Cntrol systems often operate in the presence of de-lays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions, and to execute these decisions, as shown in Figure 1. Systems with delays arise in engineering, biology, physics, operations research, and economics. In traffic-flow models, the drivers’ delayed reactions, which combine sensing, perception, response, selection, and programming delays, must be considered [1]–[3]. These delays are critical in accounting for human behavior, analyzing traffic-flow stability, and designing collision-free traffic flow using adaptive cruise controllers [4]. Material distribution and supply-chain systems are com-posed of interconnected supply-demand points that share products and information to regulate inventories and respond to customer demands [5]. Sources of delay in supply chains include decision-making, transportation-line delivery, and manufacturing facilities that work with lead times [6]. These delays, which influence every stage of the supply-demand chain, deteriorate inventory regulation, thereby causing financial losses, inefficiencies, and reduced quality-of-service [7].

Strange patterns in one ring of Chen oscillators coupled to a ‘buffer ’ cell Carla MA Pinto1,2 and Ana RM Carvalho3 We study curious dynamical patterns appearing in networks of one ring of cells coupled to a ‘buffer ’ cell. Depending on how the cells in the ring are coupled to the ‘buffer ’ cell, the full network may have a nontrivial group of symmetries or a nontrivial group of ‘interior ’ symmetries. This group is Z3 in the unidirectional case and D3 in the bidirectional case. We simulate the coupled cell systems associated with the networks and obtain steady states, rotating waves, quasiperiodic behavior, and chaos. The different patterns seem to arise through a sequence of Hopf, period-doubling, and period-halving bifurcations. The behavior of the systems with exact symmetry are similar to the ones with ‘interior ’ symmetry. The network architecture appears to explain some features, whereas the properties of the Chen oscillator, used to model cells ’ internal dynamics, may explain others. We use XPPAUT and MATLAB to numerically compute the relevant states.