Stability Of Synchronous State Research Articles

Periodic systems have new classes of synchronization stability

The master stability function is a robust and useful tool for determining the conditions of synchronization stability in a network of coupled systems. While a comprehensive classification exists in the case in which the nodes are chaotic dynamical systems, its application to periodic systems has been less explored. By studying several well-known periodic systems, we establish a comprehensive framework to understand and classify their properties of synchronizability. This allows us to define five distinct classes of synchronization stability, including some that are unique to periodic systems. Specifically, in periodic systems, the master stability function vanishes at the origin, and it can therefore display behavioral classes that are not achievable in chaotic systems, where it starts, instead, at a strictly positive value. Moreover, our results challenge the widely held belief that periodic systems are easily put in a stable synchronous state, showing, instead, the common occurrence of a lower threshold for synchronization stability. Published by the American Physical Society 2024

Synchronization and stability of a vibrating system with two rigid frames driven by two groups of coaxial rotating exciters

This article explores the synchronization, stability and motion characteristics of the generalized dynamical model with two rigid frames (RFs) driven by two groups (even number) of coaxial rotating exciters. In light of this system’s generalized coordinates, we use Lagrange’s equations to derive the generalized differential equation of motion. The responses of absolute and relative motion of the generalized system are obtained using the transfer function method. The synchronization and stability criteria of multiple exciters are derived using the average method and Hamilton’s theory, respectively. Taking a dynamical model driven by two pairs of exciters as an example, the localized studies of generalized results are carried out. The stable synchronous solutions for phase differences and excitation frequency, the stability ability coefficient curves and the response curves are graphically presented considering the effect of two crucial dimensionless parameters on the stable synchronous states. The simulation results of the specific system are obtained using the fourth-order Runge-Kutta algorithms and compared with the numerical qualitative analysis results to reveal the high consistency between them and clarify the used methods’ effectiveness. The strength of this work stems from its use in the field of high power and large scale self-synchronization vibrating machines.

Global synchronization in generalized multilayer higher-order networks

Networks incorporating higher-order interactions are increasingly recognized for their ability to introduce novel dynamics into various processes, including synchronization. Previous studies on synchronization within multilayer networks have often been limited to specific models, such as the Kuramoto model, or have focused solely on higher-order interactions within individual layers. Here, we present a comprehensive framework for investigating synchronization, particularly global synchronization, in multilayer networks with higher-order interactions. Our framework considers interactions beyond pairwise connections, both within and across layers. We demonstrate the existence of a stable global synchronous state, with a condition resembling the master stability function, contingent on the choice of coupling functions. Our theoretical findings are supported by simulations using Hindmarsh-Rose neuronal and Rössler oscillators. These simulations illustrate how synchronization is facilitated by higher-order interactions, both within and across layers, highlighting the advantages over scenarios involving interactions within single layers. Published by the American Physical Society 2024

Multistable chimera states in a smallest population of three coupled oscillators.

We uncover the emergence of distinct sets of multistable chimera states in addition to chimera death and synchronized states in a smallest population of three globally coupled oscillators with mean-field diffusive coupling. Sequence of torus bifurcations result in the manifestation of distinct periodic orbits as a function of the coupling strength, which in turn result in the genesis of distinct chimera states constituted by two synchronized oscillators coexisting with an asynchronous oscillator. Two subsequent Hopf bifurcations result in homogeneous and inhomogeneous steady states resulting in desynchronized steady states and chimera death state among the coupled oscillators. The periodic orbits and the steady states lose their stability via a sequence of saddle-loop and saddle-node bifurcations finally resulting in a stable synchronized state. We have generalized these results to N coupled oscillators and also deduced the variational equationscorresponding to the perturbation transverse to the synchronization manifold and corroborated the synchronized state in the two-parameter phase diagrams using its largest eigenvalue. Chimera states in three coupled oscillators emerge as a solitary state in N coupled oscillator ensemble.

Stability and multistability of synchronization in networks of coupled phase oscillators

Coupled phase oscillators usually achieve synchronization as the coupling strength among oscillators is increased beyond a critical value. The stability of synchronous state remains an open issue. In this paper, we study the stability of the synchronous state in coupled phase oscillators. It is found that numerical integration of differential equations of coupled phase oscillators with a finite time step may induce desynchronization at strong couplings. The mechanism behind this instability is that numerical accumulated errors in simulations may trigger the loss of stability of the synchronous state. Desynchronization critical couplings are found to increase and diverge as a power law with decreasing the integral time step. Theoretical analysis supports the local stability of the synchronized state. Globally the emergence of synchronous state depends on the initial conditions. Other metastable ordered states such as twisted states can coexist with the synchronous mode. These twisted states keep locally stable on a sparse network but lose their stability when the network becomes dense.

Multistability and anomalies in oscillator models of lossy power grids

The analysis of dissipatively coupled oscillators is challenging and highly relevant in power grids. Standard mathematical methods are not applicable, due to the lack of network symmetry induced by dissipative couplings. Here we demonstrate a close correspondence between stable synchronous states in dissipatively coupled oscillators, and the winding partition of their state space, a geometric notion induced by the network topology. Leveraging this winding partition, we accompany this article with an algorithms to compute all synchronous solutions of complex networks of dissipatively coupled oscillators. These geometric and computational tools allow us to identify anomalous behaviors of lossy networked systems. Counterintuitively, we show that loop flows and dissipation can increase the system’s transfer capacity, and that dissipation can promote multistability. We apply our geometric framework to compute power flows on the IEEE RTS-96 test system, where we identify two high voltage solutions with distinct loop flows.

Asymmetry underlies stability in power grids

Behavioral homogeneity is often critical for the functioning of network systems of interacting entities. In power grids, whose stable operation requires generator frequencies to be synchronized—and thus homogeneous—across the network, previous work suggests that the stability of synchronous states can be improved by making the generators homogeneous. Here, we show that a substantial additional improvement is possible by instead making the generators suitably heterogeneous. We develop a general method for attributing this counterintuitive effect to converse symmetry breaking, a recently established phenomenon in which the system must be asymmetric to maintain a stable symmetric state. These findings constitute the first demonstration of converse symmetry breaking in real-world systems, and our method promises to enable identification of this phenomenon in other networks whose functions rely on behavioral homogeneity.

Stable chimeras of non-locally coupled Kuramoto–Sakaguchi oscillators in a finite array

We consider chimera states of coupled identical phase oscillators where some oscillators are phase synchronized (have the same phase) while others are desynchronized. Chimera states of non-locally coupled Kuramoto–Sakaguchi oscillators in arrays of finite size are known to be chaotic transients; after a transient time, all the oscillators are phase synchronized, with the transient time increasing exponentially with the number of oscillators. In this work, we consider a small array of six non-locally coupled Kuramoto–Sakaguchi oscillators and modify the range of the phase lag parameter to destabilize their complete phase synchronization. Under these circumstances, we observe a chimera state spontaneously formed by the partition of oscillators into two independently synchronizable clusters of both stable and unstable synchronous states. In the chimera state, the trajectory of the phase differences of the desynchronized oscillators relative to the synchronous cluster is a stable periodic orbit, and as a result, the chimera state is a stable but not long-lived transient. We observe the chimera state with random initial conditions in a restricted range of the phase lag parameter and clarify why the state is observable in the restricted range using Floquet theory for periodic orbit stability.

Properties of generalized synchronization in smooth and non-smooth identical oscillators

This paper deals with the phenomenon of the GS only in the context of unidirectional connection between identical exciter and receivers. A special attention is focused on the properties of the GS in coupled non-smooth Chua circuits. The robustness of the synchronous state is analyzed in the presence of slight parameter mismatch. The analysis tools are transversal and response Lyapunov exponents and fractal dimension of the attractor. These studies show differences in the stability of synchronous states between smooth (Lorenz system) and non-smooth (Chua circuit) oscillators.

Stability of synchronous states in sparse neuronal networks

The stability of synchronous states is analyzed in the context of two populations of inhibitory and excitatory neurons, characterized by two different pulse-widths. The problem is reduced to that of determining the eigenvalues of a suitable class of sparse random matrices, randomness being a consequence of the network structure. A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the overall stability depends crucially on the relative pulse-width. This has implications for the overall property of the asynchronous (balanced) regime.

Delay master stability of inertial oscillator networks

Time lags occur in a vast range of real-world dynamical systems due to finite reaction times or propagation speeds. Here we derive an analytical approach to determine the asymptotic stability of synchronous states in networks of coupled inertial oscillators with constant delay. Building on the master stability formalism, our technique provides necessary and sufficient delay master stability conditions. We apply it to two classes of potential future power grids, where processing delays in control dynamics will likely pose a challenge as renewable energies proliferate. Distinguishing between phase and frequency delay, our method offers an insight into how bifurcation points depend on the network topology of these system designs.

Synchronization in networks of coupled hyperchaotic CO2 lasers

We propose a non-autonomous dynamical system for an optically modulated CO2 laser and show that it exhibits hyperchaos in presence of electro-optic feedback beams. The system is then used to study the synchronization in networks of mutually coupled hyperchaotic CO2 lasers. By the method of master stability function it is shown that the stable synchronous state can be reached for both the ring of diffusively coupled and star-coupled networks of at most 24 nodes or oscillators. However, in the former networks, high-coupling strengths (∼10) are required for synchronization compared to the latter ones (∼1). A numerical simulation of the coupled 24 hyperchaotic CO2 lasers is also performed to show that the corresponding synchronization error ≲10−6. Furthermore, the chimera states of the networks are found to coexist in some intervals of time and the coupling strengths where the networks are not synchronized, implying that the synchronization occurs only in some specific ranges of values of the coupling strengths.

Dynamics of disordered heterogeneous chains of phase oscillators

We study the problem of robustness of synchronous states to disorder in the chain of phase oscillators with local coupling. The study combines a numerical determination of the existence and stability of synchronous states with an analytical investigation of the role of the phase shift and the level of disorder in the natural frequencies in the destruction of synchrony. We show that the presence of the phase shift facilitates robustness of the synchronous regime, at least up to its certain threshold value.

Hydrodynamic synchronization and collective dynamics of colloidal particles driven along a circular path.

We study theoretically the collective dynamics of particles driven by an optical vortex along a circular path. Phase equations of N particles are derived by taking into account both hydrodynamic and repulsive interactions between them. For N=2, the particles attract with each other and synchronize, forming a doublet that moves faster than a singlet. For N=3 and 5, we find periodic rearrangement of doublets and a singlet. For N=4 and 6, the system exhibits either a periodic oscillating state or a stable synchronized state depending on the initial conditions. These results reproduce main features of previous experimental findings. We quantitatively discuss the mechanisms governing the nontrivial collective dynamics.

Brain synchronizability, a false friend

Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network is, given its topological organization, are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. Here, we argue that brain dynamics does not fulfil the formal criteria under which synchronizability is usually quantified and, perhaps more importantly, this measure refers to a global dynamical condition that never holds in the brain (not even in the most pathological conditions), and therefore no neurophysiological conclusions should be drawn based on it. We discuss the meaning of synchronizability and its applicability to neuroscience and propose alternative ways to quantify brain networks synchronization.

The Kuramoto Model on Oriented and Signed Graphs

Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.

Dynamics of Kuramoto oscillators with time-delayed positive and negative couplings

Many real-world examples of distributed oscillators involve not only time delays but also attractive (positive) and repulsive (negative) influences in their network interactions. Here, considering such examples, we generalize the Kuramoto model of globally coupled oscillators with time-delayed positive and negative couplings to explore the effects of such couplings in collective phase synchronization. We analytically derive the exact solutions for stable incoherent and coherent states in terms of the system parameters allowing us to precisely understand the interplay of time delays and couplings in collective synchronization. Dependent on these parameters, fully coherent, incoherent states and mixed states are possible. Time-delays especially in the negative coupling seem to facilitate collective synchronization. In case of a stronger negative coupling than positive one, a stable synchronized state cannot be achieved without time delays. We discuss the implications of the model and the results for natural systems, particularly neuronal network systems in the brain.

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

Optimizing synchronization stability of the Kuramoto model in complex networks and power grids.

Maintaining the stability of synchronization state is crucial for the functioning of many natural and artificial systems. In this study, we develop methods to optimize the synchronization stability of the Kuramoto model by minimizing the dominant Lyapunov exponent. Using the recently proposed cut-set space approximation of the steady states, we greatly simplify the objective function, and further derive its gradient and Hessian with respect to natural frequencies, which leads to an efficient algorithm with the quasi-Newton's method. The optimized systems are demonstrated to achieve better synchronization stability for the Kuramoto model with or without inertia in certain regimes. Hence our method is applicable in improving the stability of power grids. It is also viable to adjust the coupling strength of each link to improve the stability of the system. Various operational constraints can also be easily integrated into our scope by employing the interior point method in convex optimization. The properties of the optimized networks are also discussed.

Synchronization of cyclic power grids: Equilibria and stability of the synchronous state.

Synchronization is essential for the proper functioning of power grids; we investigate the synchronous states and their stability for cyclic power grids. We calculate the number of stable equilibria and investigate both the linear and nonlinear stabilities of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. We use the energy barrier to measure the nonlinear stability and calculate it by comparing the potential energy of the type-1 saddles with that of the stable synchronous state. We find that the energy barrier depends on the network size (N) in a more complicated fashion compared to the linear stability. In particular, when the generators and consumers are evenly distributed in an alternating way, the energy barrier decreases to a constant when N approaches infinity. For a heterogeneous distribution of generators and consumers, the energy barrier decreases with N. The more heterogeneous the distribution is, the stronger the energy barrier depends on N. Finally, we find that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of N→∞. This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.