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Convergence of chaotic attractors due to interaction based on closeness
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Convergence of chaotic attractors due to interaction based on closeness Sayantan Nag Chowdhury a , Soumen Majhi a , Dibakar Ghosh a,∗ , Awadhesh Prasad b a b
Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
a r t i c l e
i n f o
Article history: Received 24 June 2019 Received in revised form 13 September 2019 Accepted 17 September 2019 Available online xxxx Communicated by M. Perc Keywords: Convergence of attractors Temporal network Giant connected component
a b s t r a c t Exploration of coherence phenomena in ensembles of interacting dynamical systems has been in the centre of research in social, physical, biological and technological systems for decades. But, in most of the studies, either completely percolated time- and space-static networks or temporal connectivities disregarding the systems’ own dynamics have been dealt with. In this work, we examine the correlation between structural and dynamical evolution in networks of interacting dynamical systems. We specifically demonstrate the scenario of convergence of a set of chaotic attractors into a single attractor as a result of sufficient interaction based on the closeness of their own states. We characterize this occurrence through different measures, and map the collective states in network parameters’ space. We further validate our proposition while exposing the whole scenario for different chaotic systems, namely Lorenz and Rössler oscillators. © 2019 Elsevier B.V. All rights reserved.
1. Introduction One of the fundamental concern in current inter-disciplinary research is to understand the correlations between network structure and network dynamics [1,2]. In spite of the inherent contrast, from interacting populations of animals to the cells of body, numerous biological, man-made and social framework can be represented as dynamical ensembles coupled via complex architectures [3]. As far as the existing literature on interacting dynamical systems is concerned, straightforward elementary approach in networks has mostly presumed a static network constitution. But in reality, such perspective of interaction is undemanding, because in real world, dynamic connectivity [4] is further significant and practical. To confront with this pragmatic constraint, synchronization [5] have been explored in temporal networks [6–22]. To reveal the intrinsic substantial physical mechanisms governing the transfer of information in a system of networking units, the process of synchronization is perhaps premier specimen due to its entanglement in aspect of collective organizations from microscopic interactions of unitary constituents. In each time, during synchronized state, the ingredients of a system manifest indistinguishable dynamical states. On the other hand, emergence of fascinating phenomena like chimera state [23,24] and amplitude death [25,26] have also been explored in such dynamic networks.
*
Corresponding author. E-mail address: [email protected] (D. Ghosh).
https://doi.org/10.1016/j.physleta.2019.125997 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Mostly in those investigated cases for time-varying networks, in each of the node of the network, the authors have allocated a dynamical system among which interactions took place while completely ignoring the states of the individual dynamical units. In contrast, we choose a framework where the systems interact entirely based on the closeness of their trajectories [27]. The connections among those units reflected through the adjacency matrix of the associated network, is thus updated in every time step depending on the accessibility to the near-by neighboring trajectories. As a result of this formalism, we have observed a throughout collapse of all the trajectories into a single attractor [28–33], depending on the network parameters. This sort of convergence of a set of attractors into a single attractor owing to their interaction is popularly known as the phenomenon of synchronization in the literature. In our network, nodes are gradually connected with adjacent nodes with proper choice of network parameters and exhibits a transition from heterogeneous degree distributed disconnected components to formation of connected components. Finally, they generate a clique when the system is in global synchrony. This brings contrast to the previously studied cases [6,8] of synchronization, where the underlying network is always random network [34]. We have gone through rigorous analysis of paradigmatic chaotic Lorenz and Rössler systems where we have witnessed the occurrence of the same phenomenon. We have been able to characterize the observations both through synchronization error and master stability function (MSF) approach. Moreover, we have com-
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pared the process of convergence of attractors to the appearance of a fully connected component in the network. The type of controlled interactional platform we have gone through, may be useful in robotic communication [11] as well as in wireless communication systems, where transmission is possible only within a confined neighborhood [38,39]. Our results suggest that the appropriate strategy of externally imposing control [35–37] on the agents’ interaction helps to converge the systems into a single attractor. It has been observed that synchronization with chaotic and hyperchaotic systems [40,41] has effective applications in cryptography [42–47] but most of that cases requires high implementation costs. Our scheme may be beneficial in generation of synchronization from the perspective of optimal cost too. Our imposed restriction on the system’s interaction in the entire phase space is the key feature and somewhat counter-intuitive to the earlier perceived studies. Our results may help to achieve maximum communication efficiency through limited accessibility [17]. The remaining parts of the paper are assembled as follows. In Sec. 2, we describe mathematical framework of our time varying model where the network topology varies due to the intrinsic dynamics of the individual dynamical units. In Sec. 3, we explore the convergence of attractors numerically for Lorenz oscillators. Also, we examine the connectedness of the network for suitable choice of parameters. Stability of the converging manifold is established with the help of MSF approach. In Sec. 4, average degree of the network and emergence of giant connected component is investigated for Rössler oscillators. Finally, Sec. 5 provides conclusions of our findings. 2. Mathematical framework We consider N (≥ 2) independent oscillatory dynamical systems such that they are randomly distributed initially in their d-dimensional phase space. The state of each of these i (i = 1, 2, 3, ..., N ) oscillators at any time t is depicted by x˙ i = F (xi ), where F : Rd → Rd describes the vector field connected with the d-dimensional vector xi (t ) = [x1i (t ), x2i (t ), · · · , xdi (t )] T of dynamical variables for the i-th oscillator, T being used to denote the transpose of a matrix. We next assign a communication radius of fixed length r with each of these oscillators. Then an oscillator is coupled bi-directionally with any other oscillator if and only if they lie within the volume of the oscillators’ communication closed ball of dimension d. Depending on such interaction strategy over time, the dynamical equation of each agent can be written as
x˙ i = F (xi ) − k
N
j =1
g i j (t ) H (x j ), i = 1, 2, ..., N .
(1)
Here, k is the coupling constant reckoning as the strength of the interaction among those oscillators. H (x j ) is the vector coupling function for diffusive type of interaction pointing out which variables of one oscillator are coupled with which variables of the others. The time-varying Laplacian matrix G (t ) = [ g i j (t )] N × N is the matrix describing the location of all systems and hence the connectivity pattern of the network at any time t with respect to the volume of the oscillator’s communication ball. Laplacian matrix is a symmetric (for undirected case), singular, positive semi-definite matrix defined as the difference between the corresponding degree matrix and the adjacency matrix. Particularly, g i j (t ) = g ji (t ) = −1 if both i-th and j-th systems lie in the communication ball of i-th oscillator (or j-th oscillator) at time t and otherwise zero. Since, any kind of self interaction is not allowed, so we assign g ii (t ) = h, where h is the number of neighbors of the i-th oscillator at time t. To explain the interactional platform, a two dimensional ([a, b] × [c , d]) schematic representation is depicted through Fig. 1.
Fig. 1. Schematic representation of a possible interaction scenario in a twodimensional plane for N = 9 interacting nodes.
In this figure, N = 9 oscillators are considered which are interacting based on their circular communication range. In top most left (A), oscillators 1 and 2 are interacting bi-directionally but oscillator 3 situated in the bottom left (C) is completely isolated with any other oscillator at that particular instant. Moreover, in (B), oscillators 4 and 6 are communicating with oscillator 5, but those oscillators 4 and 6 are not interacting as they are beyond uniform communication radius of each other. All the oscillators 7, 8, 9 in the bottom right (D) of the figure are interacting with each other. So, k and r are the two control parameters in our described model for fixed N number of oscillators. In the next sections, our main stress will be to detect the required critical parameter range for complete convergence of the attractors and also to analyze this process from the perspective of appearance of giant connected component in the network. For this, we will be examining two paradigmatic chaotic systems, namely Lorenz and Rössler oscillators one by one. For numerical simulations, we integrate Eq. (1) using fifth order Runge-Kutta-Fehlberg method with a fixed time step δt = 0.01. The below described all results are examined for different choices of step length δt like δt = 0.05 or 0.001, but our observed results remain unaffected due to such various choices of δt. Also, without any loss of generality, we assign a spherical communication range of radius r with each identical oscillators. The j coupling function is considered as H (x j ) = (0, x2 , 0) T . 3. Lorenz oscillators 3.1. Interplay between k and r It is anticipated that with increasing value of communication radius r, the probability of interaction among the oscillators gets enhanced. As a result of that, small coupling constant k is sufficient to attain global convergence of all the attractors for a fixed N number of oscillators. To illustrate this phenomenon, N = 100 identical Lorenz oscillators [48] are considered. The state dynamics of each Lorenz oscillator is represented by
⎛ ⎜
10(x2i − x1i )
⎞ ⎟
F (xi ) = ⎝ x1i (28 − x3i ) − x2i ⎠ , x1i x2i
−
(2)
8 i x 3 3
that exhibits chaotic behavior. In Fig. 2, a range of k ∈ [0, 1] and r ∈ [0, 5] have been considered and it is clearly discernible that the critical stability curve above which global convergence of the attractors (red region) is achieved, follows an inverse proportional relation between r and k. The parameter space in Fig. 2 depicts that with increasing r, the possibility of interaction between nodes increases and smaller k values are quite sufficient in order to make all the attractors converge into single one.
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Fig. 2. k − r parameter plane is represented for N = 100 identical Lorenz oscillators. White subspace is non-convergence region, where the red space is for global convergence of the attractors. If E < 10−5 , then that state is considered in the latter region, whereas whenever E ≥ 10−5 , that state belongs to the former one. To attain a smooth repercussion, 50 independent numerical results are accumulated. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 4. (a) Evolution of GCC with increasing time is portrayed. Here, N = 100 identical Lorenz oscillators are considered with communication radius r = 0.75 and coupling constant k = 0.5. In order to illustrate the appearance of converged attractor with time, three distinct time segments of the projection of the 3-D evolution of trajectories in 2D phase plane of the Lorenz attractor are depicted in (b) for t = 40, (c) for t = 1200 and (d) for t = 1800 respectively.
Fig. 3. Left panel: N − k parameter space is represented for identical Lorenz oscillators with k ∈ [0, 1.8] and N ∈ [2, 200]. Here, r = 1.0 is taken. Right panel: N − r parameter space is shown here for fixed k = 0.50 with N ∈ [2, 200] and r ∈ [0, 10.0].
In order to plot the above figure, we use the measurement of synchronization error usually defined as N
E=
i , j =1(i = j )
j
x1 −x1i
2
2
2 j j + x2 −x2i + x3 −x3i
( N −1 )
,
(3)
in terms of the standard Euclidean norm. After an initial transient of period 1900 times, if E is less than 10−5 , then that (k, r ) point is included in the convergence region, which is represented by red zone in the Fig. 2. Otherwise, that (k, r ) point is incorporated in the white non-convergence section. 3.2. Effect of network size on the convergence Till now, we have studied the impact of alteration in various system parameters, namely, the coupling strength k and the communication radius r while keeping fixed the number of oscillators N. Here, we are interested to examine the effect of N on the appearance of complete synchronization. For this, we fix the coupling radius r = 1.0 and numerically investigate the consequence of network size varying within the range [2, 200]. In Fig. 3(a), it is observed that comparatively lesser coupling strength k is sufficient for global convergence of attractors with increasing N for suitable fixed value of r. When the number of oscillators are larger in the phase space, randomly distributed nodes are more probable to connect to other nodes in the phase space, as the phase space becomes more congested. As a result of this increment in possible connection among the nodes, required coupling strength k reduces with increasing value of density of oscillators to observe global convergence of attractors. In our perceived investigation, the nodes move in the phase-space at every iteration step so that neighborhood of each node fluctuates extremely fast and due to their such typical movement mechanism, the nodes get the window of opportunity of
interacting with most of the other nodes during the entire period with suitable sufficiently large r. This fact plays a crucial role to achieve global convergence of the attractors at a lower coupling strength as the density increases. Fig. 3(b) also demonstrates this feature, where we inspect the correlation between N and r for fixed k = 0.50. Such an N-dependent threshold for attaining global convergence of attractors is noticed to have a decisive role in quorum-sensing transition in indirectly coupled systems [49–51], bacterial infection, biofilm formation and bioluminescence [52]. 3.3. Explanation of convergence of the attractors in terms of appearance of giant connected component Here we explain the process of convergence for Lorenz attractors in detail. In real undirected networks, we typically find that there is a large component (i.e. the giant connected component) that fills most of the network usually more than half while the rest of the network is divided into a large number of small components disconnected from the rest. In other word, a giant connected component (GCC) contains a finite fraction of nodes (more than 50%) of N. For our constructed model, at the time of convergence of the systems while treating each oscillator as a node and the coupling between them as edges, we observe that this giant component resulting in a single connected network. The normalized value of giant connected component NNG increases with increasing time, is illustrated with the Fig. 4. Here, N G describes the number of nodes present in the giant connected component. Initially N identical Lorenz oscillators are randomly distributed in the phase space where the chaotic attractors reside. Now, based on their own states, a sphere of radius r (the communication ball) is assigned with each of these oscillators, which are basically the interaction zones of the attractors. As a result of that, choosing j the coupling function H (x j ) = (0, x2 , 0) T , the attractors start communicating and with growing time if the strength of the coupling constant k is sufficient, we can anticipate complete convergence of all the attractors. Consequently if these attractors are considered as being the nodes of the network, then basically the network gets fully percolated having a single connected component and the
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number of attractors get diminished with evolving time. At the time of convergence, all the attractors will be converged to a single node following the same trajectory during the entire course of future iterations. To demonstrate this fact, Fig. 4 is presented here, where in Fig. 4(a), we started with N = 100 randomly distributed Lorenz oscillators with communication radius r = 0.750 and the interaction strength k = 0.5. Almost 0.04% of these nodes are connected at the beginning as per this specific snapshot. Here, NNG is the majority of connected nodes, normalized with respect to N, present in the network. As time increases, the network grows and this quantity NNG
gradually enlarges. This raise in the value of NNG with respect to increasing time t as in the curve of Fig. 4(a), correspond to partial convergence (cluster synchronization) of the attractors, which is clarified in the following. At about t ∼ 1600 in this snapshot with particular initial conditions, this quantity reaches its maximum value and all the attractors converge into a single one. Three separated distinct time segments is depicted in the x1i − x2i plane of the Lorenz attractor in the Figs. 4(b, c, d). At t = 40, we have numerous yellow points residing in the attractor, which is exemplified in the Fig. 4(b). This is because by this time the attractors have not got the opportunity of enough communication. While at t = 1200, most of the attractors have interacted and hence have converged to a few subsets, which is depicted in Fig. 4(c). Here all the attractors have not converged to a single attractor, but subsets of attractors have converged to each other within the subsets [53]. This scenario essentially signifies the appearance of partial convergence (cluster synchronization) of the attractors. But, as time increases further, all the attractors interact and the network gets fully connected. Then the synchronization error E becomes zero, i.e. the systems evolve identically following the same trajectory, all the attractors converge into a single attractor. This scenario is displayed with the help of the Fig. 4(d). 3.4. Linear stability analysis for the convergence In order to perform linear stability analysis of the state that corresponds to the global convergence of all the attractors, we go through the MSF approach in this subsection. We assume s = x1 = x2 = · · · = x N to be that state associated to the complete con= F (s) is an exact solution of Eq. (1). Let vergence so that ds dt
ηi (t ) = xi (t ) − s(t ) be the infinitesimal perturbations about the synchronous solution added onto each oscillator i. This leads to N × d variational equations corresponding to the Eq. (1) as,
η˙ i = D F (s)ηi − k
N
j =1
g i j (t ) D H (s)η j , i = 1, 2, ..., N
(4)
where D F (s) and D H (s) are the d × d Jacobian matrices of the corresponding vector functions evaluated . For our choice of ⎛ at s(t )⎞ 0 0 0 linear coupling function H (x), D H = ⎝ 0 1 0 ⎠ and the Jacobian 0 0 0 ⎞ ⎛ −10 10 0 matrix for Lorenz system, D F = ⎝ 28 − z −1 −x ⎠. y x − 83 Now, the negativity of the MSF [54–56], i.e., the largest Lyapunov exponent of the transverse manifold determined from Eq. (4) provides the necessary conditions for the stability of the synchronous solution. In that case, an infinitesimal perturbation from the converged state (i.e., synchronization) will diminish exponentially so that this solution becomes stable, at least when the oscillators are initialized in its vicinity. In Fig. 5, MSF and synchronization error E are plotted for N = 3 identical Lorenz oscillators with communication radius r = 10.0. It
Fig. 5. (a) MSF and (b) Synchronization error E as a function of coupling parameter k.
Fig. 6. Emergence of Giant Connected Component with increasing time in the dynamical network having Rössler oscillators as their unit dynamics. Here, N = 100, k = 0.07 and r = 5.
is noticed that at k = 0.9, E reduces to zero as well as MSF changes its sign from positive to negative, which is anticipated as then the attractors start following the same trajectory. 4. Rössler oscillators We next consider another paradigmatic chaotic Rössler oscillator [57] as the local dynamical system, where state dynamics of each individual is depicted by
⎛ ⎜
F ( xi ) = ⎝
−x2i − x3i x1i + 0.2x2i 0.2 +
x3i (x1i
⎞ ⎟ ⎠
(5)
− 5.7) j
along with the same coupling function H (x j ) = (0, x2 , 0) T , as considered earlier. In our work, time-varying interaction helps the system to interact with other nodes depending on communication radius r. A giant connected component is a necessary condition to settle down those attractors into a single one. As can be seen from Fig. 6, starting with a nominal value the normalized size NNG of GCC increases as time increases, where N = 100, k = 0.07, and r = 5 have been taken. This shows that the process of convergence of attractors is taking place for the above choice of parameters. In network science, networks can have certain attributes, which can be calculated to analyze the characteristics of the network. One
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Fig. 7. Inverse proportional relation between average degree D av g and synchronization error E is depicted. Here, N = 100, k = 0.02 and r = 5.
of the key network properties is network’s average degree. Network’s average degree D av g is defined as
D av g =
2L t N
=
N
i =1t
N
,
(6)
where L t represents the total number of links in the undirected network at any instant t and < di >t is the time-averaged degree of the i-th node. In our work, based on the sufficient uniform communication radius r, the number of links among the N nodes gradually increases. As a consequence of that, network’s average degree D av g is increased with respect to time t. When the attractors get fully converged with the help of the sufficient strength of the dissipative term k, this quantity D av g attains its maximum value N − 1. This implies that each node is connected to remaining N − 1 nodes. At the same time, E descends to zero as the convergence of attractors occur. Thus, each node follow the same path with respect to time t. Therefore, the control parameter r plays the crucial role for the emergence of completely connected network through the emergence of GCC. The other control parameter k helps to converge those attractors into a single one. In Fig. 7, we consider N = 100 identical Rössler oscillators, each with a communication radius r = 5. The value of D av g gradually increases and tends to ( N − 1) as the synchronization error E approaches to zero. Here, uniform interacting radius r = 5 is sufficient enough for the appearance of completely connected network. Convergence of attractors is taken place with the help of coupling constant k = 0.02. Thus, appropriate choices of both control parameters r and k are decisive otherwise procedure of convergence can not happen. As explained earlier, the converged state is stable when all the transverse Lyapunov exponents are negative. The Lyapunov exponents are calculated from expansion around the synchronous solution xi = s, using the Eq. (4). Here,
⎛
0 −1 D F = ⎝ 0 0.2 z 0
−1
⎞
⎠. 0 x − 5.7
With N = 3 and r = 1.0, it is observed from Fig. 8(a) that MSF changes its sign from positive to negative at k = 0.05, which is a necessary condition for convergence of attractors for a coupled oscillators’ network. To illustrate this specific characteristics, in lower panel of the Fig. 8(b), we plot the synchronization error E. With increasing coupling strength k, E diminishes to zero from its positive values at k = 0.05, which is exactly the same value where MSF exhibits its negative manifestation.
Fig. 8. Variations of (a) MSF and (b) synchronization error E as functions of coupling parameter k for Rössler system.
5. Conclusions It is well known that most of the existing studies in networks of interacting oscillatory systems contemplate with either completely static and percolated network formalism or temporal framework disregarding the local dynamical units’ intrinsic dynamics. Contrary to this, in this article, we have gone through a time-varying network connectivity pattern in which local dynamical systems interact only based on the closeness of their trajectories. The attractors thus interact only when they belong to the same communication ball, and consequently the connectivity changes with respect to time. As our principal observation, set of attractors converged into a single attractor owing to this scenario depending on proximity of the trajectories with suitable choice of the network parameters. Based on the uniform communication radius r for each attractor, the time-varying network is updated in each time-step. To achieve complete convergence, suitable choice of coupling strength k played a crucial role depending on r. In order to validate our proposition, we have chosen two paradigmatic systems, namely Lorenz and Rössler oscillators. The convergence of attractors (i.e., synchronization) phenomenon has been established using synchronization error index E and verified with the help of MSF approach. Important network properties like (i) average degree, and (ii) connectedness are studied from the point of view of the convergence of attractors. We observed gradual increment in average degree from a minimal non-zero value to ( N − 1) under appropriate choice of parameters and node number N during the transition of convergence. From disconnected components to a fully connected network through the emergence of largest giant connected component, is also detected during this transition. This work indicates the process of synchronization. The results present here are important for transient dynamics, particularly in biological systems [58] where studying transient dynamics is unavoidable. Acknowledgements SNC and DG were supported by Department of Science and Technology, Government of India (Project No. EMR/2016/001039).
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[32] M. Perc, Visualizing the attraction of strange attractors, Eur. J. Phys. 26 (2005) 579. [33] P.H.O. Silva, L.G. Nardo, S.A.M. Martins, E.G. Nepomuceno, M. Perc, Graphical interface as a teaching aid for nonlinear dynamical systems, Eur. J. Phys. 39 (2018) 065105. [34] A.-L. Barabási, Network Science, Cambridge University Press, 2016. [35] P. Arena, S. Baglio, L. Fortuna, G. Manganaro, Hyperchaos from cellular neural networks, Electron. Lett. 31 (1995) 250. [36] R. Caponetto, G. Dongola, L. Fortuna, A. Gallo, New results on the synthesis of FO-PID controllers, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 997. [37] A. Vicari, A. Ciraudo, C.D. Negro, A. Herault, L. Fortuna, Lava flow simulations using discharge rates from thermal infrared satellite imagery during the 2006 Etna eruption, Nat. Hazards 50 (2009) 539. [38] M. Schroder, M. Mannattil, D. Dutta, S. Chakraborty, M. Timme, Transient uncoupling induces synchronization, Phys. Rev. Lett. 115 (2015) 054101. [39] M. Schroder, M. Mannattil, M. Timme, S. Chakraborty, Synchronizing noisy nonidentical oscillators by transient uncoupling, Chaos 26 (2016) 094817. [40] A. Buscarino, L. Fortuna, M. Frasca, Experimental robust synchronization of hyperchaotic circuits, Phys. D: Nonlinear Phenom. 238 (2009) 1917. [41] K.E. Chlouverakis, J.C. Sprott, Chaotic hyperjerk systems, Chaos Solitons Fractals 28 (2006) 739. [42] S. Banerjee, D. Ghosh, A.R. Chowdhury, Multiplexing synchronization and its applications in cryptography, Phys. Scr. 78 (2008) 015010. [43] W. Yu, J. Cao, Adaptive QS (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks, Chaos 16 (2006) 023119. [44] U.S. Freitas, E.E.N. Macau, C. Grebogi, Using geometric control and chaotic synchronization to estimate an unknown model parameter, Phys. Rev. E 71 (2005) 047203. [45] D. Ghosh, S. Banerjee, A.R. Chowdhury, Synchronization between variable timedelayed systems and cryptography, Europhys. Lett. 80 (2007) 30006. [46] H.B. Fotsin, J. Daafouz, Adaptive synchronization of uncertain chaotic colpitts oscillators based on parameter identification, Phys. Lett. A 339 (2005) 304. [47] S. Banerjee, D. Ghosh, A. Ray, A. Roy Chowdhury, Synchronization between two different time-delayed systems and image encryption, Europhys. Lett. 81 (2007) 20006. [48] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130. [49] A. Camili, B.L. Bassler, Bacterial small-molecule signaling pathways, Science 311 (2006) 1113. [50] A.F. Taylor, M.R. Tinsley, F. Wang, Z. Huang, K. Showalter, Dynamical quorum sensing and synchronization in large populations of chemical oscillators, Science 323 (2009) 614. [51] S. Majhi, M. Perc, D. Ghosh, Chimera states in uncoupled neurons induced by a multilayer structure, Sci. Rep. 6 (2016) 39033. [52] C.D. Nadell, J. Xavier, S.A. Levin, K.R. Foster, The evolution of quorum sensing in bacterial biofilms, PLoS Comput. Biol. 6 (2008) e14. [53] L.M. Pecora, T.L. Carroll, Synchronization of chaotic systems, Chaos 25 (2015) 097611. [54] D.J. Stilwell, E.M. Bollt, D.G. Roberson, Sufficient conditions for fast switching synchronization in time-varying network topologies, SIAM J. Appl. Dyn. Syst. 5 (2006) 140. [55] L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80 (1998) 2109. [56] L. Huang, Q.-F. Chen, Y.-C. Lai, L.M. Pecora, Generic behavior of master-stability functions in coupled nonlinear dynamical systems, Phys. Rev. E 80 (2009) 036204. [57] O.E. Rössler, An equation for continuous chaos, Phys. Lett. A 57 (1976) 397. [58] A. Hastings, K.C. Abbott, K. Cuddington, T. Francis, G. Gellner, Y.-C. Lai, A. Morozov, S. Petrovskii, K. Scranton, M.L. Zeeman, Transient phenomena in ecology, Science 361 (2018) 6406.
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Physics Letters A www.elsevier.com/locate/pla
Convergence of chaotic attractors due to interaction based on closeness Sayantan Nag Chowdhury a , Soumen Majhi a , Dibakar Ghosh a,∗ , Awadhesh Prasad b a b
Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700108, India Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
a r t i c l e
i n f o
Article history: Received 24 June 2019 Received in revised form 13 September 2019 Accepted 17 September 2019 Available online xxxx Communicated by M. Perc Keywords: Convergence of attractors Temporal network Giant connected component
a b s t r a c t Exploration of coherence phenomena in ensembles of interacting dynamical systems has been in the centre of research in social, physical, biological and technological systems for decades. But, in most of the studies, either completely percolated time- and space-static networks or temporal connectivities disregarding the systems’ own dynamics have been dealt with. In this work, we examine the correlation between structural and dynamical evolution in networks of interacting dynamical systems. We specifically demonstrate the scenario of convergence of a set of chaotic attractors into a single attractor as a result of sufficient interaction based on the closeness of their own states. We characterize this occurrence through different measures, and map the collective states in network parameters’ space. We further validate our proposition while exposing the whole scenario for different chaotic systems, namely Lorenz and Rössler oscillators. © 2019 Elsevier B.V. All rights reserved.
1. Introduction One of the fundamental concern in current inter-disciplinary research is to understand the correlations between network structure and network dynamics [1,2]. In spite of the inherent contrast, from interacting populations of animals to the cells of body, numerous biological, man-made and social framework can be represented as dynamical ensembles coupled via complex architectures [3]. As far as the existing literature on interacting dynamical systems is concerned, straightforward elementary approach in networks has mostly presumed a static network constitution. But in reality, such perspective of interaction is undemanding, because in real world, dynamic connectivity [4] is further significant and practical. To confront with this pragmatic constraint, synchronization [5] have been explored in temporal networks [6–22]. To reveal the intrinsic substantial physical mechanisms governing the transfer of information in a system of networking units, the process of synchronization is perhaps premier specimen due to its entanglement in aspect of collective organizations from microscopic interactions of unitary constituents. In each time, during synchronized state, the ingredients of a system manifest indistinguishable dynamical states. On the other hand, emergence of fascinating phenomena like chimera state [23,24] and amplitude death [25,26] have also been explored in such dynamic networks.
*
Corresponding author. E-mail address: [email protected] (D. Ghosh).
https://doi.org/10.1016/j.physleta.2019.125997 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Mostly in those investigated cases for time-varying networks, in each of the node of the network, the authors have allocated a dynamical system among which interactions took place while completely ignoring the states of the individual dynamical units. In contrast, we choose a framework where the systems interact entirely based on the closeness of their trajectories [27]. The connections among those units reflected through the adjacency matrix of the associated network, is thus updated in every time step depending on the accessibility to the near-by neighboring trajectories. As a result of this formalism, we have observed a throughout collapse of all the trajectories into a single attractor [28–33], depending on the network parameters. This sort of convergence of a set of attractors into a single attractor owing to their interaction is popularly known as the phenomenon of synchronization in the literature. In our network, nodes are gradually connected with adjacent nodes with proper choice of network parameters and exhibits a transition from heterogeneous degree distributed disconnected components to formation of connected components. Finally, they generate a clique when the system is in global synchrony. This brings contrast to the previously studied cases [6,8] of synchronization, where the underlying network is always random network [34]. We have gone through rigorous analysis of paradigmatic chaotic Lorenz and Rössler systems where we have witnessed the occurrence of the same phenomenon. We have been able to characterize the observations both through synchronization error and master stability function (MSF) approach. Moreover, we have com-
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pared the process of convergence of attractors to the appearance of a fully connected component in the network. The type of controlled interactional platform we have gone through, may be useful in robotic communication [11] as well as in wireless communication systems, where transmission is possible only within a confined neighborhood [38,39]. Our results suggest that the appropriate strategy of externally imposing control [35–37] on the agents’ interaction helps to converge the systems into a single attractor. It has been observed that synchronization with chaotic and hyperchaotic systems [40,41] has effective applications in cryptography [42–47] but most of that cases requires high implementation costs. Our scheme may be beneficial in generation of synchronization from the perspective of optimal cost too. Our imposed restriction on the system’s interaction in the entire phase space is the key feature and somewhat counter-intuitive to the earlier perceived studies. Our results may help to achieve maximum communication efficiency through limited accessibility [17]. The remaining parts of the paper are assembled as follows. In Sec. 2, we describe mathematical framework of our time varying model where the network topology varies due to the intrinsic dynamics of the individual dynamical units. In Sec. 3, we explore the convergence of attractors numerically for Lorenz oscillators. Also, we examine the connectedness of the network for suitable choice of parameters. Stability of the converging manifold is established with the help of MSF approach. In Sec. 4, average degree of the network and emergence of giant connected component is investigated for Rössler oscillators. Finally, Sec. 5 provides conclusions of our findings. 2. Mathematical framework We consider N (≥ 2) independent oscillatory dynamical systems such that they are randomly distributed initially in their d-dimensional phase space. The state of each of these i (i = 1, 2, 3, ..., N ) oscillators at any time t is depicted by x˙ i = F (xi ), where F : Rd → Rd describes the vector field connected with the d-dimensional vector xi (t ) = [x1i (t ), x2i (t ), · · · , xdi (t )] T of dynamical variables for the i-th oscillator, T being used to denote the transpose of a matrix. We next assign a communication radius of fixed length r with each of these oscillators. Then an oscillator is coupled bi-directionally with any other oscillator if and only if they lie within the volume of the oscillators’ communication closed ball of dimension d. Depending on such interaction strategy over time, the dynamical equation of each agent can be written as
x˙ i = F (xi ) − k
N
j =1
g i j (t ) H (x j ), i = 1, 2, ..., N .
(1)
Here, k is the coupling constant reckoning as the strength of the interaction among those oscillators. H (x j ) is the vector coupling function for diffusive type of interaction pointing out which variables of one oscillator are coupled with which variables of the others. The time-varying Laplacian matrix G (t ) = [ g i j (t )] N × N is the matrix describing the location of all systems and hence the connectivity pattern of the network at any time t with respect to the volume of the oscillator’s communication ball. Laplacian matrix is a symmetric (for undirected case), singular, positive semi-definite matrix defined as the difference between the corresponding degree matrix and the adjacency matrix. Particularly, g i j (t ) = g ji (t ) = −1 if both i-th and j-th systems lie in the communication ball of i-th oscillator (or j-th oscillator) at time t and otherwise zero. Since, any kind of self interaction is not allowed, so we assign g ii (t ) = h, where h is the number of neighbors of the i-th oscillator at time t. To explain the interactional platform, a two dimensional ([a, b] × [c , d]) schematic representation is depicted through Fig. 1.
Fig. 1. Schematic representation of a possible interaction scenario in a twodimensional plane for N = 9 interacting nodes.
In this figure, N = 9 oscillators are considered which are interacting based on their circular communication range. In top most left (A), oscillators 1 and 2 are interacting bi-directionally but oscillator 3 situated in the bottom left (C) is completely isolated with any other oscillator at that particular instant. Moreover, in (B), oscillators 4 and 6 are communicating with oscillator 5, but those oscillators 4 and 6 are not interacting as they are beyond uniform communication radius of each other. All the oscillators 7, 8, 9 in the bottom right (D) of the figure are interacting with each other. So, k and r are the two control parameters in our described model for fixed N number of oscillators. In the next sections, our main stress will be to detect the required critical parameter range for complete convergence of the attractors and also to analyze this process from the perspective of appearance of giant connected component in the network. For this, we will be examining two paradigmatic chaotic systems, namely Lorenz and Rössler oscillators one by one. For numerical simulations, we integrate Eq. (1) using fifth order Runge-Kutta-Fehlberg method with a fixed time step δt = 0.01. The below described all results are examined for different choices of step length δt like δt = 0.05 or 0.001, but our observed results remain unaffected due to such various choices of δt. Also, without any loss of generality, we assign a spherical communication range of radius r with each identical oscillators. The j coupling function is considered as H (x j ) = (0, x2 , 0) T . 3. Lorenz oscillators 3.1. Interplay between k and r It is anticipated that with increasing value of communication radius r, the probability of interaction among the oscillators gets enhanced. As a result of that, small coupling constant k is sufficient to attain global convergence of all the attractors for a fixed N number of oscillators. To illustrate this phenomenon, N = 100 identical Lorenz oscillators [48] are considered. The state dynamics of each Lorenz oscillator is represented by
⎛ ⎜
10(x2i − x1i )
⎞ ⎟
F (xi ) = ⎝ x1i (28 − x3i ) − x2i ⎠ , x1i x2i
−
(2)
8 i x 3 3
that exhibits chaotic behavior. In Fig. 2, a range of k ∈ [0, 1] and r ∈ [0, 5] have been considered and it is clearly discernible that the critical stability curve above which global convergence of the attractors (red region) is achieved, follows an inverse proportional relation between r and k. The parameter space in Fig. 2 depicts that with increasing r, the possibility of interaction between nodes increases and smaller k values are quite sufficient in order to make all the attractors converge into single one.
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Fig. 2. k − r parameter plane is represented for N = 100 identical Lorenz oscillators. White subspace is non-convergence region, where the red space is for global convergence of the attractors. If E < 10−5 , then that state is considered in the latter region, whereas whenever E ≥ 10−5 , that state belongs to the former one. To attain a smooth repercussion, 50 independent numerical results are accumulated. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 4. (a) Evolution of GCC with increasing time is portrayed. Here, N = 100 identical Lorenz oscillators are considered with communication radius r = 0.75 and coupling constant k = 0.5. In order to illustrate the appearance of converged attractor with time, three distinct time segments of the projection of the 3-D evolution of trajectories in 2D phase plane of the Lorenz attractor are depicted in (b) for t = 40, (c) for t = 1200 and (d) for t = 1800 respectively.
Fig. 3. Left panel: N − k parameter space is represented for identical Lorenz oscillators with k ∈ [0, 1.8] and N ∈ [2, 200]. Here, r = 1.0 is taken. Right panel: N − r parameter space is shown here for fixed k = 0.50 with N ∈ [2, 200] and r ∈ [0, 10.0].
In order to plot the above figure, we use the measurement of synchronization error usually defined as N
E=
i , j =1(i = j )
j
x1 −x1i
2
2
2 j j + x2 −x2i + x3 −x3i
( N −1 )
,
(3)
in terms of the standard Euclidean norm. After an initial transient of period 1900 times, if E is less than 10−5 , then that (k, r ) point is included in the convergence region, which is represented by red zone in the Fig. 2. Otherwise, that (k, r ) point is incorporated in the white non-convergence section. 3.2. Effect of network size on the convergence Till now, we have studied the impact of alteration in various system parameters, namely, the coupling strength k and the communication radius r while keeping fixed the number of oscillators N. Here, we are interested to examine the effect of N on the appearance of complete synchronization. For this, we fix the coupling radius r = 1.0 and numerically investigate the consequence of network size varying within the range [2, 200]. In Fig. 3(a), it is observed that comparatively lesser coupling strength k is sufficient for global convergence of attractors with increasing N for suitable fixed value of r. When the number of oscillators are larger in the phase space, randomly distributed nodes are more probable to connect to other nodes in the phase space, as the phase space becomes more congested. As a result of this increment in possible connection among the nodes, required coupling strength k reduces with increasing value of density of oscillators to observe global convergence of attractors. In our perceived investigation, the nodes move in the phase-space at every iteration step so that neighborhood of each node fluctuates extremely fast and due to their such typical movement mechanism, the nodes get the window of opportunity of
interacting with most of the other nodes during the entire period with suitable sufficiently large r. This fact plays a crucial role to achieve global convergence of the attractors at a lower coupling strength as the density increases. Fig. 3(b) also demonstrates this feature, where we inspect the correlation between N and r for fixed k = 0.50. Such an N-dependent threshold for attaining global convergence of attractors is noticed to have a decisive role in quorum-sensing transition in indirectly coupled systems [49–51], bacterial infection, biofilm formation and bioluminescence [52]. 3.3. Explanation of convergence of the attractors in terms of appearance of giant connected component Here we explain the process of convergence for Lorenz attractors in detail. In real undirected networks, we typically find that there is a large component (i.e. the giant connected component) that fills most of the network usually more than half while the rest of the network is divided into a large number of small components disconnected from the rest. In other word, a giant connected component (GCC) contains a finite fraction of nodes (more than 50%) of N. For our constructed model, at the time of convergence of the systems while treating each oscillator as a node and the coupling between them as edges, we observe that this giant component resulting in a single connected network. The normalized value of giant connected component NNG increases with increasing time, is illustrated with the Fig. 4. Here, N G describes the number of nodes present in the giant connected component. Initially N identical Lorenz oscillators are randomly distributed in the phase space where the chaotic attractors reside. Now, based on their own states, a sphere of radius r (the communication ball) is assigned with each of these oscillators, which are basically the interaction zones of the attractors. As a result of that, choosing j the coupling function H (x j ) = (0, x2 , 0) T , the attractors start communicating and with growing time if the strength of the coupling constant k is sufficient, we can anticipate complete convergence of all the attractors. Consequently if these attractors are considered as being the nodes of the network, then basically the network gets fully percolated having a single connected component and the
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number of attractors get diminished with evolving time. At the time of convergence, all the attractors will be converged to a single node following the same trajectory during the entire course of future iterations. To demonstrate this fact, Fig. 4 is presented here, where in Fig. 4(a), we started with N = 100 randomly distributed Lorenz oscillators with communication radius r = 0.750 and the interaction strength k = 0.5. Almost 0.04% of these nodes are connected at the beginning as per this specific snapshot. Here, NNG is the majority of connected nodes, normalized with respect to N, present in the network. As time increases, the network grows and this quantity NNG
gradually enlarges. This raise in the value of NNG with respect to increasing time t as in the curve of Fig. 4(a), correspond to partial convergence (cluster synchronization) of the attractors, which is clarified in the following. At about t ∼ 1600 in this snapshot with particular initial conditions, this quantity reaches its maximum value and all the attractors converge into a single one. Three separated distinct time segments is depicted in the x1i − x2i plane of the Lorenz attractor in the Figs. 4(b, c, d). At t = 40, we have numerous yellow points residing in the attractor, which is exemplified in the Fig. 4(b). This is because by this time the attractors have not got the opportunity of enough communication. While at t = 1200, most of the attractors have interacted and hence have converged to a few subsets, which is depicted in Fig. 4(c). Here all the attractors have not converged to a single attractor, but subsets of attractors have converged to each other within the subsets [53]. This scenario essentially signifies the appearance of partial convergence (cluster synchronization) of the attractors. But, as time increases further, all the attractors interact and the network gets fully connected. Then the synchronization error E becomes zero, i.e. the systems evolve identically following the same trajectory, all the attractors converge into a single attractor. This scenario is displayed with the help of the Fig. 4(d). 3.4. Linear stability analysis for the convergence In order to perform linear stability analysis of the state that corresponds to the global convergence of all the attractors, we go through the MSF approach in this subsection. We assume s = x1 = x2 = · · · = x N to be that state associated to the complete con= F (s) is an exact solution of Eq. (1). Let vergence so that ds dt
ηi (t ) = xi (t ) − s(t ) be the infinitesimal perturbations about the synchronous solution added onto each oscillator i. This leads to N × d variational equations corresponding to the Eq. (1) as,
η˙ i = D F (s)ηi − k
N
j =1
g i j (t ) D H (s)η j , i = 1, 2, ..., N
(4)
where D F (s) and D H (s) are the d × d Jacobian matrices of the corresponding vector functions evaluated . For our choice of ⎛ at s(t )⎞ 0 0 0 linear coupling function H (x), D H = ⎝ 0 1 0 ⎠ and the Jacobian 0 0 0 ⎞ ⎛ −10 10 0 matrix for Lorenz system, D F = ⎝ 28 − z −1 −x ⎠. y x − 83 Now, the negativity of the MSF [54–56], i.e., the largest Lyapunov exponent of the transverse manifold determined from Eq. (4) provides the necessary conditions for the stability of the synchronous solution. In that case, an infinitesimal perturbation from the converged state (i.e., synchronization) will diminish exponentially so that this solution becomes stable, at least when the oscillators are initialized in its vicinity. In Fig. 5, MSF and synchronization error E are plotted for N = 3 identical Lorenz oscillators with communication radius r = 10.0. It
Fig. 5. (a) MSF and (b) Synchronization error E as a function of coupling parameter k.
Fig. 6. Emergence of Giant Connected Component with increasing time in the dynamical network having Rössler oscillators as their unit dynamics. Here, N = 100, k = 0.07 and r = 5.
is noticed that at k = 0.9, E reduces to zero as well as MSF changes its sign from positive to negative, which is anticipated as then the attractors start following the same trajectory. 4. Rössler oscillators We next consider another paradigmatic chaotic Rössler oscillator [57] as the local dynamical system, where state dynamics of each individual is depicted by
⎛ ⎜
F ( xi ) = ⎝
−x2i − x3i x1i + 0.2x2i 0.2 +
x3i (x1i
⎞ ⎟ ⎠
(5)
− 5.7) j
along with the same coupling function H (x j ) = (0, x2 , 0) T , as considered earlier. In our work, time-varying interaction helps the system to interact with other nodes depending on communication radius r. A giant connected component is a necessary condition to settle down those attractors into a single one. As can be seen from Fig. 6, starting with a nominal value the normalized size NNG of GCC increases as time increases, where N = 100, k = 0.07, and r = 5 have been taken. This shows that the process of convergence of attractors is taking place for the above choice of parameters. In network science, networks can have certain attributes, which can be calculated to analyze the characteristics of the network. One
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Fig. 7. Inverse proportional relation between average degree D av g and synchronization error E is depicted. Here, N = 100, k = 0.02 and r = 5.
of the key network properties is network’s average degree. Network’s average degree D av g is defined as
D av g =
2L t N
=
N
i =1
N
,
(6)
where L t represents the total number of links in the undirected network at any instant t and < di >t is the time-averaged degree of the i-th node. In our work, based on the sufficient uniform communication radius r, the number of links among the N nodes gradually increases. As a consequence of that, network’s average degree D av g is increased with respect to time t. When the attractors get fully converged with the help of the sufficient strength of the dissipative term k, this quantity D av g attains its maximum value N − 1. This implies that each node is connected to remaining N − 1 nodes. At the same time, E descends to zero as the convergence of attractors occur. Thus, each node follow the same path with respect to time t. Therefore, the control parameter r plays the crucial role for the emergence of completely connected network through the emergence of GCC. The other control parameter k helps to converge those attractors into a single one. In Fig. 7, we consider N = 100 identical Rössler oscillators, each with a communication radius r = 5. The value of D av g gradually increases and tends to ( N − 1) as the synchronization error E approaches to zero. Here, uniform interacting radius r = 5 is sufficient enough for the appearance of completely connected network. Convergence of attractors is taken place with the help of coupling constant k = 0.02. Thus, appropriate choices of both control parameters r and k are decisive otherwise procedure of convergence can not happen. As explained earlier, the converged state is stable when all the transverse Lyapunov exponents are negative. The Lyapunov exponents are calculated from expansion around the synchronous solution xi = s, using the Eq. (4). Here,
⎛
0 −1 D F = ⎝ 0 0.2 z 0
−1
⎞
⎠. 0 x − 5.7
With N = 3 and r = 1.0, it is observed from Fig. 8(a) that MSF changes its sign from positive to negative at k = 0.05, which is a necessary condition for convergence of attractors for a coupled oscillators’ network. To illustrate this specific characteristics, in lower panel of the Fig. 8(b), we plot the synchronization error E. With increasing coupling strength k, E diminishes to zero from its positive values at k = 0.05, which is exactly the same value where MSF exhibits its negative manifestation.
Fig. 8. Variations of (a) MSF and (b) synchronization error E as functions of coupling parameter k for Rössler system.
5. Conclusions It is well known that most of the existing studies in networks of interacting oscillatory systems contemplate with either completely static and percolated network formalism or temporal framework disregarding the local dynamical units’ intrinsic dynamics. Contrary to this, in this article, we have gone through a time-varying network connectivity pattern in which local dynamical systems interact only based on the closeness of their trajectories. The attractors thus interact only when they belong to the same communication ball, and consequently the connectivity changes with respect to time. As our principal observation, set of attractors converged into a single attractor owing to this scenario depending on proximity of the trajectories with suitable choice of the network parameters. Based on the uniform communication radius r for each attractor, the time-varying network is updated in each time-step. To achieve complete convergence, suitable choice of coupling strength k played a crucial role depending on r. In order to validate our proposition, we have chosen two paradigmatic systems, namely Lorenz and Rössler oscillators. The convergence of attractors (i.e., synchronization) phenomenon has been established using synchronization error index E and verified with the help of MSF approach. Important network properties like (i) average degree, and (ii) connectedness are studied from the point of view of the convergence of attractors. We observed gradual increment in average degree from a minimal non-zero value to ( N − 1) under appropriate choice of parameters and node number N during the transition of convergence. From disconnected components to a fully connected network through the emergence of largest giant connected component, is also detected during this transition. This work indicates the process of synchronization. The results present here are important for transient dynamics, particularly in biological systems [58] where studying transient dynamics is unavoidable. Acknowledgements SNC and DG were supported by Department of Science and Technology, Government of India (Project No. EMR/2016/001039).
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