Nonlinear Dynamics of Oscillator Networks

许多生物振荡器种群具有惊人的自我同步能力。某些种类的蟋蟀在齐鸣；马来西亚的萤火虫在同步闪烁；在我们的一生中，我们心脏中数以千计的起搏器细胞以有节奏的同步跳动数十亿次。如果工程师和科学家能够模仿大自然的成功，设计出自动同步的网络，许多技术上的好处就会随之而来。例如，以无线传感器网络为例，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保持所有传感器的同步，协调它们之间的通信，并使它们能够在空间和时间上准确地记录数据。但是，基于交换时间戳分组的传统保持同步的方法需要大量能量。通过将传感器建模为由突然脉冲耦合的理想化萤火虫，已经开发出更有效的方法，这是一种首次在数学生物学背景下研究的同步方案。这位研究人员和他的同事们研究了这种受生物学启发的自同步网络。其目的是了解它们的数学性质，并建议它们在物理和工程中的潜在应用。对于我们理解节律活跃的细胞如何在组织和器官中协同工作，以及衍生到涉及振荡器阵列的技术应用，如传感器网络、激光和超导约瑟夫森结，有望带来好处。通过在这里提供的研究和外展机会培训四名研究生，这一努力还将有助于开发对我们国家在科学、技术、工程和数学领域的成功至关重要的人力资源。这位研究人员和他的同事们利用动力系统理论、分叉理论和统计物理的数学方法，结合数值模拟，研究了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。第一个项目解决了如果模型的交互包含现实但在数学上不方便的特征时会发生什么，例如排斥和吸引交互的随机混合，交互随着距离的减弱，或者时间延迟的交互。第二个项目的目标是找到一种转变，将物理学和工程学中研究的某些无限维振荡器网络转化为低维系统--四年前仓本模型出人意料地实现了这一壮举，这可能是即将取得突破的先兆。第三个项目研究脉冲耦合振荡器，并询问同步是如何从初始无序状态建立起来的。这里的新想法是用聚集理论来解决这个问题，这是从统计物理学借来的一种强大的技术。第四个项目使用动力系统理论来研究囚犯困境中的叛逃、报复和合作的循环以及相关的进化博弈。

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.