Pairwise particle interactions arise in diverse physical systems ranging from insect swarms to self-assembly of nanoparticles. In the presence of long-range attraction and short-range repulsion, such systems can exhibit bound states. We use linear stability analysis of a ring equilibrium to classify the morphology of patterns in two dimensions. Conditions are identified that assure the well-posedness of the ring. In addition, weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria including triangular shapes, annuli, and spot patterns with N-fold symmetry. Many of these patterns have been observed in nature, although a general theory has been lacking, in particular how small changes to the interaction potential can lead to large changes in the self-organized state.

Large systems of particles interacting pairwise in d-dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere (in 3D) or a ring (in 2D). Localized, space-filling, codimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a non-local linear stability analysis for particles uniformly distributed on a d − 1 sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in selfassembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

Abstract. In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We review and expand upon recent developments of this class of problems as well as present new results. As in previous work, we leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence of the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the "designer potential" problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model. IntroductionMathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2]. In the case of swarming, these nonlocal interactions between individuals usually consist of a shorter range repulsion to avoid collisions and medium to long range attraction to keep the swarm cohesive. While some models include anisotropy in this communication (e.g. an organism's eyes may have a limited field of vision) simplified isotropic interactions have been shown to capture many important swarming behaviors including milling [20,10]. More recently it has been shown [17,38,37] that the competition between the desire to avoid collisions and the desire to remain in a cohesive swarm can sometimes result in simple radially symmetric patterns such as rings, annuli and uniform circular patches and other times result in exceedingly complex patterns. Moreover how modelers select the strength and form of the repulsion near the origin

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This paper provides both theoretical and algorithmic results for the 1 -relaxation of the Cheeger cut problem. The 2 -relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The 1 -relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The 1 -relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for algorithms that minimize the 1 -relaxation. The second challenge entails comprehending the 1energy landscape, i.e. the set of possible points to which an algorithm might converge. We show that 1 -algorithms can get trapped in local minima that are not globally optimal and we provide a classification theorem to interpret these local minima. This classification gives meaning to these suboptimal solutions and helps to explain, in terms of graph structure, when the 1 -relaxation provides the solution of the original Cheeger cut problem.