2. Applications of delay and fluctuation

Applications of delay and fluctuation

  • Graduate School of Mathematics

Toru Ohira [Professor]


Outline of Seeds

Our research lab is studying two major topics.
(1) The mathematics of noise and delays: Noise and delays in information transmission are part of various targets of study, from biological organisms to financial markets. Both can frequently create complex and undesirable behaviors in systems. I have been studying these factors and the systems that contain them from a mathematical perspective for nearly twenty years. I have developed unique mathematical methods that incorporate delays into random walks, applying and presenting research outcomes in diverse fields that include human posture balance control, models that can be applied to network traffic issues, and foreign exchange markets, for example.
(2) Chase-and-escape is a classic mathematical problem in which a chaser pursues an escapee. It is usually considered in a one-on-one context. The question was merged with game theory in the 20th century, and a variety of application approaches have been consideredincluding the construction of a search game theory. My research in recent years has grown to integrate these problems with studies involving crowds or groups of self-driven actors (like cars, people, or animals) to propose mathematical models for group chase-and-escape. The models are likely to be effective in applications involving the control of groups of robots or group automotive systems.

Novelty and originality of this research

In researches on systems involving noise or delays, traditional approaches have studied the inclusion of each factor independently. Research on systems that include both has not really advanced due to its complexity. I have pioneered a delayed random-walk mathematical approach that independently includes delays in the random walk. This allows me to study systems that contain both noise and delay factors in a way that more closely resembles real phenomena. With this approach, I can explain the properties of statistical fluctuation in these systems, something that was not fully grasped under conventional theories. Also, with the topic of group chase-and-escape proposed in recent years, I have integrated research in different fields with the idea of contributing to applications like controlling groups of robots or learning more about crowd characteristics. I am also constantly working towards the construction of mathematical models that are as simple as possible so that they can be applied more easily. The effects of noise and delays can introduce complex behaviors even in simple systems, and I believe that searching for approaches that will let us control them is critical as we consider broad-based applications.

Application and research area for Industry collaboration

Within our research, the topics that seem to have the most potential for industrial-academic collaboration are (1) delayed stochastic resonance and (2) group chase-and-escape. With the first topic, we are working to boost control performance by intentionally adding a moderate amount of fluctuation to delayed feedback systems. For example, it is common to observe phenomena wherein control of balance can be extended by adding fluctuation such as the fluctuation a person adds with their hands to control balance. Gleaning from these possibilities, I observed that in controlling an unstable physical target, it is possible to get effective performance by adding a moderate amount of fluctuation. One possible application might be in balance control devices that prevent falls in the elderly. The second topic has to do with group behavior among self-driven particlesparticles that, unlike physical particles, move on their own. These include things like vehicles or groups of robots. In this area, we are not simply studying their characteristics as a group, but are adding chase-and-escape capabilities with the idea that our work might contribute to the natural formation of convoys, for example. We are in contact with a Hungarian research group that is adding these factors to groups of robots, and see great potential for a variety of future developments.

Key Takeaway

Although we are conducting mathematical research, we are also actively carrying out joint studies in different fields (including biology, finance, and physics) from both theoretical and experimental perspectives. For more information on our overall research approach, please see Chosen Suru Kagakusha (Nikkei Publishing 2014).


Delays, fluctuation, probability, resonance, crowd behavior, traffic, self-driven particles, pursuit/evasion


  • Mathematical insights for overall systems, including delays and noise
  • Mathematical insights related to chase-and-escape problems and to the characteristics of collective behavior