Research Initiation Awards provide support for junior and mid-career faculty at Historically Black Colleges and Universities who are building new research programs or redirecting and rebuilding existing research programs. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at the home institution, and involves undergraduate students in research experiences. The award to Clark Atlanta University (CAU) has potential broader impacts in a number of areas. The goal of this project is to advance understanding of the mechanisms for collective dynamical behaviors in networks of coupled oscillator systems with multi-time and multi-amplitude scales, and to enhance teaching and learning at the university by involving African American undergraduate students in multidisciplinary research activities. <br/><br/>An understanding of the dynamical mechanisms is crucial to make sense of experimentally observed network oscillations, for example, the neural oscillations in the brain. The proposed research will use theoretical analysis and computational modeling of networks of time-delay coupled oscillators. It aims to uncover the effects of time-delayed positive and negative couplings in synchronization; formulate a generalized theory for the origin of mixed-mode amplitude patterns; and develop a general network topology for coupled oscillators to mimic brain activity propagation and stable patterns. The project will integrate research and student training in applied mathematics by embedding research projects in courses and training students in advanced analytical and computational skills. This project will be conducted in collaboration with Georgia State University.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.