Nonlinear Dynamics of Oscillator Networks

许多生物振荡器群体都具有惊人的自我同步能力。某些种类的蟋蟀会齐声鸣叫;马来西亚的萤火虫会同步闪光;我们心脏中的数千个起搏细胞在我们的一生中以有节奏的步调跳动数十亿次。如果工程师和科学家能够模仿自然界的成功，设计出自动同步的网络，那么许多技术上的好处就会随之而来。例如，考虑无线传感器网络，其应用包括栖息地监测和入侵检测（用于生态学和反恐），医院病人的健康监测，以及跟踪煤矿工人。一个技术挑战是无线传感器网络需要保持所有传感器同步，协调它们之间的通信，并使它们能够在空间和时间上准确记录数据。但是，传统的保持同步的方法，基于交换时间戳数据包，需要大量的能量。通过将传感器建模为由突然脉冲耦合的理想化萤火虫，已经开发了更有效的方法，这是一种首先在数学生物学背景下研究的同步方案。这位研究者和他的同事研究这种受生物学启发的自同步网络。目的是了解它们的数学性质，并提出它们在物理和工程中的潜在应用。预期将有利于我们理解有节奏活性的细胞如何在组织和器官中一起工作，以及涉及振荡器阵列的技术应用的副产品，如传感器网络，激光器和超导约瑟夫森结。通过这里提供的研究和推广机会培训四名研究生，这一努力也将有助于开发对我们国家在科学、技术、工程和数学方面取得成功至关重要的人力资源。研究者和他的同事们研究振荡器网络的非线性动力学，使用动力系统理论，分叉理论和统计物理的数学方法，沿着数值模拟。其中两个项目涉及Kuramoto模型，这是自同步系统的最简单的生物启发模型。第一个项目解决了如果模型的相互作用包含现实但数学上不方便的特征，例如排斥和吸引相互作用的随机混合，距离的相互作用减弱，或时间延迟的相互作用，会发生什么。第二个项目的目标是找到一种转换，将物理学和工程学中研究的某些无限维振荡器网络减少到低维系统-这是四年前仓本模型意外实现的壮举，这可能是未来突破的预兆。第三个项目研究脉冲耦合振荡器，并询问同步如何从初始无序状态建立起来。这里的新想法是用聚集理论来处理这个问题，这是一种从统计物理学中借来的强大技术。第四个项目使用动力系统理论来研究囚徒困境和相关进化博弈中的背叛、报复和合作的循环。

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.