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Register a new userGrass-roots optimization of coupled oscillator networks
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Added by
Per Sebastian Skardal
Publication date
2021
fields
Physics
and research's language is
English
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Synchronization is critical for system function in applications ranging from cardiac pacemakers to power grids. Existing optimization techniques rely largely on global information, and while they induce certain local properties, those alone do not yield optimal systems. Therefore, while useful for designing man-made systems, existing theory provides limited insight into self-optimization of naturally-occurring systems that rely on local information and offer limited potential for decentralized optimization. Here we present a method for grass-roots optimization of synchronization, which is a multiscale mechanism involving local optimizations of smaller subsystems that are coordinated to collectively optimize an entire system, and the dynamics of such systems are particularly robust to islanding or targeted attacks. In addition to shedding light on self-optimization in natural systems, grass-roots optimization can also support the parallelizable and scalable engineering of man-made systems.
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The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques which capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to partial synchronization. Our model reduction builds upon the method of collective coordinates. We first show that standard collective coordinate reduction has difficulties capturing this bifurcation. We identify a particular topological structure at bifurcation consisting of a main synchronized cluster, the oscillator that desynchronizes at bifurcation, and an intermediary node connecting them. Utilizing this structure and ensemble averages we derive an analytic expression for the mismatch between the true bifurcation from global to partial synchronization and its estimate calculated via the collective coordinate approach. This allows to calibrate the standard collective coordinate approach without prior knowledge of which node will desynchronize. We introduce a second mesoscopic reduction, utilizing the same particular topological structure, which allows for a quantitative dynamical description of the phases near bifurcation. The mesoscopic reductions significantly reduce the computational complexity of the collective coordinate approach, reducing from $mathcal{O}(N^2)$ to $mathcal{O}(1)$. We perform numerical simulations for ErdH{o}s-Renyi networks and for modified Barabasi-Albert networks demonstrating excellent quantitative agreement at and close to bifurcation.
We show that amplitude chimeras in ring networks of Stuart-Landau oscillators with symmetry-breaking nonlocal coupling represent saddle-states in the underlying phase space of the network. Chimera states are composed of coexisting spatial domains of coherent and of incoherent oscillations. We calculate the Floquet exponents and the corresponding eigenvectors in dependence upon the coupling strength and range, and discuss the implications for the phase space structure. The existence of at least one positive real part of the Floquet exponents indicates an unstable manifold in phase space, which explains the nature of these states as long-living transients. Additionally, we find a Stuart-Landau network of minimum size $N=12$ exhibiting amplitude chimeras
The Kuramoto model, which serves as a paradigm for investigating synchronization phenomenon of oscillatory system, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here, we generalize a number of classical results by presenting a general framework for capturing, analytically, the critical scaling of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determines the scaling properties of order parameter above the criticality.
A scenario has recently been reported in which in order to stabilize complete synchronization of an oscillator network---a symmetric state---the symmetry of the system itself has to be broken by making the oscillators nonidentical. But how often does such behavior---which we term asymmetry-induced synchronization (AISync)---occur in oscillator networks? Here we present the first general scheme for constructing AISync systems and demonstrate that this behavior is the norm rather than the exception in a wide class of physical systems that can be seen as multilayer networks. Since a symmetric network in complete synchrony is the basic building block of cluster synchronization in more general networks, AISync should be common also in facilitating cluster synchronization by breaking the symmetry of the cluster subnetworks.
The stable operation of the electric power grid relies on a precisely synchronized state of all generators and machines. All machines rotate at exactly the same frequency with fixed phase differences, leading to steady power flows throughout the grid. Whether such a steady state exists for a given network is of eminent practical importance. The loss of a steady state typically leads to power outages up to a complete blackout. But also the existence of multiple steady states is undesirable, as it can lead to sudden transitions, circulating flows and eventually also to power outages. Steady states are typically calculated numerically, but this approach gives only limited insight into the existence and (non-)uniqueness of steady states. Analytic results are available only for special network configuration, in particular for grids with negligible Ohmic losses or radial networks without any loops. In this article, we introduce a method to systematically construct the solutions of the real power load-flow equations in the presence of Ohmic losses. We calculate the steady states explicitly for elementary networks demonstrating different mechanisms leading to multistability. Our results also apply to models of coupled oscillators which are widely used in theoretical physics and mathematical biology.
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