Nonlinear Dynamics of Oscillator Networks

许多生物振荡器种群具有显著的自我同步能力。某些种类的蟋蟀会齐声鸣叫；马来西亚萤火虫同步闪光；我们心脏中成千上万的起搏器细胞在我们的一生中有节奏地同步跳动数十亿次。如果工程师和科学家能够模仿大自然在设计自动同步网络方面的成功，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（用于生态和反恐），医院患者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使所有传感器保持同步，协调它们之间的通信，并使它们能够准确地记录空间和时间上的数据。但是传统的基于交换时间戳数据包的保持同步的方法需要大量的能量。更有效的方法是将传感器建模为由突然脉冲耦合的理想化萤火虫，这是一种首次在数学生物学背景下研究的同步方案。研究者和他的同事们研究这种受生物学启发的自我同步网络。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。对于我们理解节律性活跃细胞在组织和器官中是如何协同工作的，以及涉及振荡器阵列的技术应用的副产品，如传感器网络、激光和超导约瑟夫森结，这些都是有益的。通过这里提供的研究和推广机会培训四名研究生，这一努力还将有助于培养对我国在科学、技术、工程和数学领域取得成功至关重要的人力资源。研究者和他的同事们利用动力系统理论、分岔理论和统计物理的数学方法，以及数值模拟，研究振荡器网络的非线性动力学。其中两个项目涉及Kuramoto模型，这是自同步系统最简单的生物启发模型。第一个项目解决的是，如果模型的相互作用包含了现实的但在数学上不方便的特征，比如排斥和吸引相互作用的随机混合，相互作用随距离的减弱，或者延迟的相互作用，会发生什么。第二个项目的目标是找到一种转换，将某些在物理和工程中研究的无限维振荡器网络减少到低维系统——这是四年前Kuramoto模型意外实现的壮举，这可能是未来突破的预兆。第三个项目检查脉冲耦合振荡器，并询问如何从初始无序状态建立同步。这里的新想法是用聚合理论来解决这个问题，这是一种借鉴于统计物理学的强大技术。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.