Torus bifurcations of large-scale swarms having range dependent communication delay.

Dynamical emergent patterns of swarms are now fairly well established in nature and include flocking and rotational states. Recently, there has been great interest in engineering and physics to create artificial self-propelled agents that communicate over a network and operate with simple rules, with the goal of creating emergent self-organizing swarm patterns. In this paper, we show that when communicating networks have range dependent delays, rotational states, which are typically periodic, undergo a bifurcation and create swarm dynamics on a torus. The observed bifurcation yields additional frequencies into the dynamics, which may lead to quasi-periodic behavior of the swarm.

Delay induced swarm pattern bifurcations in mixed reality experiments.

Swarms of coupled mobile agents subject to inter-agent wireless communication delays are known to exhibit multiple dynamic patterns in space that depend on the strength of the interactions and the magnitude of the communication delays. We experimentally demonstrate communication delay-induced bifurcations in the spatiotemporal patterns of robot swarms using two distinct hardware platforms in a mixed reality framework. Additionally, we make steps toward experimentally validating theoretically predicted parameter regions where transitions between swarm patterns occur. We show that multiple rotation patterns persist even when collision avoidance strategies are incorporated, and we show the existence of multi-stable, co-existing rotational patterns not predicted by usual mean field dynamics. Our experiments are the first significant steps toward validating existing theory and the existence and robustness of the delay-induced patterns in real robotic swarms.

Unstable modes and bistability in delay-coupled swarms.

It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-laying the groundwork for comparisons to robotics experiments.

Approximating stable and unstable manifolds in experiments.

We introduce a procedure to reveal invariant stable and unstable manifolds, given only experimental data. We assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle. We show that the method is able to capture the fine structure of the manifold, is independent of dimension, and is efficient relative to previous techniques.

Open-loop sustained chaos and control: a manifold approach.

We present a general method for preserving chaos in nonchaotic parameter regimes as well as preserving periodic behavior in chaotic regimes using a multifrequency phase control. The systems considered are nonlinear systems driven at a base frequency. Multifrequency phase control is defined as the addition of small subharmonic amplitude modulation coupled with a phase shift. By implementing multifrequency control, stable and unstable manifold intersections in postcrisis regimes may be manipulated to sustain chaos as well as to sustain periodic behavior. The theory and a preliminary experiment are demonstrated for a CO2 driven laser.

Chaos and intermittent bursting in a reaction-diffusion process.

Karhunen-Loeve decomposition is done on a chaotic spatio-temporal solution obtained from a nonlinear reaction-diffusion model of a chemical system simulating a chemical process in an open Couette-flow reactor. Using a Galerkin projection of the dominant Karhunen-Loeve modes back onto the nonlinear partial differential system, we obtain an ordinary differential equation model of the same process. Major features such as intermittent and chaotic bursting of the nonlinear process as well as the mechanism of transition to chaos are shown to exist in the low-dimensional model as well as the PDE model. From the low-dimensional model the onset of intermittent bursts followed by small amplitude oscillations is shown to arise due to a sequence of saddle-node bifurcations.

Tracking controlled chaos: Theoretical foundations and applications.

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection.

It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.