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Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental set-up controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, additional secondary states, where phase differences differ, are detected. Previous work on this system saw the development of a mathematical model that explains how the existence, stability, and common frequency of the experimentally observed cluster states can be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals a complex interconnectedness between branches of different cluster types; in particular, we find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. Otherwise, for a smaller voltage, all the branches of secondary states are completely unstable and therefore hidden to experimentalists., Comment: 13 pages, 14 figures

Educators from the teacher, administrator, and school-board associations of the states of Alaska, Washington, Idaho, Montana, and Oregon joined with the Northwest Regional Educational Laboratory to highlight the achievements of the public schools in preparing children and youth for a successful and fulfilling life, and to note the significant participation of the associations in these endeavors. This summary is a way of sharing the practices deemed effective by the associations, and of offering them as resources for others. All five states can be justifiably proud of their students' academic achievement. On a variety of measures, students from the Northwest score as high, and usually higher, than their peers at the national level. This pattern has held and recently been strengthened, despite the challenges of persistent childhood poverty, rapidly increasing numbers of students whose first language is not English, and the mainstreaming of special-education students. The report presents the results of an international benchmarking study whose purpose is to provide useful student-achievement information about the performance of states relative to other countries. It broadly compares state 8th-grade mathematics and science performance for each of 44 states and jurisdictions participating in the National Assessment of Educational Progress with that of 41 nations who participated in the third International Mathematics and Science Study. (Contains 15 references.) (DFR)