McMahon Christopher J, Toomey Joshua P, Kane Deb M
MQ Photonics Research Centre and Department of Physics & Astronomy, Macquarie University, Sydney, NSW, Australia.
PLoS One. 2017 Aug 24;12(8):e0181559. doi: 10.1371/journal.pone.0181559. eCollection 2017.
We have analysed large data sets consisting of tens of thousands of time series from three Type B laser systems: a semiconductor laser in a photonic integrated chip, a semiconductor laser subject to optical feedback from a long free-space-external-cavity, and a solid-state laser subject to optical injection from a master laser. The lasers can deliver either constant, periodic, pulsed, or chaotic outputs when parameters such as the injection current and the level of external perturbation are varied. The systems represent examples of experimental nonlinear systems more generally and cover a broad range of complexity including systematically varying complexity in some regions.
In this work we have introduced a new procedure for semi-automatically interrogating experimental laser system output power time series to calculate the correlation dimension (CD) using the commonly adopted Grassberger-Proccacia algorithm. The new CD procedure is called the 'minimum gradient detection algorithm'. A value of minimum gradient is returned for all time series in a data set. In some cases this can be identified as a CD, with uncertainty.
Applying the new 'minimum gradient detection algorithm' CD procedure, we obtained robust measurements of the correlation dimension for many of the time series measured from each laser system. By mapping the results across an extended parameter space for operation of each laser system, we were able to confidently identify regions of low CD (CD < 3) and assign these robust values for the correlation dimension. However, in all three laser systems, we were not able to measure the correlation dimension at all parts of the parameter space. Nevertheless, by mapping the staged progress of the algorithm, we were able to broadly classify the dynamical output of the lasers at all parts of their respective parameter spaces. For two of the laser systems this included displaying regions of high-complexity chaos and dynamic noise. These high-complexity regions are differentiated from regions where the time series are dominated by technical noise. This is the first time such differentiation has been achieved using a CD analysis approach.
More can be known of the CD for a system when it is interrogated in a mapping context, than from calculations using isolated time series. This has been shown for three laser systems and the approach is expected to be useful in other areas of nonlinear science where large data sets are available and need to be semi-automatically analysed to provide real dimensional information about the complex dynamics. The CD/minimum gradient algorithm measure provides additional information that complements other measures of complexity and relative complexity, such as the permutation entropy; and conventional physical measurements.
我们分析了由来自三种B型激光系统的数万个时间序列组成的大型数据集，这三种激光系统分别是：光子集成芯片中的半导体激光器、受到来自长自由空间外腔光反馈的半导体激光器以及受到主激光器光注入的固态激光器。当注入电流和外部扰动水平等参数变化时，这些激光器可以输出恒定、周期性、脉冲或混沌信号。这些系统更广泛地代表了实验非线性系统的实例，涵盖了广泛的复杂性范围，包括某些区域中系统变化的复杂性。
在这项工作中，我们引入了一种新的程序，用于半自动询问实验激光系统的输出功率时间序列，以使用常用的格拉斯贝格 - 普罗卡恰算法计算关联维数（CD）。这种新的CD程序被称为“最小梯度检测算法”。对于数据集中的所有时间序列，都会返回一个最小梯度值。在某些情况下，这可以被确定为关联维数，但存在不确定性。
应用新的“最小梯度检测算法”CD程序，我们对从每个激光系统测量的许多时间序列获得了可靠的关联维数测量值。通过在每个激光系统运行的扩展参数空间中映射结果，我们能够自信地识别出低CD（CD < 3）区域，并为关联维数指定这些可靠值。然而，在所有三个激光系统中，我们无法在参数空间的所有部分测量关联维数。尽管如此，通过映射算法的阶段性进展，我们能够大致对激光器在其各自参数空间的所有部分的动态输出进行分类。对于其中两个激光系统，这包括显示高复杂性混沌和动态噪声区域。这些高复杂性区域与时间序列受技术噪声主导的区域不同。这是首次使用CD分析方法实现这种区分。
在映射背景下询问一个系统时，比使用孤立时间序列进行计算能更多地了解该系统的关联维数。这在三个激光系统中得到了证明，并且该方法预计在其他非线性科学领域也将有用，在这些领域中可以获得大型数据集，并且需要进行半自动分析以提供有关复杂动力学的实际维度信息。CD/最小梯度算法测量提供了补充其他复杂性和相对复杂性度量（如排列熵）以及传统物理测量的额外信息。
