July 8, 2020 12:38 WSPC/S0218-1274 2050123 International Journal of Bifurcation and Chaos, Vol. 30, No. 8 (2020) 2050123 (8 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127420501230 Is There a Relation Between Synchronization Stability and Bifurcation Type? Zahra Faghani Department of Biomedical Engineering, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran Zhen Wang Shaanxi Engineering Research Center of Controllable Neutron Source, School of Science, Xijing University, Xi’an 710123, P. R. China Fatemeh Parastesh and Sajad Jafari Department of Biomedical Engineering, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran MatjaˇzPerc Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇska cesta 160, 2000 Maribor, Slovenia Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Complexity Science Hub Vienna, Josefst¨adterstraße 39, 1080 Vienna, Austria [email protected] [email protected] Received November 11, 2019 Int. J. Bifurcation Chaos 2020.30. Downloaded from www.worldscientific.com Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation by CITY UNIVERSITY OF HONG KONG on 07/15/20. Re-use and distribution is strictly not permitted, except for Open Access articles. diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others. Keywords: Synchronization; master stability function; bifurcation diagram; chaos. 2050123-1 July 8, 2020 12:38 WSPC/S0218-1274 2050123 Z. Faghani et al. 1. Introduction collective behaviors to the local dynamics of the units of the network, raises this question: Is there For studying the complex networks, the main any relation between the stability of the synchro- specifications for describing the network are the nized behavior of the elements and their dynamics? microscale level (node dynamics) and the macro- To find the answer to this question, in this paper, scale level (the whole network characteristics) [Are- we calculate the MSF for various values of a system nas et al., 2006]. One of the most popular properties parameter that can be considered as a bifurcation at the macroscale level which is attracting the sci- parameter. Then the obtained conditions from MSF entists increasingly is the collective behavior of the solution are compared with the bifurcation diagram. complex networks [Chavez et al., 2005; Boccaletti This process is done for four different systems with et al., 2006]. One of the hottest topics in recent different bifurcation diagrams. years is the synchronized collective behavior in con- The rest of the paper is organized as follows. trol and systems, inspired in many applications In Sec. 2, the master stability function method is such as physics, biology, and engineering [Math- described. The results of calculating the MSF for eny et al., 2014; Totz et al., 2015; Bolhasani et al., four systems are presented in Sec. 3. The conclu- 2017; Chowdhury et al., 2019; Parastesh et al., 2019; sions are given in Sec. 4. Li et al., 2019a]. Synchronization is an emergent phenomenon, resulting from the interaction of two or more systems [Boccaletti et al., 2002; Pikovsky 2. Master Stability Function , , et al. 2003; Arenas et al. 2008]. But there has been In order to calculate the MSF, first consider that the a larger interest in the synchronization study of describing equation of the network can be written networks composed of a large number of elements. as follows: An important subject in this area is to determine the stability of the synchronization state [Xie & X˙ i = F (Xi)+σ GijH(Xj), (1) Chen, 1996; Gao et al., 2006; Russo & Di Bernardo, j , , 2009; Rakshit et al. 2017; Li et al. 2019b]. In 1994, where Xi is the m-dimensional vector of the dynam- Pecora and Carroll proposed the “Master Stability ical variable of the ith oscillator, F (Xi)isthe Function” (MSF) method which gives the necessary uncoupled dynamical system, and H is the coupling conditions for the linear stability of the synchroniza- function which determines which node’s variable is tion manifold in a network of identical oscillators used in the coupling. In this paper, we consider [Pecora & Carroll, 1998, 1999]. H =[100;000;000].σ is the cou- The local dynamics of the network elements pling strength and G represents the topology of the is an important factor in the emergent collective connectivity in the network. behavior. Many of the scientists have concentrated Then the relevant variational equation of on the synchronization of chaotic systems with an Eq. (1) is determined as follows: Int. J. Bifurcation Chaos 2020.30. Downloaded from www.worldscientific.com emphasis on the large-scale and complex networks ξ˙ σγ ξ , [Banerjee et al., 2017; Chen et al., 2018; Li et al., k =[DF + kDH ] k (2) 2019b]. The first chaotic system was discovered where ξk is the variations on the ith node, and γk in 1963 by Lorenz who found a three-dimensional is an eigenvalue of G, k =0, 1,...,N.Notethatin autonomous system with chaotic attractor during Sec. 3, the parameter d equals γmax × σ.Themax- his studies in atmospheric convection [Sparrow, imum Lyapunov exponent of Eq. (2) is the MSF. If 2012]. After that, R¨ossler [R¨ossler, 1976] proposed MSF is negative, the network has a stable synchro- by CITY UNIVERSITY OF HONG KONG on 07/15/20. Re-use and distribution is strictly not permitted, except for Open Access articles. another three-dimensional chaotic system, simpler nized state and if it is positive, the synchronization than Lorenz equations. Discoveries of the chaotic is unstable. systems led to a huge flow of research into the anal- ysis of these systems. Some basic tools in this field are the bifurcation diagram and the Lyapunov spec- 3. Results trum. These diagrams can exhibit different dynam- To investigate the relationship between system ics of the system and dynamical transitions with dynamics and synchronization stability of networks, respect to a system parameter [Qi et al., 2005]. we compare the bifurcation diagram of the system The dependence of the system’s dynamics to and the MSF diagram in four different systems. The the parameters’ values and also the relation of the MSF is calculated for different d values with respect 2050123-2 July 8, 2020 12:38 WSPC/S0218-1274 2050123 Is There a Relation Between Synchronization Stability and Bifurcation Type? to the bifurcation parameter. Therefore, a 2D dia- To obtain the MSF diagram, the variational gram is presented for the MSF solution. The results equations for the R¨ossler system, defined in Eq. (3), of each system are presented in the following. can be described as follows: ξ˙1 = −dξ1 − ξ2 − ξ3, 3.1. R¨ossler ξ˙ ξ aξ , The R¨ossler system is defined by three differential 2 = 1 + 2 (4) equationsasfollows[Yu, 1997]: ξ˙3 = zξ1 +(x − c)ξ3. x˙ = −y − z, Figure 1(a) shows the bifurcation diagram of y˙ = x + ay, (3) the R¨ossler system according to the parameter a. The system has a period-doubling route to chaos. z b z x − c , ˙ = + ( ) For plotting the bifurcation diagram, we have used where b =0.6, c = 6 and parameter a is considered the initial conditions with forward continuation. as the bifurcation parameter in a =[0, 0.4]. Figure 1(b) demonstrates the MSF diagram. The Int. J. Bifurcation Chaos 2020.30. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 07/15/20. Re-use and distribution is strictly not permitted, except for Open Access articles. Fig. 1. (a) The bifurcation diagram of R¨ossler system according to parameter a. (b) The MSF diagram, in the (a, d)plane. The black curve in (b) shows where MSF equals to zero. The vertical lines connect the bifurcation points to their corresponding points in the MSF diagram. 2050123-3 July 8, 2020 12:38 WSPC/S0218-1274 2050123 Z. Faghani et al. y-axis describes d,andx-axis describes the bifurca- 3.2. Hindmarsh–Rose tion parameter a. The black line shows where the The Hindmarsh–Rose neuron model is described by MSF equals to zero. The space above the black line Eq. (5) [Hindmarsh & Rose, 1984]: is MSF > 0, where the network is asynchronized < and the space below the black line is MSF 0, x˙ = y − x3 + ax2 + I − z, where the network is synchronized. Generally, as the bifurcation parameter a grows, the black line y˙ = b + cx2 − y, (5) has a downward trend. It means that for higher val- ues of a, the network is synchronized for the lower z˙ = r(s(x − x1) − z), value of d. The trend is smooth and there are no sharp jumps in the line. Therefore, the changes in where a =3,b =1,c = −5, s =4,x1 =1.6, I = the dynamics of the elements have no effect on syn- 3.25 and r =[0, 0.5] is considered as a bifurcation chronization stability.