The Boundaries of Synchronization in Oscillators Networks
Everton S Medeiros Rene O. Medrano-T Iberê L Caldas Ulrike Feudel






esm@if.usp.br
Institute of Physics, University of São Paulo, Rua do Matão, Travessa R 187, 05508-090, São Paulo, Brazil
Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, Oldenburg, Germany

Departamento de Física, Universidade Federal de São Paulo,  Diadema, São Paulo, Brazil

Institute of Physics, University of São Paulo, Rua do Matão, Travessa R 187, 05508-090, São Paulo, Brazil

Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, Oldenburg, Germany


We analyze the final state sensitivity of nonlocal networks with respect to initial conditions of their units. By changing the initial conditions of a single network unit, we perturb an initially synchronized state. Depending on the perturbation strength, we observe the existence of two possible network long-term states: (i) The network neutralizes the perturbation effects and returns to its synchronized configuration. (ii) The perturbation leads the network to an alternative desynchronized state. By computing uncertainty exponents of a two-dimensional cross section of the state space, we find the existence of fractal basin boundaries separating synchronized solutions from desynchronized ones. We attribute these features to an unstable chaotic set in which trajectories persist for times indefinitely long in the network.

The Boundaries of Synchronization in Oscillator Networks
    Ulrike Feudel
    December 30, 2023 (Original: July 27, 2018)
========================================================




Synchronization is a universal concept in nonlinear dynamics characterizing the emergence of correlations between connected subsystems <cit.>. The ubiquity of synchronization transcends limits of different fields of science. In ecology, synchronization of populations in different habitats may be crucial for species persistence <cit.>. In cardiology, the loss of synchronicity of the heart rhythms leads to the onset of ventricular fibrillation causing death <cit.>. In neurodynamics, synchronicity of rhythms is a key concept to understand how processes performed in different parts of the neural system can be integrated <cit.>. Also, many applications have been reported, e.g., mechanical oscillators <cit.>, laser physics <cit.>, phase oscillators <cit.> and complex networks <cit.>. 

The asymptotic stability of synchronized states has been addressed in networks by the master stability function formalism <cit.>. This method accounts for a linear analysis of perturbations applied transversely to the synchronized state. It provides satisfactory insights about the local stability of the synchronized state, however, it does not provide any information about its whole basin, i.e. the impact of larger perturbations beyond the linear limit. More recent approaches have focused on such global issues by estimating the relative size of the synchronization basin <cit.>. Although such analyses address a nonlocal character of stability in networks, yet, it does not give any insights about the boundaries of the basins of the synchronized state. The characteristics of such boundaries has practical consequences for the synchronization in networks. For instance, for fractal basin boundaries  <cit.>, the synchronized states would be sensitive to small perturbations in their initial conditions, even when these states are classified as asymptotically stable by the previously mentioned formalisms. Surprisingly, final state sensitivity due to fractal basin boundaries has not been addressed for synchronized states so far.

We investigate the sensitivity of synchronized states with respect to initial conditions in homogeneous networks and show that a small perturbation in the initial condition of a single unit is capable to desynchronize the whole network. The sole requirement for each network unit is the coexistence of an unstable chaotic set with a periodic attractor. As consequence, two long-term network states are available, a complete synchronized state at the periodic attractor and a fully desynchronized one around the chaotic set which appears to turn into an attractor. We explain this behavior by employing the concept of final state sensitivity. The final state sensitivity appears in two regimes: A less uncertain regime corresponding to the boundaries of a continuous region of perturbations leading to synchronization; and an extremely sensitive regime corresponding to an intricated structure of perturbations. The uncertainty exponents of both regimes indicates the occurrence of fractal basin boundaries between synchronized and desynchronized states. We point out the generality of this behavior, since coexistence of unique attractors and chaotic saddles embedded in their basins of attraction is a generic feature of nonlinear oscillators. 

We study a network composed of N units arranged in a ring. For the dynamics in each unit we chose the Duffing oscillator: 

    ẋ_i   =    y_i + σ/2R∑_j=i-R^j=i+R(x_j-x_i), 
    ẏ_i   =    -γ y_i + x_i - x_i^3+A cos(ω t) +σ/2R∑_j=i-R^j=i+R(y_j-y_i).

The pair (x_i,y_i) defines the state space of each oscillator i with i=1,...,N.  The parameter σ controls the coupling strength, R>1 specifies the number of first neighbors connected to the ith unit, realizing a nonlocal coupling of the units. The parameter γ denotes the damping in the system, while A and ω are the forcing amplitude and frequency, respectively. We fix the parameters of each unit at ω=0.5, γ=0.24, and A=13.633, where the Duffing oscillator exhibits a globally stable period-3 attractor, A, and an unstable chaotic set, Λ. The states (x_i,y_i) are sampled at multiples of the forcing period, T=2π/ω. The coupling parameters R and σ as well as the network size N are specified for each simulation.    

In Fig. <ref>, we show the dynamics of the network in space and time described by system (<ref>). In these simulations, all network units, but one, are set to the same initial condition defining a synchronization manifold, S. All units that compose the synchronization manifold started at (-5.0, 5.0). In Fig. <ref>(top), the unique unit starting outside the synchronization manifold receives (-2.25, -11.00) as initial condition. This setup leads to synchronization of the whole network after some transient time, as illustrated with snapshots (i × y_i) in the right. Next, the initial condition of the unique unit is slightly perturbed to (-2.25, -11.00+ϵ) with ϵ=10^-4. As demonstrated in Fig. <ref>(bottom), the network converge to a completely desynchronized state, see snapshot in the right. Hence, Fig. <ref> demonstrated the coexistence of two different global network behaviors, a fully synchronized and a completely incoherent one. Moreover, it suggests that the network's final state is highly sensitive with respect to small changes in its initial conditions. 



The synchronized solution shown in Fig. <ref>(top) possesses a basin of synchronization, i.e., a set of initial conditions that leads the network to the synchronized state. We compute a slice of such basin by keeping all initial conditions of (N-1) constant at the chosen synchronization manifold and vary only the initial conditions attributed to the perturbed unit. In order to identify such synchronization basin, we define an order parameter 𝒵 by analyzing a next neighbor error E_i=r_i-r_i-1, where r_i is the state vector of the dynamical variables (x_i,y_i);

    𝒵=1/N∑_i=1^NK_i,     K_i= 1,   if   E_i > δ
    
        0,                 if   E_i < δ.

The parameter δ establishes the synchronization quality, which we fixed to δ=0.01. Hence, the completely synchronized state returns an order parameter 𝒵=0, while the completely desynchronized state implies an order parameter 𝒵=1.

The order parameter 𝒵 is computed for a 2-dimensional grid of initial conditions (x_12^0, y_12^0) attributed to the  single perturbed unit of a network composed of N=25 units. Accordingly, the plane spanned by x_12^0 and y_12^0 forms a two-dimensional section of the 50-dimensional state space. For each initial condition on this plane, a transient initial time of 2 × 10^4T is discarded before computing the order parameter shown in Fig. <ref>(a). The initial conditions colored in blue (dark gray) correspond to the basin of the synchronized state, while the ones colored in yellow (light gray) correspond to trajectories for which the whole network desynchronizes. The white cross indicates the initial condition attributed to the synchronization manifold, i.e., the (N-1) network units defined as synchronized in the initial instant of time. Since we have fixed the overall integration time to t_end= 2 × 10^4T, some order parameters, white in Figs. <ref>(a) and (c), can take values between 0 and 1 this corresponds to network realizations for which full (de)synchronization has not been reached yet. Those are initial conditions corresponding to unstable chimera states <cit.>.
 
As we observe in Fig. <ref>(a), the basin of the synchronized state contains open sets of initial conditions in which all of them converge to synchronization in the period-3 orbit; they appear as blue (dark gray) continuous areas. However, we note that these open sets are separated by a nontrivial boundary from complex regions, where a mixture of blue (dark gray) and yellow (light gray) points occurs indicating high final state sensitivity. In those mixed regions, Fig. <ref>(c), small changes in the initial condition of the perturbed unit lead to the incoherent state of the whole network. This part of the basin corresponds to a cross section of the extended state space in which the stable manifold (foliation) of the chaotic set is close.   




We computed the fraction of initial conditions that are uncertain with respect to perturbations ε <cit.>. An initial condition is called uncertain if it converges to some specific attractor when unperturbed, but converges to another when a perturbation of size ε is applied. Considering a large interval for such perturbations, ε∈ [10^-8,10^0], the fraction of uncertain initial conditions f(ε) scales with ε as f(ε) ∼ε^α, where α is the uncertainty exponent. Fig. <ref>(b) shows the relation log_10(f(ε)) versus log_10(ε) for the domain of attraction shown in Fig. <ref>(a). We find two different uncertainty regimes in the interval of ε. For small values of ε, including the limit ε→ 0, the uncertainty exponent is significantly lower than the one for larger perturbations. The lower exponent, α=0.019 ± 0.002, indicates extreme sensitivity to perturbations expected for the complex region, while the larger exponent, α=0.370 ± 0.002, indicates lower uncertainty of the open sets. This scale separation becomes more clear if we restrict the region in state space for which we compute the uncertainty exponent, shown in Fig. <ref>(c) (the rectangle in Fig. <ref>(a)), yielding α=0.011 ± 0.002, (Fig. <ref>(d)). The proximity of the uncertainty exponent to zero, indicates the occurrence of riddled basins <cit.>. However, the strict definition of riddled basins is valid for the coexistence of at least two attractors, where the probability of reaching one of them is positive within any neighborhood of the other. Here, due to the difficulty of strictly ascertain the desynchronized state as an attractor of the system, we classify the proximity of the uncertainty exponent to zero, α≈ 0, as an indication of riddled-like basins <cit.>.

The mechanism leading to the two distinct behaviors of the network lies in the existence of an unstable chaotic set Λ, a chaotic saddle, which is embedded in the state space of each unit of the uncoupled system and also persists in the high dimensional state space of the network. As pointed out previously, each unit possesses a globally stable attractor – a period-3 orbit (black circles in Figs. <ref>(a) and  <ref>(b)) – besides the chaotic saddle (black dots in Fig. <ref>(a) and Fig. <ref>(b)). Though all initial conditions ultimately converge to the period-3 orbit, the time they need to reach the neighborhood of the attractor can be quite different, as demonstrated by the shades of gray according to the length of their transient time. We find that those long convergence points (colored black) belong to a set very close to the stable manifold of the chaotic saddle embedded in the basin of attraction. Using the sprinkler method <cit.>, we obtain an approximation of that stable manifolds, gray dots in Fig. <ref>(b). All trajectories starting from initial conditions very close to it will first approach the chaotic saddle, dwell close to it for some time interval and finally will be ejected along its unstable manifold (green (light gray) in Fig. <ref>(b)) reaching the period-3 attractor. This dynamical feature of the state space of a single unit has striking consequences for the network dynamics.


All network units starting with synchronized initial conditions that approach the chaotic set, i.e, they are close to its stable manifold, would first approach the chaotic set in a synchronized manner and then converge to the period-3 orbit through its unstable manifold. For such features, if the unique perturbed unit approaches the chaotic saddle, its originally synchronized neighbors will be distorted due to exponential separation of nearby trajectories on the chaotic saddle leading to their desynchronization. This mechanism is shown in the sketch of Fig. <ref>(a), where a cluster of units is close to the set Λ, and the other units have reached the attractor A. Every network unit in the cluster is subject to two different influences: their local dynamics represented by T⃗_i (see arrows in Fig. <ref>(a)); and the resulting coupling, C⃗_i, due to the coupling to R neighbors on each side (arrow in Fig. <ref>(a)). Since the overall coupling on any unit depends on the instantaneous trajectories of its neighbors, the trajectory of every unit in the cluster is constantly influenced by the motion on the chaotic set. Such sensitive disturbances may retard the escape of units from the chaotic saddle especially by decreasing the likelihood of approaching its unstable manifold <cit.>. Due to the coupling with units outside the initial cluster, even more units can be pulled into the chaotic saddle increasing the cluster of desynchronized units. Once a critical number of desynchronized units is reached, the convergence to full desynchronization becomes possible (Fig. <ref>(c)) <cit.>. Otherwise, all network units end up completely synchronized at the periodic attractor, Fig. <ref>(b). 



Now, we investigate the structure of the fully incoherent state. Fig. <ref>(a) shows the state variables of each unit after a very long time of network iterations, 10^6 cycles of the forcing T. We observe that in the fully desynchronized state, each network unit is tracing out the chaotic set, indicating its dominance in the dynamics. This result suggests that the chaotic set, unstable for a single unit, plays the role of an attractor in the network. 

Finally, we address the question how robust this phenomenon is with respect to the variation of the coupling strength. To this end, we vary σ in the interval [0,1], and perform network simulations with different initial conditions attributed to the perturbed unit. We compute the probability of convergence to the synchronized state, P_S=n_S/n, as a function of σ for n=182 realizations discarding a large interval of transients (Fig. <ref>(b)). Here, n_S is the number of realizations that leads to network synchronization. We remark that in the first interval, σ ∈ [0,0.0036], for very small values of coupling constants σ, network units may reach the attractor with different phases. However, even for intermediate values of σ in this interval, complete synchronization is already observed. As we increase further the coupling constant, we find an approximated interval,  σ ∈ [0.0036,0.18], in which the probability of observing the completely synchronized state is lower than 1. This interval corresponds to the coupling regime for which the loss of synchrony leading to an incoherent state is observed. For values of σ even larger, the coupling easily overcomes the chaotic dynamics in the neighborhood of the chaotic set and the network completely synchronizes for all network realizations, P_S=1.0. 



To check the persistence of the incoherent behavior, and to distinguish it from chaotic transients, we again consider n=182 simulations, each one starting with different, randomly chosen, initial conditions attributed to the perturbed unit. By computing the number of networks, n_S, for which the network synchronizes as a function of different simulation times, t_end, we obtain the probability of complete synchronization, P_S=n_S/n. Fig. <ref> shows that P_S saturates at a fixed value, P_S≈0.77, while the complementary probability of P_S corresponding to the likelihood of desynchronization, reaches P_Λ=1-P_S≈0.23. For conventional transient dynamics, the probability P_S is expected to approach 1 as the simulation time, t_end, increases. In our case, the stability of P_S≈0.77 for t_end≳ 311 T, indicates that incoherent state is present for arbitrarily long times. Hence, in this case, it is impossible to ascertain this behavior as a transient as done for spatially extended system in Refs. <cit.>. Here, the chaotic saddle of the single unit appears to turn into an attractor. Numerically, this is difficult to distinguish from a super-persistent chaotic saddle which would be characterized by a large unlikeliness of unstable directions <cit.>. 




We reported the existence of final state sensitivity between synchronized and desynchronized states in networks with nonlocal coupling. It indicates the occurrence of fractal basin boundaries between these two solutions. The observed network sensitivity appears in two regimes: a moderate sensitivity corresponding to fractal boundaries of a continuous region of initial conditions leading to synchronization; and a very sensitive regime corresponding to riddled-like region of initial conditions leading to synchronized and desynchronized states. This is due to the interplay of the coupling and the unstable chaotic sets. Furthermore, we observe that the desynchronized state appears to be an attractor or a super-persistent chaotic saddle with a low likelihood of unstable directions. Finally, we emphasize that these findings are independent of a particular choice of the dynamics for the network units, since the coexistence between an unstable chaotic set and a periodic attractor is sufficient for the occurrence of the results reported here. Such a situation is generic for almost all chaotic systems, since it is present in any periodic window interspersed into chaotic dynamics. The same behavior can be expected for excitable systems possessing period-adding cascades in which parameter ranges of mixed mode oscillations are interspersed with chaotic regions. Examples include the spike adding dynamics in bursting neuron models like the Hindmarsh-Rose <cit.> and a Hodgkin-Huxley model for thermally sensitive neurons <cit.>.   



The authors thank Professor T. Tél for useful discussions. This work was supported by FAPESP (Process: 2011/19296-1, 2013/26598-0, 2015/50122-0 and 2017/05521-0 ).

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@bib@innerbibempty

[Pikovsky et al.(2003)Pikovsky, Rosenblum, and Kurths]Kurths2003
  author author A. Pikovsky, author M. Rosenblum,
   and author J. Kurths,  @noop  title Synchronization: A Universal Concept
  in Nonlinear Sciences (publisher Cambridge University
  Press, address United Kingdom, year
  2003)NoStop
[Blasius et al.(1999)Blasius, Huppert, and Stone]Stone1999
  author author B. Blasius, author A. Huppert,   and author L. Stone,  @noop  journal journal Nature volume 399, pages 354 (year
  1999)NoStop
[Witkowski et al.(1998)Witkowski, Leon, Penkoske, Giles, Spano, Ditto, and Winfree]Witkowski1998
  author author F. X. Witkowski, author L. J. Leon,
  author P. A. Penkoske, author W. R. Giles, author
  M. L. Spano, author
  W. L. Ditto,  and author
  A. T. Winfree, @noop  journal journal Nature volume 392,  pages 78 (year 1998)NoStop
[Gray(1994)]Gray1994
  author author C. M. Gray, @noop  journal journal J. of
  Comp. Neurosc. volume 1, pages 11
  (year 1994)NoStop
[Kapitaniak et al.(2014)Kapitaniak, Czolczynski, Perlikowski,
  Stefanski, and Kapitaniak]Kapitaniak2014
  author author M. Kapitaniak, author K. Czolczynski, author P. Perlikowski, author A. Stefanski,  and author T. Kapitaniak, https://doi.org/10.1016/j.physrep.2014.02.008 journal
  journal Physics Reports volume 541,  pages 1  (year 2014), note
  synchronous states of slowly rotating pendulaNoStop
[Roy and Thornburg(1994)]Thornburg1994
  author author R. Roy and author K. S. Thornburg, 10.1103/PhysRevLett.72.2009 journal journal Phys. Rev. Lett. volume 72, pages 2009 (year
  1994)NoStop
[Acebrón et al.(2005)Acebrón, Bonilla, Pérez Vicente,
  Ritort, and Spigler]Acebron2005
  author author J. A. Acebrón, author L. L. Bonilla, author C. J. Pérez Vicente, author F. Ritort,  and author R. Spigler, 10.1103/RevModPhys.77.137 journal journal Rev. Mod. Phys. volume
  77, pages 137 (year 2005)NoStop
[Osipov et al.(2007)Osipov,
  Kurths, and Zhou]Osipov2007
  author author G. Osipov, author J. Kurths,   and author C. Zhou, @noop
   title Synchronization in Oscillatory Networks  (publisher Springer, address United States,  year 2007)NoStop
[Arenas et al.(2008)Arenas,
  Díaz-Guilera, Kurths, Moreno, and Zhou]Arenas2008
  author author A. Arenas, author A. Díaz-Guilera, author J. Kurths, author Y. Moreno,   and author C. Zhou, https://doi.org/10.1016/j.physrep.2008.09.002 journal journal Physics Reports volume
  469, pages 93  (year 2008)NoStop
[Pecora and Carroll(1998)]Pecora1998
  author author L. M. Pecora and author T. L. Carroll, @noop  journal journal Phys.
  Rev. Lett. volume 80, pages 2109
  (year 1998)NoStop
[Pecora et al.(2000)Pecora,
  Carrol, Johnson, Mar, and  Fink]Pecora2000
  author author L. Pecora, author T. Carrol,
  author G. Johnson, author D. Mar,  and author
  K. S. Fink, @noop  journal journal Int. J. of Bif. and Chaos volume 10, pages 273 (year
  2000)NoStop
[Wiley et al.(2006)Wiley,
  Strogatz, and Girvan]Girvan2006
  author author D. A. Wiley, author S. H. Strogatz,
   and author M. Girvan,  @noop  journal journal Chaos: An
  Interdisciplinary Journal of Nonlinear Science volume
  16, pages 015103 (year 2006)NoStop
[Menck et al.(2013)Menck,
  Heitzig, Marwan, and Kurths]Menck2013
  author author P. J. Menck, author J. Heitzig,
  author N. Marwan,  and author J. Kurths, @noop 
  journal journal Nat. Phys. volume 9, pages 89 (year 2013)NoStop
[Grebogi et al.(1985)Grebogi, McDonald, Ott, and Yorke]Grebogi1985
  author author C. Grebogi, author S. W. McDonald, author E. Ott,   and author J. A. Yorke,  @noop  journal journal Phys. Lett. A  volume 110, pages 01 (year
  1985)NoStop
[McDonald et al.(1985)McDonald, Grebogi, Ott, and Yorke]McDonald1985
  author author S. W. McDonald, author C. Grebogi,
  author E. Ott,  and author J. A. Yorke, @noop 
  journal journal Physica D volume 17, pages 125 (year 1985)NoStop
[Kuramoto and Battogtokh(2002)]Kuramoto2002
  author author Y. Kuramoto and author D. Battogtokh, @noop  journal journal
  Nonlinear Phenom. Complex Syst. volume 5,  pages 380 (year 2002)NoStop
[Abrams and Strogatz(2004)]Strogatz2004
  author author D. M. Abrams and author S. H. Strogatz, @noop  journal journal
  Phys. Rev. Lett. volume 93, pages
  174102 (year 2004)NoStop
[Omelchenko et al.(2011)Omelchenko, Maistrenko, Hövel, and  Schöll]Omelchenko2011
  author author I. Omelchenko, author Y. Maistrenko, author P. Hövel,
   and author E. Schöll,  @noop  journal journal Phys. Rev. Lett.  volume 106, pages 234102 (year 2011)NoStop
[Alexander et al.(1992)Alexander, Yorke, You, and Kan]Alexander1992
  author author J. Alexander, author J. A. Yorke, author Z. You,  and  author I. Kan, @noop 
  journal journal Int. J. of Bif. and Chaos  volume 02, pages 795 (year
  1992)NoStop
[Woltering and Markus(2000)]Woltering2000
  author author M. Woltering and author M. Markus, @noop  journal journal Phys.
  Rev. Lett. volume 84, pages 630
  (year 2000)NoStop
[Lai and Tél(2011)]Tamas2011
  author author Y.-C. Lai and author T. Tél,  @noop  title Transient Chaos (publisher Springer-Verlag New York, address United States Of
  America, year 2011)NoStop
[Kantz and Grassberger(1985)]Kantz1985
  author author H. Kantz and author P. Grassberger, @noop  journal journal
  Physica D: Nonlinear Phenomena volume 17,  pages 75 (year 1985)NoStop
[Lai and Winslow(1995)]Lai1995
  author author Y.-C. Lai and author R. L. Winslow, @noop  journal journal Phys.
  Rev. Lett. volume 74, pages 5208
  (year 1995)NoStop
[Not()]Note1
  @noop  note See Supplemental Material at [] for an
  estimate of the critical number of units in the saddle.Stop
[Lilienkamp et al.(2017)Lilienkamp, Christoph, and Parlitz]Lilienkamp2017
  author author T. Lilienkamp, author J. Christoph,  and author U. Parlitz, 10.1103/PhysRevLett.119.054101 journal journal Phys. Rev. Lett. volume 119, pages 054101 (year
  2017)NoStop
[Lilienkamp and Parlitz(2018)]Lilienkamp2018
  author author T. Lilienkamp and author U. Parlitz, 10.1103/PhysRevLett.120.094101 journal journal Phys. Rev. Lett. volume 120, pages 094101 (year
  2018)NoStop
[Crutchfield and Kaneko(1988)]Crutchfield1988
  author author J. P. Crutchfield and author K. Kaneko, @noop  journal journal Phys.
  Rev. Lett. volume 60, pages 2715
  (year 1988)NoStop
[Barrio et al.(2014)Barrio,
  Angeles Martínez, Serrano, and  Shilnikov]Barrio2014
  author author R. Barrio, author M. Angeles Martínez, author S. Serrano,  and author A. Shilnikov, @noop  journal journal
  Chaos: An Interdisciplinary Journal of Nonlinear Science volume 24, pages 023128 (year
  2014)NoStop
[Feudel et al.(2000)Feudel,
  Neiman, Pei, Wojtenek,
  Braun, Huber, and Moss]Feudel2000
  author author U. Feudel, author A. Neiman,
  author X. Pei, author
  W. Wojtenek, author
  H. Braun, author M. Huber,  and author F. Moss, @noop  journal journal Chaos: An Interdisciplinary Journal of Nonlinear Science  volume 10, pages 231 (year
  2000)NoStop



