Reconstructing the states of a nonlinear dynamical system

Many frequently observed real-world phenomena are nonlinear in nature. This means that their output does not change in a manner that is proportional to their input. These models have a degree of unpredictability, where it is unclear how the system will respond to any changes in its input. This is especially important in the case of dynamical systems, where the output of the model changes with time. For such systems, the time series data, or the measurements from the system over time, have to be analyzed to determine how the system changes or evolves with time.

Due to the commonality of the problem, many solutions have been proposed to analyze time-series data to gain an understanding of the system. One method of reconstructing the state of a system based on time series data is state space reconstruction, which can be used to reconstruct those states where the system remains stable or unchanged with time. Such states are known as “attractors.” However, the accuracy of the reconstructed attractors depends on the parameters used for reconstruction, and due to the finite nature of the data, such parameters are difficult to ascertain, resulting in inaccurate reconstructions.

Now, in a new study to be published on April 1, 2022, in Nonlinear Theory and Its Applications, IEICE, Professor Tohru Ikeguchi from Tokyo University of Science, his PhD student Mr. Kazuya Sawada from Tokyo University of Science, and Prof. Yutaka Shimada from Saitama University, Japan, have used the geometric structure of the attractor to estimate the reconstruction parameters.

“To reconstruct the state space using time-delay coordinate systems, two parameters, the dimension of the state space and the delay time, must be set appropriately, which is an important issue that is still being actively studied in this field. We discuss how to set these parameters optimally by focusing on the geometric structure of the attractor as one way to solve this problem,” explains Prof. Ikeguchi.

To obtain the optimal values of the parameters, the researchers used five three-dimensional nonlinear dynamical systems and maximized the similarity of the inter-point distance distributions between the reconstructed attractor and the original attractor. As a result, the parameters were obtained in a way that produced a reconstructed attractor which was geometrically as close as possible to the original.

While the method was able to generate the appropriate reconstruction parameters, the researchers did not factor in the noise that is normally encountered in real-world data, which can significantly affect the reconstruction. “Mathematically, this method has been proven to be a good one, but there are many considerations that need to be made before applying this method to real-world data analysis. This is because real-world data contains noise, and the length and accuracy of the observed data is finite,” explains Prof. Ikeguchi. 

Despite this, the method resolves one of the limitations involved in determining the state of nonlinear dynamical systems that are encountered in various fields of science, economics, and engineering. “This research has yielded an important analysis technique in the current data science field, and we believe that it is important for handling a wide variety of data in the real world,” concludes Prof. Ikeguchi.

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Reference

DOI: https://doi.org/10.1587/nolta.13.385

About The Tokyo University of Science

Tokyo University of Science (TUS) is a well-known and respected university, and the largest science-specialized private research university in Japan, with four campuses in central Tokyo and its suburbs and in Hokkaido. Established in 1881, the university has continually contributed to Japan's development in science through inculcating the love for science in researchers, technicians, and educators.

With a mission of “Creating science and technology for the harmonious development of nature, human beings, and society", TUS has undertaken a wide range of research from basic to applied science. TUS has embraced a multidisciplinary approach to research and undertaken intensive study in some of today's most vital fields. TUS is a meritocracy where the best in science is recognized and nurtured. It is the only private university in Japan that has produced a Nobel Prize winner and the only private university in Asia to produce Nobel Prize winners within the natural sciences field.

Website: https://www.tus.ac.jp/en/mediarelations/

About Professor Tohru Ikeguchi from Tokyo University of Science

Tohru Ikeguchi received M.E. and Ph.D degrees from Tokyo University of Science, Japan. After working for nearly a decade as Full Professor at Saitama University, Japan, he worked at Tokyo University of Science as Full Professor at the Department of Management Science from 2014 to 2016. Since then, he has been a Full Professor at the Department of Information and Computer Technology in Tokyo University of Science. His research interests include nonlinear time series analysis, computational neuroscience, application of chaotic dynamics to solving combinatorial optimization problems, and complex network theory. He has published over 230 papers and proceedings.

Funding information

This study is partially supported by JSPS KAKENHI Grant Numbers JP17K00348, JP20H00596, JP21H03508, and JP21H03514.