Quinn John C, Bryant Paul H, Creveling Daniel R, Klein Sallee R, Abarbanel Henry D I
Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jul;80(1 Pt 2):016201. doi: 10.1103/PhysRevE.80.016201. Epub 2009 Jul 6.
We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time-dependent and time-independent coupling and discuss the use of periodic impulse coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time-dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. For time-independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time-dependent coupling appears to have benefits in synchronizing long data sets with minimal impact, while the initial value method with time-independent coupling tends to be substantially faster, more flexible, and easier to use. We also describe a method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time-dependent injected current. They are expected to be of value in nonchaotic systems as well. Software is available on request.
我们研究了将同步作为从实验系统中提取参数和状态信息的一种机制。我们关注这个问题中以前很少受到关注的重要方面，并以混沌科尔皮茨振荡器作为示例系统，通过实验和模拟来探索这些方面。我们探讨了模型不完善对从实验系统中提取有效信息能力的影响。我们比较了两种优化方法：初始值方法和约束方法。这两种方法都涉及将模型方程与实验数据耦合，以便在同步流形上使混沌运动正则化。我们研究了与时间相关和与时间无关的耦合，并讨论了周期性脉冲耦合的使用。我们还研究了优化耦合和固定（或手动调整）耦合。对于优化的与时间相关的耦合函数u(t)的情况，我们发现了一种稳健的结构，其中包括尖锐的峰值和为零的区间。这种结构与相空间中的位置有很强的相关性，并且似乎取决于噪声、模型的不完善以及李雅普诺夫方向向量。对于与时间无关的耦合，我们发现了一个违反直觉的结果，即通常在将模型拟合到数据时，初始的均方根误差会随着耦合强度的增加而增大。将这个结果与使用模拟数据得到的结果进行比较，可能会提供一种衡量模型不完善程度的方法。具有与时间相关耦合的约束方法在同步长数据集时似乎具有优势，影响最小，而具有与时间无关耦合的初始值方法往往速度更快、更灵活且更易于使用。我们还描述了一种对稀疏实验数据集有用的耦合方法。我们对科尔皮茨振荡器的使用使我们能够详细探索具有一个正李雅普诺夫指数的系统的情况。我们探索的方法很容易扩展到诸如具有随时间变化注入电流的神经元等驱动系统。预计它们在非混沌系统中也有价值。如有需要可提供软件。
