Copyright c 29 by ABCM PHASE-LOCKED SOLUTIONS IN A HUB CONNECTED OSCILLATOR RING NETWORK Jacqueline Bridge, Jacqueline.Bridge@sta.uwi.edu Department of Mechanical Engineering, The University of the West Indies, St. Augustine, Trinidad Abstract. In this work, we analyze a type of system called the hub connected oscillator ring (HCOR) network. The model consists of 3 sub-rings each coupled to a central hub oscillator with a bi-directional coupling constant, α. When uncoupled from the hub, each substructure is characterized by 3 identical oscillators with natural frequency ω i and bi-directional coupling constant, α i. The conditions necessary for existence of phase-locked solutions were derived and their associated stability criteria were determined. The synchronization tree for phase-locked solutions was also developed. A bifurcation analysis of the system was conducted. Finally simulations were carried out to validate the analytically derived results; these results agreed well with the theory. Keywords: Phase-only oscillators, nonlinear dynamics, bifurcations 1. INTRODUCTION Coupled limit cycle oscillators have been widely used to model both physical (laser arrays, Josephson junctions) and engineering systems (communications and electricity distribution networks). In particular, there has been much interest in the development of collective behavior of these systems, particularly phase-locking ( Bridge et.al., 29; Chopra and Spong, 25; D Huys et.al.,28; Jadbabaie et.al., 24; Jeanne et.al., 25; Kuramoto, 1974; Mirollo and Strogatz, 25; Rogge and Aeyels, 24; Winfree, 198). Several factors dictate the type of responses that arise in a given system of oscillators, the most significant being the nature of the individual oscillators and the topology of their connections. Researchers have noted that many of the key phenomena observed in these systems may be recovered by investigating a phase-only equivalent systems. The Kuramoto (phase-only) model has therefore been widely researched. The most common topologies examined have been the (a) all-to-all (or global network) where each oscillator is coupled to every other oscillator in the system, (b) nearest neighbor coupled rings and (c) chains of coupled oscillators. However, a typical engineering topology would be more likely to consist of several groups of interacting motifs (D Huys et.al.,28) with limited communication via a hub. This is particularly important in communications. To begin to understand the dynamics of such systems, we analyze a type of system called the hub connected oscillator ring (HCOR) network. We look at a model which consists of 3 sub-rings each coupled to a central hub oscillator with a bi-directional coupling constant, α. When uncoupled from the hub, each substructure is characterized by 3 identical oscillators with natural frequency ω i and bi-directional coupling constant, α i. In this work, it was shown that phase-locked solutions only exist when each intra-ring coupling coefficients (α i ) exceeds a ring-specific critical value. Using numericall integration, we show that there is a heirarchy of synchronization of phase-locked solutions. A bifurcation analysis of the system was conducted. Finally simulations were carried out to validate the analytically derived results; these results agreed well with the theory. 2. EXISTENCE OF PHASE-LOCKED SOLUTIONS Consider 3 coupled oscillator rings, each coupled to a central hub oscillator with a bidirectional coupling constant α, as shown in Fig. (1). When uncoupled from the other rings, each substructure consists of three identical oscillators with natural frequency (ω i ) and distinct bidirectional coupling constant (α i ). We consider the case of phase only oscillators which evolve in time according to the differential equation: θ i,j = ω i + α sin(θ θ i,j )δ j1 + α i (sin(θ i,j+1 θ i,j ) + sin(θ i,j 1 θ i,j )) (1) where i, j = 1, 2, 3. The hub oscillator has its own natural frequency of ω and its phase is denoted by θ, so θ = ω + α 3 sin(θ i,1 θ ) 1 where δ j1 = 1 if j = 1 and otherwise. If we define the phase differences by: φ i,j = (θ i,j+1 θ i,j ) mod 2π φ i,3 = (θ i,1 θ i,3 ) mod 2π and ψ i = (θ θ i,1 ) mod 2π then the equations may be rewritten as: θ i,j = ω i + α sin(ψ i )δ j1 + α i (sin(φ i,j ) sin(φ i,j 1 )) (3) (2)
Copyright c 29 by ABCM Figure 1. An example of a hub connected topology. Each dot is a particle and each line is a bi-directional coupling link. and θ = ω α 3 sin(ψ i ) 1 (4) 2.1 The Phase-locked frequency A network is said to be phase locked when the phase difference between any two oscillators is constant with time, i.e. we can define θ = Ωt θ i,j (t) = Ωt + k i,j (5) (6) where k i,j is the steady-state phase difference and Ω is the phase locked frequency. If we sum the governing equations (1) and (2) for the oscillators, we determine that the phase-locked frequency is the weighted average of the oscillator frequencies, i.e. Ω = ω + 3ω i 1 (7) 2.2 Critical Intra-Ring Coupling Coefficients, α i s Differentiating equations (5) and (6) and substituting into (3) and (4), yields the phase locking equation Ω = ω i + α sin(ψ i )δ j1 + α i (sin(φ i,j ) sin(φ i,j 1 )) (8) For the three element subring, we may rearrange equation (8) to give: Ω ω i = α i (sin(φ i,2 ) sin(φ i,1 )) (9) Ω ω i = α i (sin(φ i,3 ) sin(φ i,2 )) (1) But φ i,3 + φ i,2 + φ i,1 = mod 2π (11) so substituting this relationship in equation (9) and (1) and simplifying the trigonometric relationships we have, (α i s + i ) 2 ( 4 α 2 i s 4 + 16 α i i s 3 + 16 2 i s 2 3 α 2 i s 2 6 α i i s + 9 2 i ) = (12) where s = sin(φ i,1 ) and i = (Ω ω i )
Copyright c 29 by ABCM Figure 2. (a) Typical bifurcation diagram showing the emergence of "possible" phase-locked solutions in a sub-ring. (b) Synchronization tree as α varies for fixed parameters ω 1 = 2, ω 2 = 3, ω 3 = 4, ω = 1, α 1 = α 2 = 1, α 3 = 2 When α i =, equation (12) has no real roots for s, while as α i, the equation yields 6 real roots (2 simple roots at ± 3/2 and a quadruple root at ). Therefore, as α i is increased the number of real roots of the equation increases from to 6. There is therefore a minimum value of α i for which a real solution will exist. Note the number of roots changes when there is a double root. We differentiate equation (12) to obtain 4 (α i s + i ) ( 6 αi 3 s 4 + 24 αi 2 i s 3 + 28 α i 2 i s 2 3 αi 3 s 2 + 8 3 i s 6 αi 2 i s + 3 α i 2 ) i (13) The term (s + i α i ) = satisfies both (12) and (13). This has a real solution iff sin(φ i,j ) 1 i.e. Ω ωi α i 1, implying that a pair of critical values of α i occur at ±(Ω ω i ). To determine when the two other pairs of roots emerge, we eliminate s from (12) using the relationship in (13). The resulting equation is The equation has double roots, indicating that at two pairs of new roots emerge. 2.2.1 α i,cra = (Ω ω i ) α 6 i 2 i (4 i α i ) 2 (4 i + α i ) 2 ( 2 i + 3α 2 i ) 2 = α i,cr = ±4(Ω ω i ) For this case, s is given by sin(φ i,1 ) = (Ω ω i )/α i, substituting into equation (9), we obtain sin(φ i,2 ) = i.e. φ i,2 =, π and (14) sin(φ i,3 ) = (Ω ω i) α i = sin(φ i,1 ) (15) The solutions corresponding to φ i,2 = are equivalent to θ 1,2 = θ 1,3, i.e. the solution is spatially symmetric. For one of these symmetric solutions all the cos φ i,j terms were positive while the other has two negative cos φ i,j terms. The other solution φ i,2 = π and the consistency condition, implies that φ i,3 = π φ i,1. But for this relationship, sin(φ i,3 ) = sin(φ i,1 ) which negates equation(15); hence these solutions cannot exist. Thus on exceeding the lower critical coupling coefficient, only two symmetricphase-locked solution candidates appear. 2.2.2 α i,crb = 4(Ω ω i ) At the second critical value of α i, the resulting trajectory is not spatially symmetric. The corresponding bifurcation diagram for the onset of synchronization within a sub-ring is as shown in Figure 2(a). However, imposing the consistency equation results yields a combination of phase differences within the subring with at least one term φ i,j < π 2.
Copyright c 29 by ABCM 2.3 Critical Hub Coupling Coefficients, α The equation governing the asscoiated phase differrences of the j = 1 oscillators which are connected to the hub oscillator is Ω = ω i + α sin(ψ i ) + α i (sin(φ i,1 ) sin(φ i,3 )) (16) But from summing equations (1) and (11), we know that sin(φ i,1 ) sin(φ i,3 ) = 2(Ω ω i ), so the equation is transformed into: 3(Ω ω i ) = α sin(ψ i ) (17) These equations then give us the following value for the critical link coupling coefficient: α cr = max {3(Ω ω i )} for i = 1, 2, 3 (18) 3. STABILITY OF PHASE-LOCKED SOLUTIONS In order to study the stability of the phase-locked solutions, we transform the equations using θ i,j = θ i,j Ωt. The periodic orbits are thus transformed into equilibrium points. Linearizing about these points and substituting the corresponding phase locked conditions yields a 1 x 1 matrix, [B], whose eigenvalues will yield the stability of the original system. Note that α N j=1 cos ψ i α cos ψ 1 α cos ψ 2 α cos ψ 3 [B] = α cos ψ 1 α cos ψ 2 α cos ψ 3 [B 1 ] [O] [O] [O] [B 2 ] [O] [O] [O] [B 3 ] with [O] being the 3 x 3 null matrix and α i (cos φ i,1 + cos φ i,3 ) α i cos φ i,1 α i cos φ i,3 [Bi α ] = α i cos φ i,1 α i (cos φ i,2 + cos φ i,1 ) α i cos φ i,2 + α i cos φ i,3 α i cos φ i,2 α i (cos φ i,3 + cos φ i,2 ) α cos ψ i We describe an oscillator ring with phase difference vector, Φ i = {φ i,1, φ i,2, φ i,3 } as a potential sink, source or saddle, respectively if the eigenvalues of [Bi ] = [Bα= i ] evaluated at Φ are all negative semi-definite, all positive semi-definite or mixed, respectively; we describe the hub connections with phase difference vector, Ψ = {ψ 1, ψ 2, ψ 3 } similarly. Using graph theory, it can be shown that the complete system will be stable iff all the oscillator rings and the hub connections are potential sinks. Using Gershgorin s theorem, for α i >, the symmetric solution with φ i,1 < π 2 is a potential sink and the other symmetric solution is a potential saddle; conversely, when α i <, the symmetric solution with φ i,1 < π 2 is a potential source and the other symmetric solution is a potential saddle. Thus for positive α i, there is only one potential sink in each sub-ring, yielding one stable limit cycle. For the asymmetric solutions, the phase-locked solutions have at least one of φ i,j > π 2. Gershgorin s theorem implies that with α i >, these will correspond to either potential sources (all φ i,j > π 2 ) or potential saddles, while α i < yields either potential sinks (all φ i,j > π 2 ) or potential saddles. Numerical investigations show that there are two potential sources per subring for α i > and two potential sinks per subring for α i < yielding two stable limit cycles. 4. NUMERICAL RESULTS In this section we simulate the HCOR network with 3 sub-rings each characterised by its own natural frequency and containing 3 elements. We use the following values in the simulation ω 1 = 2, ω 2 = 3, ω 3 = 4, ω = 1 (units in rad/s). 4.1 Parameters associated with the onset of synchronization From the theory developed above, the critical parameters associated with the onset of synchronization are: Ω = 2.8, α 1,cr =.8, α 2,cr =.2, α 3,cr = 1.2, while α cr = 3.6
Copyright c 29 by ABCM Table 1. Possible Phase-Locked Combinations per component Hub OSC. Subring 1 Subring 2 Subring 3 { Symmetric solutions Symmetric solutions Symmetric solutions.6435 ψ 1 = (.9273,,.9273) (.214,,.214) (.6435,,.6435) 2.4981 (4.689,, 4.689) (2.942,, 2.942) (2.4981,, 2.4981) {.156 ψ 2 = Asymmetric solutions Asymmetric solutions 3.2922 (1.578, 2.2143, 2.4881) { ( 2.4881, 2.2143, 1.578) 1.1198 ψ 3 = (.6892, 2.696, 2.935) 4.2614 ( 2.935, 2.696,.6892) Asymmetric solutions Note that α 2,cr < α 1,cr < α 3,cr. The individual sub-ring coefficient couplings were chosen to be α 1 = 1, α 2 = 1 and α 3 = 2, respectively. These values are greater than the individual critical synchronization values for their respective oscillator rings. The hub coupling coefficient was chosen to vary with α (, 4) and simulations were performed. The corresponding synchronization tree is shown in Fig. 2(b). Before the first partial synchronization coupling coefficient α = α 1,ps =.6, we note that there is no synchronization between the different sub-rings. Although the phases within each sub-ring remain close to each other, the phases between different sub-rings diverge. As α is increased above α 1,ps = 3α 2,cr =.6, there is partial synchronization; the oscillators in sub-ring 2 and the hub oscillator converge to the same operating frequency, while the other two sub-rings operate at their own separate frequencies. Further increases in the hub coupling coefficient (to α = α 2,ps = 3α 1,cr = 2.4) result in partial synchronization of sub-rings 1 and 2 with the hub oscillator and finally (at α = α cr = 3α 3,cr = 3.6) full synchronization. At full synchronization, the angular frequencies of all oscillators converge on the mean frequency Ω = 2.8. 4.2 Analysis of Phase-locked solution Let us consider the system with α 1 = 1, α 2 = 1, α 3 = 2, α = 4. We will determine the phase-locked solutions and investigate the stability of each system. Based on the analysis of the previous section, Note that we know that at least one solution exists. Within a sub-ring, the symmetric phase-locked solution corresponds to sin(φ i,2 ) = ; sin(φ i,3 ) = sin(φ i,1 ) = Ω ω i ; sin(ψ i ) = 3(Ω ω i) α i α while the asymmetric solutions solve (12). Only complex conjugate pairs of roots exist for subrings 1 and 3 (no asymmetric phase-locked solutions exist), while subring 2 has real roots; consequently, the possible combinations of phase differences which correspond to the frequency and coupling coefficient values given above are given in Table 1. Any combination of an element from each column gives a phase-locked solution. These phase differences represent 252 (=2*2*2*6*2*2) possible limit cycle solutions for the given combination of frequencies and coupling coefficients. Since α i >, i = 1, 2, 3, there is only one stable trajectory ; the stable orbit can be seen from numerical integration (see Fig. 3). Note that the values obtained for the relative phases in the numerical integration correspond to the values determined analytically (mod 2π). 5. CONCLUDING REMARKS In this work, we analyzed a hub connected oscillator ring (HCOR) network consisting of 3 sub-rings each coupled to a central hub oscillator with a bi-directional coupling constant, α. We showed that if a phase-locked solution exists, its frequency is Ω =.1ω +.3 3 i=1 ω i. We determined that a pair of symmetric phase-locked solutions emerge within each subring when the coupling coefficient, α i exceeds Ω ω i. When α i > 4 Ω ω i, four additional asymmetric phase locked solutions emerge. For positive α i, we noted that one of the symmetric solutions was stable while the other was a saddle; for negative α i, the solutions were both unstable, one being a saddle, the other a source. The synchronization tree for phase-locked solutions was also developed numerically. Finally simulations were carried out to validate the analytically derived results; these results agreed well with the theory.
Copyright c 29 by ABCM Figure 3. Phase differences relative to the hub oscillator for arbitrary input. (a) Phase differences for the communication oscillators; (b), (c), (d) Phase differences in subrings 1, 2 and 3, respectively. Note that the system is symmetric since θ i,2 = θ i,3 6. REFERENCES Bridge, J., Rand R., and Sah, S. Dynamics of a ring network of phase-only oscillators, Communications in Nonlinear Science and Numerical Simulation 14 (29), 391-3913. Chopra, N. and Spong, M.W., On synchronization of Kuramoto oscillators, Proc. of 44 th IEEE Conf. Decision and Control and European Control Conference (25), 3916-3922 D Huys, O., Vicente, R., Erneux, T., Danckaert, J. and Fischer, I., Synchronization properties of network motifs: Influence of coupling delay and symmetry, Chaos (18), 37116 (28) Jadbabaie, A., Motee, N., Barahona, M. On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. of the ACC. (24), Jeanne, J., Leonard, N.E., Paley, D. Collective motion of ring-coupled planar particles, Proc. of 44 th IEEE Conf. Decision and Control and European Control Conference (25), 3929-3934 Kuramoto, Y., in International Symposium on Mathematical Problems in Theoretical Physics, ed. by H. Araki, Lecture Notes in Physics, vol. 39, (Springer, New York, 1975), 42. Mirollo, R.E., Strogatz, S.H. The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D 25 (25) 249-266 Rogge, J.A., and Aeyels, D., Stability of phase locking in a ring of unidirectionally coupled oscillators, J. Phys. A: Math. Gen. 37, 11135-11148 (24) Winfree, A.T., The Geometry of Biological Time, Springer, New York (198) 7. RESPONSIBILITY NOTICE The author is the only responsible for the printed material included in this paper