MEASURE SYNCHRONIZATION IN A COUPLED HAMILTONIAN

August 11, 2005 13:33 WSPC/147-MPLB

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Modern Physics Letters B, Vol. 19, No. 15 (2005) 737–742 c World Scientific Publishing Company

MEASURE SYNCHRONIZATION IN A COUPLED HAMILTONIAN SYSTEM ASSOCIATED WITH ¨ NONLINEAR SCHRODINGER EQUATION

U. E. VINCENT∗,†,‡ , A. N. NJAH∗ and O. AKINLADE∗ ∗ Department

of Physics, University of Agriculture, P.M.B. 2240, Abeokuta, Nigeria † Department of Physics and Solar Energy, Bowen University, Iwo, Nigeria ‡ rebuche [email protected] Received 15 January 2005 We present preliminary numerical findings concerning measure synchronization in a pair of coupled Nonlinear Hamiltonian Systems (NLHS) derived from a Nonlinear Schr¨ odinger Equation (NLSE). The dynamics of the two coupled NLHS were found to exhibit a transition to coherent invariant measure; their orbits sharing the same phase space as the coupling strength is increased. Transitions from quasiperiodicity (QP) measure desynchronization to QP measure synchronization and QP measure desynchronization to chaotic (CH) measure synchronization were observed. Keywords: Measure synchronization; Schr¨ odinger equation; Hamiltonian systems. PACS Number(s): 05.45.Pq, 05.45.Xt, 05.45.-a

1. Introduction Synchronization is a basic phenomena in physics discovered at the beginning of the modern age of science by Huygens.1 Classically, synchronization implies the frequency and phase locking of periodic oscillators due to weak interaction — this formed the main focus of early studies. In the recent years, due to considerable enlargement in the field, chaos synchronization has attracted much attention and has been extensively studied.1 – 17 One of the major important motivations is to understand the coherent dynamical behavior of coupled systems. In this light, various types of synchronization has been identified. These include complete synchronization (CS),2,3 phase synchronization (PS),4,5 lag synchronization (LS),6,7 generalized synchronization (GS)8,9 and anticipated synchronization (AS).10,11 Most work on synchronization to date have focused mainly on dissipative systems (DS), the reason is that synchronization between two trajectories is normally related to the contraction of phase space volume. Recently, it has been reported that ‡ Corresponding

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coupled Hamiltonian systems can exhibit a kind of weak synchronization and this has attracted the attention of some researchers.15– 17 Hamiltonian systems (HS) behave quite differently from dissipative systems (DS). Unlike DS, HS conserve phase volumes12 and do not (in the original sense) allow synchronization, i.e. exhibit a situation in which two nonidentical trajectories approach an identical one asymptotically. Hamiltonian systems are very significant because they serve as typical models in classical and quantum mechanics; and numerous practical systems can be well approximated by Hamiltonian formalism even with weak dissipation.13,14 For this reason, exploring the synchronization phenomenon in HS is a very crucial step towards understanding the significance of the concept, mode and possible application of synchronization in quantum systems. In Ref. 15, Hampton and Zanette observed certain collective behavior between two mutually coupled identical HS. The coupled systems were found to exhibit a transition in phase space from a “nonsynchronous” state to a “synchronous” state, the synchronous state being referred to as measure synchronization (MS). The main characteristic of MS is that two orbits share the same phase space with the same identical invariant measure, though the two systems are not strictly synchronized. In a recent paper, Xingang et al.16 reported the existence of partial measure synchronization in a system of coupled φ4 Hamiltonian systems. In a more recent study, Xingang and Zhang17 considered the transition phenomenon to MS both in the quasiperiodic and chaotic cases. The study and analysis of collective behavior in Hamiltonian systems has received relatively less attention. In this present letter, we report the existence of measure synchronization in a classical Nonlinear Schr¨ odinger Hamiltonian systems. The paper is organized as follows. In Sec. 2, we describe our Nonlinear Schr¨ odinger (NLS) model. In Sec. 3, measure synchronization in NLS Hamiltonian is discussed. Conclusions and extension of the present study are finally given in Sec. 4. 2. The Model The Nonlinear Schr¨ odinger Equation (NLSE) in its various versions is one of the most important models of mathematical physics, with applications to different fields such as plasma physics,18 nonlinear optics, water waves and biomolecular dynamics,19 to cite only a few cases. In this work we study Hamiltonian systems governed by the NLSE given by 2 2 ∂ψ 2 = − ∇ + |ψ| ψ (1) i ∂t 2M where ψ is the eigenstate of a particle of mass, M, and = 1. Recently, Njah and Akin-Ojo20 obtained a nonlinear hamiltonian for the NLSE (Eq. (1)) with two degrees of freedom using a Fourier analysis in a bounded interval. Our hamiltonian model is a modified version of the nonlinear hamiltonian defined in Ref. 20, i.e. two


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linearly coupled identical systems with the hamiltonian H=

1 1 [s1 (p21 + q12 ) + s2 (p22 + q22 )] − [p21 + q12 + p22 + q22 + 2(p1 p2 + q1 q2 )]2 4M 2 + K(q1 − q2 )2 ,

(2)

where pi and qi (i = 1, 2) represent positions and momenta for any particle, i, of mass, M. K is the coupling strength. Numerically, we simulate the corresponding canonical equations: si q1 + 2[p21 + q12 + p22 + q22 + 2(p1 p2 + q1 q2 )](q1 + q2 ) + 2K(q2 − q1 ) p˙1 = − 2M s2 q2 + 2[p21 + q12 + p22 + q22 + 2(p1 p2 + q1 q2 )](q1 + q2 ) + 2K(q1 − q2 ) p˙2 = − 2M s1 q˙1 = p1 + 2[p21 + q12 + p22 + q22 + 2(p1 p2 + q1 q2 )](p1 + p2 ) (3) 2M s2 p2 + 2[p21 + q12 + p22 + q22 + 2(p1 p2 + q1 q2 )](p1 + p2 ) . q˙2 = 2M si in the first term on the right stands for the total energy, E of the system. The 2M In our computation, both K and E were used as control parameters to achieve measure synchronization. The initial conditions for pi and qi were fixed. Our choice of control parameters is evident from a simple analysis in Ref. 20 for which variety of periodic and chaotic solutions were observed as the energy, E was varied. We note that in Ref. 7 the initial conditions and the coupling constant were used as control parameters following the work of Hampton and Zanette,15 while the energy was fixed.

3. Measure Synchronization We have studied Measure Synchronization (MS) in the coupled Nonlinear Hamiltonian System (NLHS) given by Eq. (2) using the total energy, E of the system as well as the coupling strength, K as the control parameters from (K, E) = (0, 0) up to (K, E) = (1, 10). With these parameter variations varieties of dynamical behaviors (transition) were found, including MS. As stated in the introduction, our main objective is in the existence of MS. Thus for brevity we report only on this phenomenon both for the quasiperiodic and the chaotic cases. We begin by considering a case where the coupled NLHS is in the zone of regular nonchaotic evolution. In this direction, we study the low energy region, which quarantees regular solutions. Thus, we present in Fig. 1 the result for energy, E = 1.5, and initial conditions (q1 (0), q2 (0)) = (0, 0) and (p1 (0), p2 (0)) = (0.04, 0.06). In the absence of coupling, K = 0, these conditions correspond to two different periodic orbits (PO), which cover closed curves in the (qi , pi ) phase plane as shown in Fig. 1(a). When a small nonzero coupling is switched on (K = 0.0025), the two periodic orbits are replaced by two smooth quasiperiodic (QP) orbits as shown in


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Fig. 1. (a) The periodic orbits of the two oscillators defined by Eq. (3) in the (q, p) plane. E = 0; K = 0. No measure synchronization exsit between the two oscillators. (b) As in (a), but for a small nonzero coupling, K = 0.0025. The motion becomes quasiperiodic. No MS exists between them. (c) and (d) same as (b) with coupling increased to k = 0.01 beyond a critical value (Kc = 0.008). The two trajectories share the same phase space and MS is reached.

Fig. 1(b). These orbits wander in two distinct tori as a result of the increase in the effective dimension of the whole Hamiltonian system.7 By further increasing K (Figs. 1(c) and 1(d)), the two distinct tori of Fig. 1(b) widens in such a way that the external border of the inner tori approaches the internal border of the outer tori and vice versa. Hence, the two regions merge into a single and enlarged identical tori, sharing the same phase domain with their quasiperiodic orbits, signalling the phenomenon of measure synchronization. MS in the QP zone is maintained for higher value of K and was also observed within range E = (0.1, 3). Using the same initial conditions for qi and pi , we plot the trajectories for E = 5.4. In this case the result are quite different from those of Fig. 1 and there were no periodic orbits for all values of K studied. For K = 0, the orbits are quasiperiodic with the outer part of the oscillator 1 tending to approach the inner part of oscillator 2 (Fig. 2(a)). This behavior is maintained up to K < 0.0025 after which the boundaries of the two orbits collides at K ≈ 0.00447; and the orbits begins to merge as shown in Fig. 2(b) for K = 0.005. With further increase in the coupling strength, the coupled system undergoes several transformations in phase


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Fig. 2. (a) The same as in Fig. 1(a), but with E = 5.4. The motion is quasiperiodic. (b) Same as in Fig. 1(b) but for E = 5.4 and K = 0.005. oscillator 1 orbit begins to merge with oscillator 2 orbit. No MS is achieved. (c) and (d) same as in Figs. 1(c) and 1(d) but for E = 5.4 and K = 0.076. Measure synchronization is achieved. The motion is chaotic in both orbits. (e) and (f) same as in Figs. 2(c) and 2(d) but for K = 1.0. The region of chaos is increased as K increases further beyond the critical value Kc = 0.07.

space and for a critical value of K about Kc = 0.07, a change from QP orbits to chaotic (CH) orbits is observed with the two orbits sharing an identical invariant measure (Figs. 2(c) and 2(d) for K = 0.076). Just above Kc , the chaotic regions in the phase space increases with much stronger synchronism as shown in Figs. 2(e) and 2(f).


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4. Conclusion In summary, we have shown the existence of measure synchronization in a Hamiltonian system which is related to the Nonlinear Schr¨ odinger Equation. Our numerical findings reveals that transition from quasiperiodicity (QP) measure desynchronization to QP measure synchronization and QP measure desynchronization to chaotic (CH) measure synchronization are the generic features characterizing the relation among coupled NLHS subsystems. The characterization of measure synchronization in this coupled NLHS with higher degree of freedom and the study of the transition mechanisn to MS is the subject of our future work which is in progress. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

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