We analyze the stability of synchronized state for coupled nearly identical dynamical systems on networks by deriving an approximate master stability function (MSF). Using this MSF we treat the problem of designing a network having the best synchronizability properties. We find that the edges which connect nodes with a larger relative parameter mismatch are preferred. Also, the nodes having values at one extreme of the parameter mismatch are preferred as hubs where the extreme is the one which gives a better stability according to the MSF curve.

We study the generalized synchronization and its stability using master stability function (MSF), in a network of coupled nearly identical dynamical systems. We extend the MSF approach for the case of degenerate eigenvalues of the coupling matrix. Using the MSF we study the size-instability in star and ring networks for coupled nearly identical dynamical systems. In the star network of coupled Rössler systems we show that the critical size beyond which synchronization is unstable, can be increased by having a larger frequency for the central node of the star. For the ring network we show that the critical size is not significantly affected by parameter variations. The results are verified by explicit numerical calculations.

Abstract. In this paper we briefly report some recent developments on generalized synchronization. We discuss different methods of detecting generalized synchronization. We first consider two unidirectionally coupled systems and then two mutually coupled systems. We then extend the study to a network of coupled systems. In the study of generalized synchronization of coupled nonidentical systems we discuss the Master Stability Function (MSF) formalism for coupled nearly identical systems. Later we use this MSF to construct synchronized optimized networks. In the optimized networks the nodes which have parameter value at one extreme are chosen as hubs and the pair of nodes with larger difference in parameter are chosen to create links.

We study the mechanisms of frequency synchronized cluster formation in coupled non-identical oscillators and investigate the impact of presence of a leader on the cluster synchronization. We find that the introduction of a leader, node having large parameter mismatch, induces a profound change in the cluster pattern as well as in the mechanism of the cluster formation. The emergence of a leader generates a transition from the driven to the mixed cluster state. The frequency mismatch turns out to be responsible for this transition. Additionally, for a chaotic evolution, the driven mechanism stands as a primary mechanism for the cluster formation, whereas for a periodic evolution the self-organization mechanism becomes equally responsible. The interaction between individual units of a system leads to many emerging behaviors, among which synchronization is one of the most fascinating phenomenon which has been gaining tremendous attention since the first experimental demonstration of the phenomena by Christian Huygens [1]. The unexpected sway and twist of the Millennium bridge, synchronization of the neurons in the brain and synchronous fire flies [2][3][4] are few examples of the synchronization in the real world systems. Synchronization is defined as an emergence of some relation between the functional of two processes due to interaction [3]. Earlier studies on coupled non-identical oscillators have shown that the exact synchronization is hard to achieve when there is a parameter mismatch in the local dynamics [3], rather they exhibit phase or generalized synchronization [5,6]. Additionally, there exists a nontrivial transition to the global phase synchronization in a population of globally coupled chaotic non-identical oscillators [7]. Moreover, cluster pattern and frequency of the nodes in a cluster have been shown to be controlled by local external forcing as well as by changing the network architecture [8]. Furthermore, the coupled oscillators with heterogeneous coupling have been reported to exhibit synchronization triggered by the oscillators having strong couplings, further facilitating the synchronization among the nodes having weak coupling [9].We present results of cluster synchronization due to its importance and occurrence in various real world systems represented in terms of interacting units [10]. We study mechanisms of formation of frequency synchronized clusters in coupled non-identical oscillators and investigate the influence of a leader on the dynamical evolution of other nodes. One possible way of defining a leader is to make the natural frequency of a node much higher than * sarika@iiti.ac.in that of other nodes in the network [11,12]. This is one of the traditional way to define a leader in the coupled dynamics on network models. Some other ways of defining leaders are those which depend upon the application and motivation of the problem, e. g., Ref.[13] considers a leader which exchanges information with its neighbors as well as has access to its own state. In Ref.[14] the neuron which fires...

Will a large complex system be stable? This question, first posed by May in 1972, captures a long standing challenge, fueled by a seeming contradiction between theory and practice. While empirical reality answers with an astounding yes, the mathematical analysis, based on linear stability theory, seems to suggest the contraryhence, the diversity-stability paradox. Here we settle this dichotomy, by considering the interplay between topology and dynamics. We show that this interplay leads to the emergence of non-random patterns in the system's stability matrix, leading us to relinquish the prevailing random matrix-based paradigm. Instead, we offer a new matrix ensemble, which captures the dynamic stability of real-world systems. This ensemble helps us analytically identify the relevant control parameters that predict a system's stability, exposing three broad dynamic classes: In the asymptotically unstable class, diversity, indeed, leads to instability à la May's paradox. However, we also expose an asymptotically stable class, the class in which most real systems reside, in which diversity not only does not prohibit, but, in fact, enhances dynamic stability. Finally, in the sensitively stable class diversity plays no role, and hence stability is driven by the system's microscopic parameters. Together, our theory uncovers the naturally emerging rules of complex system stability, helping us reconcile the paradox that has eluded us for decades.-all naturally emerging features of real-world networks -indeed, impact the system's dynamics [25][26][27][28][29][30] . However, on their own, these topological features are insufficient to ensure stability 31,32 , leaving May's mathematical challenge unresolved.Here, we show that the desired devious strategies arise not just from the network topology, but mainly from its interplay with the system's intrinsic, often nonlinear, interaction dynamics. Constrained by the physical mechanisms that drive the system's interacting components, these dynamics provide the desired, naturally emerging, design principles that determine the system's stability. Our analysis shows that, under real-world dynamics, stability can be fully predicted from a small set of relevant parameters that capture the combined contribution of both topology and dynamics. Most crucially, we identify a broad class of frequently encountered mechanisms, for which a large complex system can, and, at times, even must be stable, therefore settling the long overdue diversity-stability debate 33,34 .