Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states
We study synchronization in delay-coupled oscillator networks, using a master stability function approach. Within a generic model of Stuart-Landau oscillators (normal form of super- or subcritical Hopf bifurcation) we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies, or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network.
pacs:05.45.Xt, 05.45.Gg, 02.30.Ks, 89.75.-k
Over the last decade, control of dynamical systems and stabilization of unstable states have become a central issue in nonlinear science SCH07 (). In parallel, the study of coupled systems ranging from a few elements to large networks has evolved into a rapidly expanding field WAT98 (). To determine the stability of synchronized oscillations in networks, Pecora and Carroll introduced a technique called master stability function (MSF) PEC98 (), which allows one to separate the local dynamics of the individual nodes from the network topology. Although some recent approaches have tried to extend this theory in the presence of time delays DHA04 (); KIN09 (), up to now control and design of dynamic behavior in complex networks with time delay is still in its infancy.
In this Letter, we aim to fill this gap by developing analytical tools for a large class of delay-coupled networks and deriving analytical conditions for controlling the different states of synchrony. We identify the coupling phase as a crucial control parameter and demonstrate that by adjusting this phase one can deliberately switch between different synchronous oscillatory states of the network. We use a generic model describing a wide range of systems near a Hopf bifurcation, which allows for an analytical treatment, including the calculation of the Floquet exponents. These results promise broad applicability, since the presence of time delays is of crucial importance in a variety of physical, biological, technological, social, ecological, or economic networks where they occur, e.g., as propagation delays in communication networks and laser arrays FIS06 (); VIC08 (); FLU09 (), electronic circuits RAM00 (), neural systems GAS07b (); BON07 (); SCH08 (), or coupled Kuramoto phase oscillators EAR03 (); DHU08 (); SET08 (), or in time-delayed feedback control loops PYR92 ().
We consider -dimensional networks of delay-coupled Stuart-Landau oscillators ()
with , time delay , and complex coupling strength . Such phase-dependent couplings have been shown to be important in overcoming the odd-number limitation of time-delayed feedback control FIE07 () and in anticipating chaos synchronization PYR08 (). The topology of the network is determined by the real-valued adjacency matrix . Nonzero diagonal elements, for instance, correspond to networks with delayed self-feedback. In the following, we consider only constant row sum such that each node is subject to the same input for complete synchronization. This generalizes the common assumption of zero row sum in the MSF approach. The local dynamics of each element is given by the normal form of a supercritical () or subcritical () Hopf bifurcation:
with real constants , , and . This system arises naturally as a generic expansion near a Hopf bifurcation, and is therefore often used as a paradigm for oscillators.
In the following, we focus on synchronous in-phase, cluster, and splay states with a common amplitude and phases given by with . The integer determines the specific state: in-phase oscillations correspond to , while cluster and splay states correspond to . The cluster number , which determines how many clusters of oscillators exist, is given by the least common multiple of and divided by . corresponds to a splay state ZIL07 (). With the above notation, we obtain for in-phase oscillation in general networks and for splay and cluster states in ring configurations
as invariant solutions of and using the abbreviation , which is independent of in the cases mentioned above footnote (). The following discussion focuses on the supercritical case (upper signs), but a similar argument holds also for the subcritical Hopf normal form (see discussion at the end of this Letter).
Figure 1 shows solutions of () and for in-phase oscillations () according to Eqs. (3) in dependence on the time delay for fixed feedback strength and in panels (a),(b) and (c),(d), respectively. The black (blue) lines show the behavior for the coupling phase . The collective frequency is distributed around the intrinsic frequency , where multiple solutions are obtained with increasing time delay . This behavior becomes more pronounced for higher (c). The collective amplitude also shows multivalued behavior; spurious solutions with , which correspond to amplitude death, are indicated as dashed curves. For a coupling phase , these unphysical solutions do not occur since as shown by the gray (yellow) curves in (b),(d). Note that for the shape of the curve in (a),(c) is unchanged, but now all points are valid solutions.
Considering small deviations and , i.e, , , , yields a variational equation for the synchronized state
with the -dimensional vector , the identity matrix , and matrices , , , which is an important generalization of the usual MSF approach.
In order to derive an analytical expression for stability, Eq. (4) has to be diagonalized in terms of . To succeed, the rotational matrix must not depend on . This is achieved in two cases: (i) By considering only in-phase synchronization (), or (ii) by considering special network configurations.
In case (i), the matrix with does not depend on and with Eq. (4) simplifies to
Diagonalizing , we arrive at the block-diagonalized variational equation:
where is an eigenvalue of , , and corresponds to the dynamics in the synchronization manifold. Since the coefficient matrices in Eq. (6) do not depend on time, the Floquet exponents of the synchronized periodic state are given by the eigenvalues of the characteristic equation
Figures 2(a),(b) depict the MSF, i.e., the largest real part of the Floquet exponents, calculated from Eq. (7) for different coupling parameters. Note that for a unidirectionally coupled ring all eigenvalues, i.e., with , are located on the black circle. Hence, for the choice of parameters in panel (a) all eigenvalues lie in the region of negative maximum real part of the Floquet exponent (stable in-phase solution), whereas the parameters in panel (b) do not allow for synchronization of the unidirectional ring. Furthermore, it can be shown using Gerschgörin’s disk theorem EAR03 () that the eigenvalues are located on or inside this circle centered at with radius for any network topology without self-feedback (diagonal elements ). The same holds for the circle centered at if self-feedback (constant , ) is added while keeping the constant row sum condition. Note that the MSF is symmetric with respect to a change of sign of .
For coupling phases with integer , i.e., , the characteristic equation (7) for the Floquet exponents factorizes:
In the supercritical case (upper sign), the dominant Floquet exponent is determined by the second factor in Eq. (8) which gives Re for for any eigenvalue on or inside , taking into account , and hence stable in-phase synchronization for any network topology without or with self-feedback, observing the constant row sum conditions. A similar equation arises for with obtained from Eq. (3b), and it can be shown by analogous arguments that there exist exponents with positive real part, which results in desynchronization. In conclusion, the synchronous (in-phase) dynamics can be stabilized or destabilized by proper choice of the coupling phase . These results are robust even if slightly nonindentical elements are considered. Fig. 2(c) shows numerical simulations for all-to-all coupling of 200 elements with Gaussian frequency distributions around with different standard deviations . For the relative phases of the individual oscillators around the order parameter (in rotating coordinates) are distributed around a maximum (in-phase) with different sharpness according to , whereas for the phase is uniformly distributed regardless of (desynchronization).
We now consider case (ii), i.e., special network configurations, and exemplarily choose a unidirectional ring with and all other . Then the matrix with does not depend on and the diagonalization of Eq. (4) yields the same form as Eq. (6) with , and replaced by . The eigenvalues of are explicitly given by , , and the Floquet exponents can be calculated from the corresponding characteristic equation (7).
Figure 3(a) shows the stability boundaries of different dynamical scenarios in the -plane for unidirectional coupling of oscillators. The coupling phase is fixed at . The gray scale (color code) indicates regions of different multistability of in-phase (), 2-cluster (), and splay states (, ): Black (blue), dark gray (red), light gray (green), and yellow (white) color corresponds to regions where one, two, three, or four of these dynamical states are stable, respectively.
Let us now consider the effects of the coupling phase . The specific choice of , , , and enlarges the stability regime of the in-phase, splay (), cluster, and splay () states, respectively, to the complete -plane. This can be understood as follows. For , i.e., with integer , the characteristic equation can again be factorized as Eq.(8). For the supercritical case, taking into account at , it follows again that the dominant Floquet exponents have negative real part for any and . Therefore the unidirectional ring configuration of Stuart-Landau oscillators exhibits in-phase synchrony, splay state and clustering according to the choice of the control parameter , , or , respectively, for any values of the coupling strength and time-delay.
To illustrate this further and demonstrate the robustness of our stability results for slightly nonidentical oscillators, we choose a set of control parameters and , denoted by the black cross in Fig. 3(a), for which multistability of all four possible synchronization states is found for the coupling phase . Figures 3(b)-(e) show time series from numerical simulations of four Stuart-Landau oscillators in a unidirectional ring configuration with slightly different frequencies . For each choice of in panel (b) - (e) the solutions were obtained by solving Eqs. (3) such that the solution of closest to unity was chosen. The differences of the phases () relative to the first oscillator phase are plotted. After transients (note that the transient oscillations are not resolved on the time scale chosen), the oscillators behave exactly as predicted by our theory, i.e., they lock into in-phase synchronization for (b), into a splay state for (c), into a 2-cluster state for , where and (d), and again into a splay state, albeit with inverted ordering of the phases, for (e).
Finally, for the subcritical Hopf normal form, it can be shown that the periodic orbit, which is unstable in the uncoupled case, can be stabilized in-phase synchronously by, e.g., bidirectional ring, star, or all-to-all coupling without self-feedback. In these cases, both the synchronization manifold and the transversal modes are stable. There, Floquet exponents satisfy again Eq. (7) with . For proper coupling phases , we find a finite interval of feedback gain for which the real parts of all Floquet exponents are negative. Note that the in-phase synchronization manifold coincides with a single oscillator with delayed self-feedback as considered in Ref. FIE07 () to refute the alleged odd number limitation.
For all-to-all coupling the adjacency matrix is given by and for and , while for star coupling all , except and for . Figure 4 shows Re in the subcritical case as a function of coupling strength for all-to-all and star coupling. The solid (blue) lines corresponds to Re inside the synchronization manifold ( for both coupling configurations). For all-to-all coupling the transversal eigenvalues of are given by for and the corresponding largest Re are denoted by the dashed (red) lines in Fig. 4 for different . For star coupling, the transversal eigenvalues of are for and , and the corresponding largest Re are marked schematically. We stress that for both all-to-all and star coupling there exists an interval of feedback strength in which all Re . Thus, time delayed coupling results in stabilization and in-phase synchronization.
In conclusion, we have shown that by tuning the coupling phase in delay-coupled networks one can easily control the stability of synchronous periodic states, and we have specified analytic conditions. In general networks in-phase synchronization or desynchronization can be chosen, and in unidirectional rings either in-phase, cluster or splay states can be selected. The coupling phase is a parameter which is readily accessible, e.g., in optical experiments SCH07 (). Our results are robust even for slightly nonidentical elements of the network.
Acknowledgements.C.-U. C. acknowledges support from Alexander von Humboldt Foundation. This work was also supported by DFG in the framework of Sfb 555. We thank A. Amann for valuable discussions.
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