Oscillation death in a coupled van der Pol Mathieu system

Transcription

1 PRAMANA c Indian Academy of Sciences Vol. 81, No. 4 journal of October 2013 physics pp Oscillation death in a coupled van der Pol Mathieu system MADHURJYA P BORA and DIPAK SARMAH Physics Department, Gauhati University, Guwahati , India Corresponding author. mpbora@gauhati.ac.in MS received 4 July 2012; revised 26 June 2013; accepted 4 July 2013 DOI: /s ; epublication: 26 September 2013 Abstract. We report an investigation of the oscillation death (OD) of a parametrically excited coupled van der Pol Mathieu (vdpm) system. The system can be considered as a pair of harmonically forced van der Pol oscillators under a double-well potential. The two oscillators are coupled with a cubic nonlinearity. We have shown that the system arrives at an OD regime when coupling strength crosses a threshold value at which the system undergoes saddle-node bifurcation and two limit cycles coalesce onto a fixed point of the system. We have further shown that this nonautonomous system possesses a centre manifold corresponding to the OD regime. Keywords. Nonautonomous oscillator; coupled oscillator; oscillation death; centre manifold. PACS Nos Bj; a 1. Introduction Coupled nonlinear systems have been a subject of extensive theoretical and experimental research [1 3]. Coupled systems, naturally, offer a rich variety of dynamical phenomena as their behaviours depend on a wide range of parameter regime. As natural dynamical systems are seldom isolated, coupled dynamical systems can mimic a variety of natural phenomena in physical, chemical, biological, and other systems including social sciences. Out of the various dynamical behaviours of coupled nonlinear system, the phenomenon of oscillation death (OD) or amplitude death [4 6], has received wide attention in the past few decades. In general, oscillation death or amplitude death refers to a state of two or more oscillators, when a multidimensional fixed point of the system becomes stable and the variables of the system become independent of time, pushing the system to a state when the variables cease to oscillate. It can also be termed as a special case of synchronization, when the system synchronizes to a steady state [7]. In the simplest case, OD usually occurs in a pair of mismatched oscillators with largely separated natural frequencies, when the coupling strength becomes large [8]. However, it can also occur in the case of two identical oscillators, when coupling between them Pramana J. Phys., Vol. 81, No. 4, October

2 Madhurjya P Bora and Dipak Sarmah is delayed [9]. Further studies revealed that OD can arise in the case of two identical oscillators, even in the absence of time delay, when the coupling is through dissimilar variables [10]. Recently, there are reports about driving the system to an OD regime, by designing a proper nonlinear coupling function [11]. One can also visualize the OD in a scene, where the oscillators are mediated by an indirect coupling through an environment or an external system [12,13]. Some recent studies [14,15] investigated the occurrence of phase-flip bifurcation and occurrence of OD in timedelay systems. Oscillation death also occurs in higher-dimensional system [16,17]. For a comprehensive analysis of the phenomenon of amplitude death in coupled nonlinear systems and its applications in different experiments, please see a recent review by Saxena et al [18]. Much of these studies involving OD phenomena of coupled oscillators, however, are devoted to autonomous or self-oscillatory systems [5,6]. It may also appear that there cannot be a similar phenomenon in a nonautonomous or a driven system, which is being parametrically modulated [19]. Nonautonomous systems are also closer to natural dynamical systems, as in such systems, different parameters are strongly dependent on time. In a parametrically modulated nonautonomous system, the OD can occur, even when the coupling strength is weak. This happens as there exist two additional parameters, the amplitude and frequency of modulation, apart from the other controllable parameter, the coupling strength [11]. In this paper, we report the OD phenomena in a system of two coupled van der Pol Mathieu (vdpm) oscillators [20]. This vdpm system is inspired by the low-dimensional model of a dusty plasma system [21]. In a dusty plasma system, the charge on a dust particle, in general, is dynamic due to the fluctuating dust potential [22]. This gives rise to parametric modulation in the system. We show that, under suitable circumstances, this system can be driven to a state of mutual death or OD owing to a saddle-node bifurcation of its limit cycles. We have shown that the system possesses a centre manifold, corresponding to the OD state. In 2, we explore the parameter space for the existence of the OD regime of the coupled system. We study the bifurcation and stability of the system in 3. In 4, we have derived the centre manifold of the coupled vdpm system. We conclude the findings of our study in The coupled vdpm system and parameter regime of oscillation death We consider the following coupled vdpm system: ẍ + (α + βx 2 )ẋ σ x(1 μ cos ωt) + κx 3 + δxy 2 = 0 ÿ + (α + βy 2 )ẏ σ y + κy 3 + δx 2, (1) y = 0 where α, σ, κ, and μ are positive constants, β is a constant which can either be positive or negative, and δ is the coupling strength. The constant μ determines the strength of the parametric forcing or amplitude of modulation. The nonlinear term (α + βx 2 )ẋ is like a van der Pol term and the parametric modulation term is like the Mathieu term. The 678 Pramana J. Phys., Vol. 81, No. 4, October 2013

3 Oscillation death in a coupled van der Pol Mathieu system coupled system, without parametric forcing (μ = 0) can be considered as a system under the an-harmonic double-well potential, ẍ + (α + βx 2 dv (x) )ẋ = dx, (2) for x = (x, y) and the potential V (x) = σ 2 x2 + κ 4 x4 + δ 2 x 2 y 2. (3) In figure 1, we have plotted this double-well potential. To give a mechanical analogy of eq. 2), we write the equation for a single oscillator (without the coupling term) as, d 2 x + f (x)dx 2 + F r(x) = F p (x, t), (4) which describes the motion of a particle of unit mass under a restoring force F r (x) = κx 3 σ x and a parametrically modulated force F p (x, t) x cos(ωt). The function f (x) represents nonlinear damping in the system. The above equation actually is a modified van der Pol equation with positive damping if f (x) >0, so that in the absence of the restoring force, oscillations always damp out. However, the restoring force can be compared to a spring which can also pump energy to the system when F r (x) <0, which happens at low x. So, we can conclude that at large x, oscillations always damp out and energy is taken out of the system and as x reaches a threshold, energy is pumped back into the system through the nonlinear restoring force. This can be compared to a dusty plasma system, where the dust number density is continuously being depleted (through the damping term f (x)) and periodically restored externally (through the nonlinear restoring term). However, we note that in actual systems, these terms may be quite different, but the essential dynamical behaviour should be reflected in our model. Figure 1. The double well potential V (x, y). Pramana J. Phys., Vol. 81, No. 4, October

4 Madhurjya P Bora and Dipak Sarmah In order to get an idea of the parameter regime of oscillation death for this coupled system, we write down the coupled system, eq. (1), as four first-order differential equations: dx 1 dx 2 dy 1 = x 2, (5) = (α + βx 2 1 )x 2 + σ x 1 (1 μ cos ωt) κx 3 1 δx 1y 2 1, (6) = y 2, (7) dy 2 = (α + βy1 2 )y 2 + σ y 1 κy1 3 δy 1x1 2. (8) The equivalent autonomous system (μ = 0) has nine equilibrium or fixed points A = (0, 0, 0, 0), B 1 = (± σ/κ,0, 0, 0), B 2 = (0, 0, ± σ/κ,0), and C = (± σ/(κ + δ), 0, ± σ/(κ + δ), 0). So the system is likely to arrive at a death regime, i.e. either a periodic or chaotic orbit being attracted by a fixed point, only when both σ and κ are positive. We note that both the self-excited (autonomous) and the parametrically modulated (nonautonomous) coupled systems can have multiple attractors which can be periodic, chaotic, or fixed points. Pisarchick and Goswami [23] have demonstrated that two or more attractors in a coupled system can be annihilated through a parametric modulation and in general they can annihilate to a death for a sufficiently large coupling strength [11]. Experimentally, such an annihilation can be found in cavity-loss driven CO 2 laser [23]. 2.1 Fixed points of the coupled vdpm system We analyse the behaviour of the fixed points of the coupled vdpm system for positive values of α, β, σ, μ, ω, κ, and δ. By looking at the eigenvalues of the linearized Jacobian of eqs (5) (8), we can make the following comments. Fixed point A, the origin, is a saddle point with 2D unstable and 2D stable nodes. The behaviour of the fixed points B 1,2 depends on the value of the coupling strength δ, ) δ>κ (1 + α2 2D stable spiral, (9) 4σ ) κ (1 + α2 >δ>κ 2D stable node, (10) 4σ κ>δ 1D stable and 1D unstable manifolds, (11) δ = κ 2D stable node on a line of zero. (12) Besides, there is a 2D stable node or spiral depending on whether the quantity (κα + σβ) 2 8κ 2 σ is positive or negative. The fixed point C becomes a line of zero when δ = κ, otherwise it becomes a stable spiral or node. So, OD should critically depend on the coupling strength δ as it approaches the parameter κ. 680 Pramana J. Phys., Vol. 81, No. 4, October 2013

5 Oscillation death in a coupled van der Pol Mathieu system Figure 2. Oscillation death of the coupled vdpm system. A numerical integration of eqs (5) (8) confirms this. In figure 2, we show the death regime of the coupled vdpm system, for the parameters α = 0.4,β = 0.1,σ = 0.25,μ= 0.8, f = 0.2,κ = 0.5, where ω = 2π f, and for different δ, where the system reaches the oscillation death regime at the fixed point B Bifurcation and stability The bifurcation diagram for the coupled system, eqs (5) (8), is shown in figure 3a. These curves are calculated using AUTO [24] (as a part of the XPPAUT [25] software package). As can be seen from the figure, two periodic branches (denoted by a +) annihilate at the limit point (LP), δ LP 0.42 as the coupling strength δ δ LP, from the given initial value of 1.5, marking the onset of the OD regime. As δ increases further and approaches κ, the system arrives at a crisis point, as the corresponding linear system has a line of zeros at δ = κ. Beyond this point, the regime cannot actually be called an OD regime, as the second oscillator [eqs (7) and (8)] arrives at the equilibrium point B 2 and continues to remain so whereas the first oscillator [eqs (5) and (6)] undergoes a periodic oscillation due to its parametric forcing, independent of the second one owing to the fact that the coupling strength is now essentially zero, signifying two decoupled oscillators. On the other hand, if δ is decreased beyond 0.07, the system undergoes a period-doubling (PD) bifurcation, finally making a transition to chaos [20]. We can get some information about the type of bifurcation at δ = δ LP by looking at the Floquet multipliers λ i [26,27]. In figure 3b, we have shown the evolution of the corresponding Floquet multipliers, Re(λ i ) with δ, fori = 1 4. The vertical line in the figure denotes the location of δ LP. As we can see, the Floquet multipliers approach the Pramana J. Phys., Vol. 81, No. 4, October

6 Madhurjya P Bora and Dipak Sarmah (a) (b) Figure 3. (a) Bifurcation curves for eqs (5) (8) with respect to the coupling strength δ. The strength of the parametric forcing μ is set to 1.5. Other parameters are the same as before. (b) Floquet multipliers (λ i ) of the periodic orbits of eqs (5) (8), as a function of δ, corresponding to (a). boundary of the unit circle from inside, indicating gradual loss of stability as δ δ LP and becoming exactly equal to unity at δ = δ LP. This implies that in the corresponding (x 1 x 2 ) and (y 1 y 2 ) phase-spaces, two limit cycles (isolated periodic orbits) coalesce onto a fixed point, which can be termed as saddle-node bifurcation of the periodic orbits [28,29]. In figure 4, we show the loss of stability of the limit cycles in the (x 1 x 2 ) phasespace as δ κ. In the top panel, the thick blue (left) and red (right) circles represent two limit cycles of the systems. In the bottom panel, the annihilation of the limit cycles to the origin is shown, as the bifurcation parameter δ is increased, signifying oscillation death. 682 Pramana J. Phys., Vol. 81, No. 4, October 2013

7 Oscillation death in a coupled van der Pol Mathieu system (a) (b) Figure 4. Saddle-node bifurcation of limit cycles of the coupled vdpm system in the x 1 x 2 phase space. (a) The periodic flow of the system. Two thick blue and red lines indicate the limit cycles of the system. The values of the parameters μ = 1.5 and δ = 0.1. (b) The annihilation of the limit cycles for δ = 0.5 onto the fixed point B 2. We however note here that unlike a pair of conjugate Landau Stuart oscillators, which typically shows a saddle-node bifurcation route to the OD regime, the chaotic regime in our coupled vdpm oscillators is well separated from the OD regime, which in the former case, lies very near to the OD [14]. It is interesting to examine the existence of the OD regime in the parameter space of (δ μ) as we can assume that the onset of the OD regime should be quicker (as δ κ), for increasing strength of the parametric forcing (μ). In order to establish this, we calculate the maximal Lyapunov exponent (m L ) for eqs (5) (8). A near-zero value of m L within the periodic region of the coupled vdpm system would signify a saddle-node bifurcation of the limit cycles and arrival of the system at the fixed point B 2 indicating OD. In figure 5, we have shown the spectrum of the maximal Lyapunov exponent in the δ μ space and the solid line in the δ μ plane indicates the onset of the OD regime. We can see that as μ increases, the onset of the OD regime occurs for smaller and smaller δ. Pramana J. Phys., Vol. 81, No. 4, October

8 Madhurjya P Bora and Dipak Sarmah Figure 5. The maximal Lyapunov spectrum in the δ μ phase space. The blue contour lines at the bottom of the figure indicate the merging line along which the maximal Lyapunov exponent m L 0 showing the line on which saddle-node bifurcation occurs. 3.1 Effect of the van der Pol term: Hysteresis For the van der Pol term (α + βx 2 )ẋ to have any effect on OD, the magnitude of the term must be large. The van der Pol term, being a purely nonlinear term involving only one oscillator, does not have any effect on the fixed points and apparently should not have any significant effect on the OD regime. However, it would be interesting to see its effect on OD regime, when the parameter β 1. In figure 6a, we have shown the bifurcation diagram of eqs (5) (8) with respect to β. Thered + represents the bifurcation of the system as β (left axis) is decreased from a large value, while the green represents the corresponding Floquet multiplier λ (right axis). The coupling strength δ is set to 0.48, which lies within the OD regime. When the initial β>β LP and approaches β LP from the positive side, the oscillation remains almost periodic (note the corresponding λs, which are 1). At β β LP 9.617, the system undergoes a bifurcation when the OD regime actually sets in. An interesting phenomenon seen here is that once the OD regime sets in, the system undergoes a hysteresis and does not come back to the periodic regime, even if β is increased beyond β LP. This is clearly reflected in the corresponding values of the Floquet multipliers λ, which now have crossed the unit circle and continue to remain so, showing that the system cannot recover from the OD regime. As we know, hysteresis can occur in bistable or multistable systems [30], where the system jumps to different stable branches with the variation of certain control parameter. In this case, we can consider the OD regime to be another stable branch of the system which causes the hysteresis. However, we note that this is different from the hysteresis in the usual sense that in this case, the system does not come back to a periodic state once it reaches the OD regime. In figure 6b, we have shown the effect of β on periodic and OD regimes. 4. Centre manifold approximation of the OD In this section, we show that the coupled vdpm system represented by eq. (1) possesses a centre manifold [27] corresponding to the OD regime. In order to facilitate the standard 684 Pramana J. Phys., Vol. 81, No. 4, October 2013

9 Oscillation death in a coupled van der Pol Mathieu system (a) (b) Figure 6. (a) The bifurcation curves for eqs (5) (8) with respect to β. Asβ decreases from a large value, we see that there is a bifurcation occurring at β LP 9.617, when the OD sets in. While the red + shows the bifurcation point (left axis), the green (right axis) shows the corresponding Floquet multiplier. (b) Effect of β on OD. As β becomes less than β LP, the system enters the OD regime. Other parameters are the same as before. Centre Manifold Theorem [27], we write the nonautonomous term of eqs (5) (8) asa solution of a pair of autonomous differential equation, by introducing two functions u(t) and v(t), u(t) = ωv(t), v(t) = ωu(t). (13) Pramana J. Phys., Vol. 81, No. 4, October

10 Madhurjya P Bora and Dipak Sarmah We further make a scale transformation, y 1 y 1 + σ/κ, δ ρ + κ. (14) The resultant dynamical system is a set of six autonomous differential equations, which can be written as dx 1 dx 2 = x 2, (15) = (α + βx 2 1 )x 2 + σ x 1 (1 μu) κx 3 1 (ρ + κ)x 1 (y 1 + σ/κ) 2, (16) dy 1 = y 2, (17) dy 2 = [α + β(y 1 + σ/κ) 2 ]y 2 + σ(y 1 + σ/κ) κ(y 1 + σ/κ) 3 (ρ + κ)(y 1 + σ/κ)x1 2, (18) du = ωv, (19) dv = ωu, (20) in the parameter space of x = (x 1, x 2, y 1, y 2, u,v) T. The resultant system, eqs (15) (20), has a linearized Jacobian with the eigenvalues ξ = (ξ i ) = (0, iω, iω, α, ξ 5,ξ 6 ) for ρ = 0, corresponding to the fixed point (0, 0, 0, 0, 0, 0) at the origin of the scaled system, where ξ 5,6 = (κα + σβ) (κα + σβ) 2 8σκ 2. (21) 2κ Note that this fixed point of the transformed system eqs (15) (20) corresponds to the fixed point B 2 of the original system for OD. So, eqs (15) (20) have a 3D centre eigenspace E c and 3D stable eigenspace E s, corresponding to the first three and last three eigenvalues, as Re(ξ 4,5,6 )<0for the chosen parameter space. To apply the Centre Manifold Theorem, we need to transform eqs (15) (20) tothe Jordan canonical form. We do so, first by forming a matrix A spanned by the eigenvectors of the linearized Jacobian of eqs (15) (20) [27], /α A = ξ 6 /(2σ) ξ 5 /(2σ) , (22) 0 i i and define a transformation, x = Au, u = A 1 x, (23) 686 Pramana J. Phys., Vol. 81, No. 4, October 2013

11 Oscillation death in a coupled van der Pol Mathieu system where the vectors u = (u i ) T, i = 1 6. After some obvious algebra (these calculations are carried out with the help of the Mathematica algebra system [31]), we can finally write the transformed dynamical system as u A = J c u A + f(u A, u A ), (24) u B = J s u B + g(u B, u B ), (25) where the vector u = (u A, u B ) T C n with J c C m, m < n, an(m m) matrix in the Jordan canonical form having eigenvalues with zero real parts and J s C n m, another (n m) (n m) Jordan matrix having eigenvalues with negative real parts, representing respectively the centre and the stable eigenspaces. In our case, n = 6 and m = 3 and J c = diag(0, iω,iω), J s = diag( α, ξ 5,ξ 6 ), (26) and u A = (u 1, u 2, u 3 ) T, u B = (u 4, u 5, u 6 ) T. The nonlinear functions f and g are given by f 1 = 1 α u 14[ βu 4 u 14 κu 2 14 κu σ u 14 {1 + iμ(u 2 u 3 )}], (27) f 2,3 = 0, (28) g 1 = α f 1, (29) g 2 = p ξ 5,6 (q + u 6,5 ), (30) where f(u A, u B ) = ( f 1,2,3 ) T, g(u A, u B ) = (g 1, g 2, g 2 ) T, and u 14 = (u 1 u 4 /α) (31) u 56 = 1 2σ (ξ 5u 6 + ξ 6 u 5 ) + σ/κ (32) p = 2κσ (κα + σβ)2 8σκ 2, (33) q = p [ u56 (σ κu σ κu2 56 ) (u 5 + u 6 )(α + βu 2 56 )]. (34) The dynamical system in its canonical form, eqs (24) and (25), has a 3D (u 1 u 2 u 3 ) centre eigenspace. So, by the Centre Manifold Theorem [27], the centre manifold of the system is tangent to the (u 1 u 2 u 3 ) hyperplane. Following standard procedure [27], we expand u 4,5,6 in a Taylor series in terms of u 1,2,3, u k = a (k) 11 u2 1 + a(k) 12 u 1u 2 + a (k) 13 u 1u 3 + a (k) 22 u2 2 + a(k) 23 u 2u 3 + a (k) 33 u2 3, k = 4, 5, 6, (35) where a (k) ij are the constants to be determined. In order to simplify the analysis, we have retained the expansion only up to the quadratic terms, though the method could be extended up to any order [27]. We now use these expansions in eq. (25) and equating Pramana J. Phys., Vol. 81, No. 4, October

12 Madhurjya P Bora and Dipak Sarmah (a) (b) Figure 7. (a) The flows on the centre manifold for the variable x 1 (t) of the coupled vdpm system. The parameters are the same as before. (b) The flows on the centre manifold for the variable y 1 (t) of the coupled vdpm system. the coefficients of quadratic powers of u 1,2,3, we would have 18 equations for the 18 coefficients a (k) jk, which determine the centre manifold as σ u 4 = (ω 2 + α 2 ) u 1[iμ(u 2 u 3 ) ω(u 2 + u 3 )], (36) [ u 5,6 = u 2 σκ 3 ] 1/2 1. (37) (κα + σβ) 2 8σκ Pramana J. Phys., Vol. 81, No. 4, October 2013

13 Oscillation death in a coupled van der Pol Mathieu system Inserting these expressions into the differential equation, eq. (24), of the centre eigenspace would finally yield the flow on the centre manifold of the transformed system, 4σα 4 T 3 1 (αt 1 + 2σμT 2 ) 1 du 1 u 1 = 8μ(σαT 1 ) 2 sin(ωt) 8μσ 2 T 2 [2κ(αT 1 + σμt 2 ) αβ(αt 1 + 2σμT 2 )u 2 1 (ακt 1) 2 u 4 1 ], (38) with T 1 = (ω 2 + α 2 ), T 2 = ω cos(ωt) α sin(ωt). We need to numerically solve eq. (38) for u 1, which can then be used to obtain the centre manifold flow of the original system from the transformation eq. (23). The flows on the centre manifold for the variables x 1 and y 1 are shown in figure 7, which indicate that the coupled vdpm system possesses a centre manifold corresponding to the OD regime at the fixed point B 2 when δ>δ LP. 5. Conclusion In this work, we have studied the dynamics of a parametrically modulated coupled system. Specifically, we have investigated the region of amplitude death of this coupled system. It is important to note that these oscillators are identical, even though they exhibit welldefined OD regime. We have shown that when the coupling parameter exceeds a certain threshold, the oscillators pull each other to a steady state. The van der Pol term used in these oscillators is responsible for a hysteresis leading to the OD regime. We have used the Centre Manifold Theorem to show that this coupled system possesses an invariant centre manifold corresponding to the OD region. References [1] A T Winfree, J. Theor. Biol. 16, 15 (1967) [2] M G Rosenblum and A S Pikovsky, Phys. Rev. Lett. 92, (2004) [3] S H Strogatz, Physica D 143, 1 (2000) [4] G B Ermentrout and N Kopell, SIAM J. Appl. Math. 50, 125 (1990) [5] K Konishi, Phys. Rev. E68, (2003) [6] A Prasad, M Dhamala, N M Adhikari and R Ramaswamy, Phys. Rev. E81, (2010) [7] Y Kuramoto, Chemical oscillations, waves, and turbulence (Dover, 2003) [8] K Bar-Eli, Physica D 14, 242 (1985) [9] D V Ramana Reddy, A Sen and G L Johnston, Phys. Rev. Lett. 80, 5109 (1998) [10] R Karnatak, R Ramaswamy and A Prasad, Phys. Rev.E76, (R) (2007) [11] A N Pisarchik, Phys. Lett. A 318, 65 (2003) [12] V Resmi, G Ambika and R E Amritkar, Phys. Rev. E84, (2011) [13] V Resmi, G Ambika, R E Amritkar and G Rangarajan, Phys. Rev.E85, (2012) [14] R Karnatak, N Punetha, A Prasad and R Ramaswamy, Phys. Rev.E82, (2010) [15] A Prasad, S K Dana, R Karnatak, J Kurths and B Blasius et al, Chaos 18, (2008) [16] M Stich, A C Casal and J I Diaz, Phys. Rev. E76, (2007) [17] H Sakaguchi and D Tanaka, Phys. Rev.E76, (2007) [18] G Saxena, A Prasad and R Ramaswamy, Phys. Rep. 521, 205 (2012) and references therein. [19] A Pikovsky, M Rosenblum and J Kurths, Synchronization: A universal concept in nonlinear science (Cambridge, New York, 2001) Pramana J. Phys., Vol. 81, No. 4, October

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