Synchronization in delaycoupled bipartite networks Ram Ramaswamy School of Physical Sciences Jawaharlal Nehru University, New Delhi February 20, 2015
Outline Ø Bipartite networks and delay-coupled phase oscillators Ø Globally synchronized solutions (GS) Ø Mismatch between the partitions Ø Time-varying frequencies and Remote Synchronization (RS) Ø Chimera states (CS) and Individual Synchronization (IS) Ø Summary
Networks Most natural systems are nonlinear and rarely isolated, functioning through a collective behavior of many interacting subsystems. These are networks, which have been extensively studied to describe wide variety of phenomena in physics, biology, chemistry, ecology, sociology etc. Examples -- networks Biological networks, social
Bipartite topology The bipartite topology describes a connection structure where the nodes of a network can be divided into two groups such that any two nodes that are coupled must belong to different groups. Different Immune cells on a bipartite network Biological systems (Immune system)
Bipartite Topology Different birds and plants relationship. Ecological systems (Birds and plants system)
File sharing protocols Computer network (Bit torrent) Bipartite network reduces search traffic cost and query response time.
Metabolic networks are bipartite
Other cases The bipartite network also covers some of the well studied topologies- Ring with nearest neighbor coupling and even number of oscillators. Star topology
Collective synchronization Ensembles of coupled oscillators typically show emergent properties, namely, the features that are not shown by the individual units. An example of such emergent behavior is collective synchronization which is observed in the systems ranging from simple mechanical oscillator to pacemaker heart cells, circadian rhythms, flashing-fireflies, clapping of the large audience, chemical, biological oscillators, laser arrays and Josephson junctions.
The Kuramoto Model The Kuramoto model describes a large population of coupled limit cycle oscillators, used to understand collective behavior of populations in several disciplines. The model equations are given by θ i (t) = ω i + K N N a ij sin(θ j (t) θ i (t)), i =1,...,N j =1 Here θ i is the phase of ith oscillator, K is the coupling strength. ω s are the natural frequencies drawn from a distribution g(ω). For simplicity, the distribution g(ω) is unimodal and symmetric around the mean frequency Ω. The factor (1/N) ensures that the model is well behaved as N à.
The Kuramoto order parameter The model is known to exhibit a spontaneous second order phase transition: with increasing coupling K, the system abruptly goes from the incoherent state to a synchronized one at the critical coupling strength K c. To visualize the dynamics of the phases, a complex order parameter is defined as: re iψ = 1 N N j =1 This value can be interpreted as the collective macroscopic rhythm produced by the whole population. r(t) measures the phase coherence and ψ(t) is the average phase. e iθ j
θ i = ω i + Krsin(ψ θ i ), i =1,...,N Phase transition Rewriting the system equation in terms of complex order parameter This indicates that the interactions among the oscillators are through the mean field quantities (ψ and r). The effective coupling strength (Kr) in the equation works as a positive feedback between coupling and coherence. The oscillators are pulled towards the mean phase ψ, the ones merging in the synchronized set are termed as phase locked while others are drifting oscillators. The competition between the two makes the population selflimiting.
Time Delay Coupling Time-delay is inherent in the couplings due to finite speed of signal propagation. Depending upon the time-scale of the system, the time delay coupling may modify the resulting dynamics of the system, including aspects of synchronization, AD, PF, multistability etc.
Time-delay coupling In order to investigate the effect of time delay on synchronization of nonlinear oscillators, Schuster and Wagner took two limit cycle oscillators coupled with time delay θ 1 = ω 1 + K sin(θ 2 (t τ) θ 1 (t)) θ 2 = ω 2 + K sin(θ 1 (t τ) θ 2 (t)) Here θ 1 and θ 2 are the phases of oscillators with individual frequencies ω 1 and ω 2. K is the coupling strength and τ is time delay. The interaction term tend to synchronize both oscillators.
Synchronized solutions The most general synchronized solution contains a common time dependent phase part ( Ψ(t) ) and a part corresponding to the constant phase difference between the oscillators ( α ) i.e. θ 1 (t) =ψ(t) + α 2 θ 2 (t) =ψ(t) α 2 Putting these θ s in the system equations, leads to a condition- cos( ψ(t) ψ(t τ) ) = ω 1 ω 2 2K sin(φ) = const.
Synchronized solutions The condition cos( ψ(t) ψ(t τ) ) = const. is satisfied if, Ψ(t) = Ωt, so the only possible synchronized solutions for the given set of parameters are those for which both oscillators have same common frequency Ω and they differ only by a constant phase shift α. i.e. θ 1,2 (t) = Ωt ± α 2
Synchronized solutions: Collective frequency and Synchronized solutions phase difference In contrast to the case with delay time = 0 where there exists only one synchronization frequency (Ω=ϖ), here we see from the transcendental equation that there can be multiple solutions for a given set of parameters. The number of solutions increases with coupling strength and time delay. ω =1, Δω = 0.4, K = 2
Synchronized solutions The frequencies and phase differences between the solutions vary with coupling strength. The number of solutions increases with coupling strength and the system jumps to new stable frequency- ω =1, Δω = 0.4, τ =1.0
Multiple frequency solutions: basins of attraction In the time interval [ -τ, 0 ], starting with the function θ 1,2 (t) = Ωt ± α 2 With different values of Ω and α as starting point, the basin of attraction: ω =1, Δω = 0.4 0.2 1.4 2.6 K = 2.0, τ = 2.0 For the set of parameter values, there are total five possible frequencies out of which three are stable as can be seen in the figure.
Relay Synchronization with delay Network of oscillators with time delayed coupling are studied theoretically and experimentally because of their applications to neurobiology, laser arrays, microwave devices, electronic circuits and also because of their inherent mathematical interest. If there is time delay in the interaction, oscillators 1 and 3 become phase synchronized and 1, 2 and 2, 3 are lag synchronized. [ ε, [ ε, 1 τ ] 2 τ ] 3
Relay Synchronization with delay This is a simple bipartite network. How general are such solutions for other bipartite networks?
A Bipartite network of Delay coupled oscillators Consider a system of N coupled phase oscillators on a bipartite topology with distributed delay coupling, θ i = ω i + ε N ( + a k ij g* θ j (t s) f (s)ds θ i (t)-, i ), j =1 0 i =1,...,N Here θ s are the corresponding phases, ω the natural frequencies, ε denotes the coupling strength, τ is the delay. k i is the number of inputs received by the i th oscillator and a ij is the adjacency matrix which reflects the connection topology.
Synchronization For synchronized solutions, θ A,B (t) = Ωt ± φ /2 where Ω is the common frequency and ϕ is the phase difference between the partitions. The necessary condition for the existence of phase-locked solutions of this form is Ω = ω +εg( Ωτ φ) = ω +εg( Ωτ + φ). The function g is periodic with period 2π, the two solutions are Ω 0 = ω +εg( Ω 0 τ) Ω π = ω +εg( Ω π τ π) = ω +εg( Ω π τ + π)
For stability With stability conditions, ε g #( Ω 0 τ) > 0 for in phase solutions ε g #( Ω π τ + π) > 0 for anti phase solutions
Bipartite Kuramoto Consider delay coupled phase oscillators on a bipartite network, with standard sinusoidal coupling function. θ i = ) ω + ε + k i * + ω + ε, + k i sin( θ j (t τ) θ i (t)), i A, j B sin( θ j (t τ) θ i (t)), j A i B
Bipartite Kuramoto The collective frequencies and phase differences between the partitions is given by And the stability conditions are Ω = ω ε sin(ωτ) ) 0 if cos(ωτ) > 0 Δφ = * + π otherwise cos(ωτ) > 0 for cos(ωτ) < 0 for in phase solutions anti phase solutions
Bipartite Kuramoto For different number of oscillators in two partitions, if the number of connections for every oscillator is same, the results for common frequencies and phase differences remain unaltered. A π Complete bipartite: different number of oscillators in each partition. B Each partition works as a big oscillator. 0 The phase difference between the partition flips between zero and π
Collective frequencies Stable collective frequencies and corresponding phase difference. From the transcendental equation (ω = 1): Ω =1 ε sin(ωτ ) The blue lines are solutions corresponding to in-phase and red lines are anti-phase solutions. The green boxes are the regions where more than one frequencies are stable. ε = 0.05
Numerical results: collective frequencies Collective frequencies As the value of delay increases, the number of solution frequencies also increases. However, irrespective to the number of solutions, we can have only two phase solutions (Φ = 0, π). ε = 0.05 Green boxes are the regions where same phase solutions (in- or antiphase) can be obtained from different collective frequency solutions.
Numerical results: collective frequencies Solutions for different coupling strengths. The overlapping region (multistable region) increases with increasing value of coupling strength. ε = 0.05, 0.125, 0.25, 0.5
Phase-diagram for N=6 Numerical frequencies for 6 oscillators N A = N B = 3, plotted with stable analytical solutions from the transcendental equation: in-phase anti-phase multistable region Ω =1 ε sin(ωτ )
Hysterisis and Multistability Numerical frequencies plotted with stable analytical solutions.
Solution Branches Analytic forms of solutions branches The width of multistable regions increase linearly with time-delay: Stable solution branches are indicated by B n, where even and odd values of n indicate in-phase and anti-phase branches, respectively. The dashed boxes are the region of multistability and the dotted boxes, indicated by the arrows, show highest number of overlaps three overlapping branches in this case.
Stable Solutions The number of stable frequencies increases approximately linearly with parameters. Ω =1 ε sin(ωτ) Ω max =1+ε Ω min =1 ε
Stable Solutions As the time delay determines the actual value of collective frequency, it effectively acts as a frequency selector in a range that is determined by the coupling strength. Ω =1 ε sin(ωτ ) Ω max =1+ε Ω min =1 ε ε = 0.5
Other oscillators on a bipartite network: Other oscillators on bipartite networks Landau-Stuart (L-S) oscillators Rössler oscillators ( 2 ( A + iω A Z i )Z ii + ε a k ij Z j (t τ) Z i (t) Z i ( ), i A, * j B i = ) 2 *( A + iω B Z i )Z ii + ε a k ij Z j (t τ) Z i (t) i ( ), i B + * j A With some approximations, we can reduce the phase dynamics of coupled L-S oscillators and Rössler oscillators on the bipartite networks to that of Kuramoto phase oscillators. ( ω A y i z i + ε a k ij i ( x j (t τ) x i (t)), i A, * j B x i = ) * ω B y i z i + ε a k ij x j (t τ) x i (t) i ( ), i B + * j A ( y i = ω A x i + ay i i A, ) + ω B x i + ay i i B z i = f + z i (x i c) ( ω A + K * θ k i i = ) * ω B + K + * k i a ij sin( θ j (t τ) θ i (t)), i A, j B a ij sin( θ j (t τ) θ i (t)), j A i B K = ε K = ε 2 For L-S oscillators For Rössler oscillators
Landau Stuart Frequencies and time series: Parameter values: A =1,ω A = ω B =1.0 ε =1.0 K = ε =1
Rössler Oscillators Parameter values: a = b = 0.2,c =1.0 ω A = ω B =1.0 ε = 2.0 K = ε 2 =1.0 Frequencies and time series.
Mismatched Partitions
We study a system of delay-coupled phase oscillators on a bipartite topology, when there is a mismatch between the partitions - Mismatched oscillators θ i = $ & & % & & ' ω A + ε k i ω B + ε k i a ij sin( θ j (t τ ) θ i (t)), i A, j B a ij sin( θ j (t τ ) θ i (t)), j A i B ω A,B are natural frequencies of the two partitions.
Global Synchrony $ & θ i = % & '& Ωt + φ 2 Ωt φ 2 if if i A i B Ω is the collective frequency and ϕ is the constant phase difference between the partitions.
Collective frequencies and phases Making the ansatz that there is a common oscillation for the oscillators in the two partitions, one can get a transcendental equation for the collective frequency and for the corresponding phase difference. F (Ω) = ω Ω ε tan(ωτ ) cos 2 (Ωτ ) (Δω)2 4ε 2 $ Δω ' φ = arcsin& ); if cos(ωτ ) > 0 % 2ε cos(ωτ )( $ Δω ' φ = π arcsin& ); otherwise % 2ε cos(ωτ )(
GS solutions and asynchrony Frequencies and phase differences between the partitions (N = 64) Ω = ω ε tan(ω τ ) cos 2 (Ω τ ) (Δω)2 4ε 2 Condition for globally synchronized solutions to exist $ cos 2 (Ωτ ) Δω ' & ) % 2ε ( 2 We observe windows (W 1, W 2 ) where globally synchronized solutions do not exist.
Global behaviour Phase behavior in the parameter space: Δω = 0.08 F (Ω) = 0 F + (Ω) = 0 multistable region desynchronized region Δω = 0.2 F (Ω) = 0 F + (Ω) = 0 multistable region desynchronized region
Global frequencies Time-averaged frequencies for the two partitions (N = 64) In regions (I) and (V): globally synchronized solutions exist Ω A = Ω B = Ω In regions (II), (III) and (IV): globally synchronized solutions do not exist Ω A Ω B $ & Ω i = Ω A if i A % ' & Ω i = Ω B if i B Time-averaged frequencies of the oscillators from the same partitions are equal.
Partition frequencies Time-averaged frequencies for the two partitions for different coupling strength and frequency mismatch θ A,B = Ωt ± Δφ 2 θ i = G i (t)+ c i Ω i (t) = G i (t) Ω i = Ω A if i A Ω i = Ω B if i B
Time-dependent frequencies and RS Remote synchronization due to locking of time-dependent frequencies (N = 64) Ω j A = Ω j B = Ω + Ω j A = f (t) Ω j B = g(t)+ c j Ω j A = f (t) Ω j B = g(t) Ω j A = f (t)+ c j Ω j B = g(t) Ω j A = Ω j B = Ω
Remote Synchronization (RS) In a system of coupled oscillator networks, RS is said to occur when indirectly coupled oscillators are phase synchronized, while not being in synchrony with the relaying unit(s), Remote synchronization: Chimera States (CS), and Individual Synchronization (IS) Remote synchronization has not been shown in a network of phase oscillators without time-delay
Phase evolution of oscillators Time evolution of oscillator phases at different values of time delay showing global synchronization (GS), chimera states (CS) and individual synchronization (IS).
Partition order-parameters A complex order parameter can be defined for each partition: z A,B = r A,B e iψ = 1 N j A,B e iθ j chimera states: One of r A and r B is 1 and other is 0 individual synchronization: r A = r B = 1 incoherent behavior: r A = r B = 0 Variation of the real parts r A and r B of the partition order parameters as a function of time delay.
Order parameter regimes Level sets of the average order parameter, r = (r A +r B )/2 as a function of ε and τ Chimera states Incoherent regions Individual synchronization Global Sync. solutions
Order parameter regimes Figure showing Arnold t o n g u e o f g l o b a l l y synchronized solutions and scenarios of RS in! Δω ε space. Chimera states Incoherent regions Individual synchronization Global Sync. solutions
Time-dependence W h e n t h e i n t r i n s i c frequencies of oscillators in each partition are drawn from two Gaussian distributions with mean values µ 1,2 = ϖ ± Δω and variance σ = Δω/(2 10 2 ).
The order parameter Chimera states Incoherent regions Individual synchronization Global Sync. solutions
Summary Ø In complete bipartite case, identical oscillators in the same partitions are always in-phase synchronized and are either in inphase or anti-phase synchronization state with the oscillators in the other partition. Ø The system flips between in-phase and anti-phase states when it jumps from one collective frequency state to another. The number of possible stable frequency solutions increase linearly when delay or coupling is varied. Ø Stability for in as well as anti phase states are obtained. There can be several stable collective frequencies indicating multistability. The system can switch from in-phase state to anti-phase state depending upon the initial conditions.
Summary Ø Reducing the phase dynamics of Landau-Stuart oscillators and Rössler oscillators to simple phase oscillators, we can predict the phase behavior of these rather complex systems in parameter space. Ø The nonlinearity and the periodic nature of the coupling is responsible for the stabilization of the anti-phase states Ø Delay coupled phase oscillators on a bipartite network with frequency mismatch show interesting and novel collective states. In addition to global synchrony when all the oscillators in the network lock onto a common frequency, there can also be remote synchronization, with different groups of indirectly connected oscillators displaying distinct patterns of phase coherence.
Summary Ø There can be several stable collective frequencies indicating multi-stability. The system can switch from in-phase state to antiphase state depending upon the initial conditions. Ø For a range of parameters, where global synchrony exist, we find solutions such that systems within a partition are in complete synchrony, while there is lag synchronization between the partitions. Ø Outside this range, one observes scenarios of remote synchronization namely, chimeras and individual synchronization, where either one or both of the partitions are synchronized independently. In the absence of time-delay such states are not observed in phase oscillators.
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