Nonlinear Dynamics of Oscillator Networks

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Many populations of biological oscillators have the remarkable ability to synchronize themselves. Certain species of crickets chirp in unison; Malaysian fireflies flash in sync; and the thousands of pacemaker cells in our hearts beat in rhythmic lockstep billions of times during our lives. If engineers and scientists could imitate nature's success at designing networks that automatically synchronize themselves, many technological benefits would follow. For instance, consider wireless sensor networks, whose applications include habitat monitoring and intrusion detection (both for ecology and anti-terrorism), health monitoring of patients in hospitals, and keeping track of workers in coal mines. One technical challenge is that a wireless sensor network needs to keep all its sensors in sync, to coordinate communication between them and to enable them to record data accurately in space and time. But traditional methods of maintaining synchrony, based on exchanging timestamp packets, require large amounts of energy. More efficient methods have been developed by modeling the sensors as idealized fireflies coupled by sudden pulses, a synchronization scheme first studied in the context of mathematical biology. The investigator and his colleagues study such self-synchronizing networks inspired by biology. The objective is to understand their mathematical properties and to suggest potential applications of them in physics and engineering. Benefits are expected for our understanding of how rhythmically active cells work together in tissues and organs, and for spin-offs to technological applications involving arrays of oscillators, such as sensor networks, lasers, and superconducting Josephson junctions. By training four graduate students through the research and outreach opportunities offered here, this effort will also help to develop human resources that are vital to our nation's success in science, technology, engineering, and mathematics. The investigator and his colleagues study the nonlinear dynamics of oscillator networks, using mathematical methods of dynamical systems theory, bifurcation theory, and statistical physics, along with numerical simulation. Two of the projects concern the Kuramoto model, the simplest bio-inspired model of a self-synchronizing system. The first project addresses what happens if the model's interactions incorporate realistic but mathematically inconvenient features, such as a random mix of repulsive and attractive interactions, a weakening of the interactions with distance, or time-delayed interactions. The goal of the second project is to find a transformation that will reduce certain infinite-dimensional oscillator networks studied in physics and engineering to low-dimensional systems - a feat that was achieved unexpectedly for the Kuramoto model four years ago, and that may be a harbinger of breakthroughs to come. The third project examines pulse-coupled oscillators, and asks how synchrony builds up from a state of initial disorder. The new idea here is to approach the question with aggregation theory, a powerful technique borrowed from statistical physics. The fourth project uses dynamical systems theory to study cycles of defection, retaliation, and cooperation in the Prisoner's Dilemma and related evolutionary games.