaotic attractors shown in figure 7(d). For ω = 0.4215 in figure 7(d), two asynchronized chaotic attractors are visible in the (x1 , v1 ) and (x2 , v2 ) sub-spaces.
Figure 6. Poincaré sections (x2 , v2 ) showing the tori-breakdown route to a strange non-chaotic attractor; (a) attractor (folded tori) ω = 1.38, (b) a strange non-chaotic attractor (destroyed tori) ω = 1.412, λmax = −0.0174.
oscillators is periodically forced. Thus, we present here the dominant structures for moderate and strong couplings. In this case, the supercritical Neimark bifurcation is also found in addition to phenomena previously observed when only one oscillator is driven. This is illustrated in figure 7. The system actually moves outward from near an unstable fixed point, figure 7(a), towards the attracting 2T invariant
4. Parameter phase diagrams and synchronization The parameter space of the full non-linear system is far too large for a systematic numerical analysis. Here, we want to make use of the two-parameter phase diagrams to globally 5
Phys. Scr. 77 (2008) 045005
U E Vincent and A Kenfack
Figure 9. Parameter space (k, f ) for several values of ω labelled on panels showing the quantity p that indicates where the synchronization happens. The colourbar takes on values ranging from 1 (red) for full synchronization to 0 (blue) for zero synchronization. Each point of the plot is obtained after averaging p over 50 realizations of our system with random initial conditions.
attempt to detect: (i) regions of chaos and periodicity, and also (ii) regimes of synchronization.
the one corresponding to the case where both oscillators are driven ( f 1 = f 2 = f = 1). Throughout one can clearly distinguish regions of chaos (yellow) and quasiperiodicity with period-1 (magenta), period-2 (green) and period-3 (red). The white colour corresponds to non-attraction region. Notice that similar structures are also observed in ( f − ω) planes when both oscillators are periodically driven.
4.1. Parameter phase diagrams Using the Software Dynamics [27], we were able to explore the bifurcations in two-parameter plane, identifying appropriately quasiperiodic and chaotic regions. Typical twoparameter phase diagrams are displayed in figure 8, for one externally driven case (i.e. f 1 = f, f 2 = 0), in both ( f − ω) and (ω − k) planes. Panels (a) and (b) are plotted in ( f − ω) planes for k = 1 and k = 5, respectively. In (ω − k) plane, however, we plot in panel (c) the phase diagram for which only oscillator is driven ( f 1 = f, f 2 = 0) and in panel (d)
4.2. Synchronization In this work another point we are interested in is in which parameter ranges the system may synchronize. To proceed we use the quantity which has been intuitively defined by Kostur et al [28]. The idea is to evaluate the fraction of 6
Phys. Scr. 77 (2008) 045005
U E Vincent and A Kenfack
time during which the two oscillators are synchronized within a desired accuracy ; that is when the trajectories x(t) and y(t) are such that the following condition is fulfilled |x(t) − y(t)| 6 . In practice, this amounts to computing the time that they stay synchronized and then determine the ratio p between the synchronized and the total time span of the considered trajectory. In all our calculations we have set = 0.01, a reasonable tolerance for synchronization. In figure 9, we show this quantity p in (k − f ) planes, when only one oscillator is driven, and for several values of the driving frequency ω labelled on panels. Each value of this figure is obtained after averaging p over many trajectories of our system generated with random initial conditions. In principle for a system exhibiting only one attractor, a single trajectory (x(t), y(t)) would suffice. The associated colourbar of figure 9 takes on values of p ranging from 1 (red) for full synchronization to zero (blue) corresponding to no synchronization and is consistent with the above definition of p. Regions of synchronization are clearly visible as we move from one panel to another. Remarkably, larger values of the driving amplitude f ( f > f thr = 1.2) do not lead to any synchronization, regardless of the driven frequency and the coupling parameter k. However, the synchronization almost always occurs if f 6 f thr and k > kthr ≈ 5.5. Though the nature of the synchronization observed here has not been ascertained, we expect it to be of phase synchronization (PS) as this has been very predominant in coupled non-identical chaotic systems [29, 30].
complement previous ones and provide a more general overview, especially as far as bifurcation structures are concerned, in two coupled driven Duffing oscillators.
Acknowledgments We thank Kamal P Singh and T Bartsch for illuminating discussions and for critically reading the manuscript. AK acknowledges the Reimar Lüst grant and the financial support by the Alexander von Humboldt foundation in Bonn.
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5. Concluding remarks In summary, we have introduced a model of periodically driven oscillators consisting of a double-well Duffing oscillator linearly coupled to a single-well Duffing oscillator. We have shown that the model admits a very complex dynamical structure that strongly depends on the control parameters. In addition to the variety of phenomena earlier reported in the coupled Duffing oscillators with identical potentials [6, 9], our model exhibits several other interesting features which are basically attributed to the difference between the two oscillators’ potentials. Essentially, we found attractor bubbling (bubbles) and chains of sb bifurcations occurring in a relatively weak coupling regime; while for much stronger couplings, Hopf bifurcations born from higher periodic orbits, subcritical and supercritical Neimark bifurcations emerge. With the help of Poincaré cross-sections, periodic, quasiperiodic and chaotic attractors of the system were visualized and in particular a tori-breakdown route to strange non-chaotic attractors was found. A perusal of the two-parameter bifurcation diagrams has revealed chaos and quasiperiodicity regions. Furthermore, the structures found in this model can be perfectly traced by the Lyapunov spectrum. This spectrum shows spikes at the bifuraction points reflecting a sudden change in the system. In ratchet-like models, such bifuractions may lead to dramatic changes in transport properties of the system such as current reversals [31, 32]. Besides the synchronization of the system has been studied using an intuitive quantity defined by Kostur et al [28]. We found parameters threshold for which the system do synchronize. The present results, however, should sufficiently 7
