Nonlinear Dynamics and Its Applications

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.

Dimension Reduction of Nonlinear Dynamical Systems and Its Applications

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)

Search algorithm and sampling algorithm using nonlinear dynamics

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.

Keywords

• Information processing with nonlinear dynamics: soft computing, analog computing
• Combinatorial optimization: Boolean satisfiability problem, Ising problem, simulated annealing
• Probability modeling and numerical methods: Boltzmann machine, Markov chain Monte Carlo, the herding algorithm, quasi-Monte Carlo method
• Numerical simulation of dynamical systems