###### Research

### Professor Hideyuki Suzuki

- Theme
- Nonlinear Dynamics and Its Applications
- Content
- Nonlinear dynamical systems are known to exhibit complex and interesting behavior such as chaos. Even in systems described by very simple equations, we can find a rich variety of fascinating phenomena. Especially, I am studying nonlinear dynamics of large-scale dynamical systems and hybrid dynamical systems. I am also working on mathematical models of real-world phenomena in various research fields including biological systems, such as the brain and nervous system, and engineering systems, such as electric power systems. These models can be used for understanding the real-world phenomena and solving problems from the viewpoint of nonlinear dynamics. Furthermore, it is often the case that such mathematical models have novel and interesting nonlinear dynamics.
- Keywords
- ・Hybrid systems: switched systems, piecewise isometries, billiard systems, etc.
- ・Many-body dynamics: phase transitions, spatio-temporal chaos, etc.
- ・Dynamics and computation: deterministic sampling algorithms
- ・Applications to mathematical modeling: brain and nervous systems, electric power systems, epidemic models, etc.

### Associate Professor Sho Shirasaka

- Theme
- Dimension Reduction of Nonlinear Dynamical Systems and Its Applications
- Content
- The vast majority of dynamic behavior in natural world involve large degrees of freedom and nonlinearities, hence they are hard to analyze in practice. However, it is frequently observed that the dynamic behavior of functional elements found in natural sciences such as neurons and proteins and their ensembles evolves on a latent low-dimensional structure. Therefore, by exploiting the dominant low-dimensional dynamics of the nonlinear systems, we can drastically simplify the analysis of complex systems such as neuronal networks. This facilitates the analysis not only of natural systems but of engineered systems such as power grids. In view of the above, we develop a systematic framework, called dimension reduction method, to extract such structures and apply the method to real world problems. We extend the dimension reduction method to new classes of systems such as hybrid dynamical systems and analyze their collective behavior such as synchronization. Also, we develop methods to design cooperative behavior of nonlinear elements, such as optimization methods of synchronization stability, as novel application of the dimension reduction method. Further, we are working on establishing and enhancing links between the dimension reduction method and data-driven science based on operator-theoretic analysis of dynamical systems.
- Keywords
- ・Nonlinear dynamics
- ・Dynamical Systems with multiple time scales
- ・Optimization and Control
- ・Dimensionality reduction for dynamical systems
- ・Synchronization
- ・Mathematical and theoretical Biology (neuronal system, circulatory system, sensorimotor system)

### Assistant Professor Hiroshi Yamashita

- Theme
- Search algorithm and sampling algorithm using nonlinear dynamics
- Content
- I am studying the use of complexity that is found in large-scale nonlinear dynamical systems for information processing and problem-solving. Modern computation and information processing are performed more diversely, for example, with specialized devices such as GPUs and quantum computers. In addition, other than in artificial computational devices, information processing can be found in other places such as brain activity and intracellular information transmission. Computation and information processing are realized in various forms, but all of which are supported by nonlinear and dynamic phenomena. By studying nonlinear dynamical systems for computation and information processing through mathematical models, I expect that we can obtain a model of dynamics that guides the designing of new computational devices or is itself useful as an algorithm through efficient numerical simulations. Specifically, I am working on optimization problems such as the satisfiability problem and Ising problem, and numerical integration for complex probability distributions. For these problems, I am developing search and sampling algorithms, analyzing their performance, and studying numerical methods. These problems are important because they frequently appear as mathematical expressions of engineering problems and they are also known as problems for which simple and regular enumeration and search become inefficient. For this reason, it is popular using stochastic noise to increase the complexity of the search, and I believe that using the complex behavior of nonlinear dynamical systems is effective for such problems. I am also involved in collaborative research in various fields to apply nonlinear mathematics more widely to solve problems in modern society.
- Keywords
- ・Information processing with nonlinear dynamics: soft computing, unconventional computing, analog computing
- ・Combinatorial optimization: Boolean satisfiability problem, Ising problem, metaheuristics, simulated annealing
- ・Probability modeling and numerical methods: Boltzmann machine, Markov chain Monte Carlo, the herding algorithm, quasi-Monte Carlo method, statistical causal analysis
- ・Numerical simulation of dynamical systems

### Specially-Appointed Assistant Professor Ken-ichi Okubo

- Theme
- Nonlinear Phenomena and its Application to Information Processing
- Content
- I have studied mainly chaotic phenomena. Chaotic phenomena are observed in a wide range of nonlinear systems such as atmospheric phenomena, chemical reactions, electronic circuits, etc. Although they are based on deterministic dynamics, they seem to be disordered at first glance. That is, it is difficult to predict their behavior over a long time because of the sensitivity to initial conditions to be described as the butterfly effect. However, they show beautifully ordered behavior statistically and probabilistically. Then, by dealing with chaotic phenomena probabilistically with the Ergodic theorem, I have approached analytically the chaotic phenomena which are often dealt with by numerical simulations. In particular, if we prove the ergodicity of target systems, we can analyze various observables using the ergodic invariant density and it is a big advantage. Specifically, I have derived the critical exponents of the Lyapunov exponents and proven the relaxation of the density functions to the equilibrium distribution by showing the ergodic properties. Historically, in addition to studies on the chaotic phenomena themselves, studies that try to solve various problems using chaos have been actively carried out. In my future research, I would like to proceed with the same research as before, and at the same time, based on my knowledge so far, I would like to tackle problems such as information processing.
- Keywords
- ・Nonlinear Phenomena: chaotic dynamics, route to chaos (universality), ergodicity, arrow of time, infinite measure
- ・Application: time series analysis, chaos indicator, metaheuristic approach, combinatorial optimization problem