One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) solutions but they all exhibit characteristic problems. Moreover, traditional methods can break down if the system contains singular solutions. Here, we propose an efficient implementation of Newton homotopies, which can sample a large number of the stationary points of complicated many-body potentials. We demonstrate how the procedure works by applying it to the nearest-neighbor ϕ(4) model and atomic clusters.
The Kuramoto model is one of the most widely studied models for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform, heterogeneous, and sparse network is important yet challenging due to complicated nonlinear interactions. From the view point of homotopy deformation, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks, which lays the foundation for a divideand-conquer approach to studying the configurations of frequency synchronization of large Kuramoto networks.The spontaneous synchronization of a network of oscillators is an emergent phenomenon that naturally appears in many seemingly independent complex systems including mechanical, chemical, biological, and even social systems. The Kuramoto model is one of the most widely studied and successful mathematical models for analyzing synchronization behaviors. While much is known about the macro-scale question of whether or not a Kuramoto network can be synchronized, detailed analysis of the possible configurations of the oscillator once it has reached synchronization remains difficult for large networks partly due to the nonlinear interactions involved. In this work, we demonstrate that by dividing the link between two oscillators into two one-way interactions, complex networks can indeed be decomposed into much simpler subnetwork. This is a crucial step toward fully understanding synchronization configurations in large networks. arXiv:1903.04492v2 [math.CO]
A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions. Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.2. Algebraic formulation. In this paper, we focus on the mathematical abstraction of a power network which is captured by a graph G = (B, E) together with a complex matrix Y = (Y ij ). Here B is the finite set of nodes representing the "buses", E is the set of edges (also called lines or branches) representing the connections between buses, and the matrix Y is the nodal admittance matrix 1 which assigns a nonzero †
Locating the stationary points of a real-valued multivariate potential energy function is an important problem in many areas of science. This task generally amounts to solving simultaneous nonlinear systems of equations. While there are several numerical methods that can find many or all stationary points, they each exhibit characteristic problems. Moreover, traditional methods tend to perform poorly near degenerate stationary points with additional zero Hessian eigenvalues. We propose an efficient and robust implementation of the Newton homotopy method, which is capable of quickly sampling a large number of stationary points of a wide range of indices, as well as degenerate stationary points. We demonstrate our approach by applying it to the Thomson problem. We also briefly discuss a possible connection between the present work and Smale's 7th problem.
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