In the 17th century, Isaac Newton formulated equations to describe laws of force and motion, using the elegant new calculus. Because calculus generally hates singularities, discontinuities and instabilities, most classical studies on equations concentrated on solutions that were smooth and stable. People thus came to conceive of a universe like a precision machine: based on the initial conditions, one could predict future behavior, such as planets orbiting the Sun, or a pendulum clock.
   However, in the 1960s and '70s, "catastrophe theory" and "chaos theory" were formulated, which are special branches of dynamical systems theory that study phenomena with abrupt changes in behavior, or with unstable and unpredictable dynamics. Suddenly, it was noticed that nature also has behaviors beyond the smooth and stable. Catastrophe theory was subsequently applied to many phenomena in different disciplines, including social science. This very popular approach, though, turned out to be superficial, because it was applied with oversimplified generality, rather than being based on sound science and engineering inputs from specific fields.
   Catastrophe theory is included in the more general theory of bifurcations, which mean that when some parameter values are changed, at a point the qualitative structure of the solutions also changes. There are presently many advanced theorems on bifurcations, some of which can explain the emergence of chaos. Chaos is now a mathematical term that represents complex behaviors with instability and long-term unpredictability in deterministic dynamical systems. Extensive studies on chaos in the 1980s clarified that it involves ubiquitous phenomena both in mathematical models and in real-world systems.
   Kazu Aihara, who has been working on chaos for more than 25 years, entered this field as a graduate student by studying the nonlinear dynamics of nerve membranes both experimentally with squid giant axons and theoretically with nerve equations, and found chaotic responses as well as bifurcations. The important implication was that the single neurons could respond chaotically even if the input was simply periodic. Thus, for real, large neural systems, the behavior must be much more dynamic and complex. Based on the discovery of nerve chaos, he began developing mathematical models of the dynamical brain, possibly with spatio-temporal chaos.
   Aihara coined the term "chaos engineering" in 1990 to develop theoretical and technological foundations for possible applications of chaos, fractals and complex systems from the viewpoint of engineering. During the course of his many theoretical and engineering investigations on modeling various complex systems, he also developed practical applications of chaos in cooperation with industry. One adventure was a dishwasher that generates chaotic water jets in the washing chamber, resulting in a better cleaning effect. Further, his research on chaos engineering has involved chaotic computation using special IC chips, including chaotic neurochips and fractal chips, modeling of chaotic engineering systems like industrial blast furnaces, utility power systems and flooded ships, and chaotic cryptography. This work advanced the theoretical framework of such investigations and provided opportunities for practical applications.

 Research Strategy

The Aihara Complexity Modelling Project is pursuing research on complex phenomena from two distinct, but complementary and supplementary perspectives: generality and individuality. Regarding generality, efforts are being made to formulate mathematical theory as well as analysis methodology for modelling complex systems, putting great emphasis on bifurcation analysis and nonlinear time series analysis. However, the inputs to make such theory are being pursued in many individual real-world systems, from the viewpoint of mathematical engineering and chaos engineering. Both aspects of generalized theory and individual systems analysis are necessary and indispensable. Thus, information is being obtained from and fed back to a wide range of disciplines: nonlinear science, information science, life science, engineering, social science and economics. Regarding individual studies, the following three major research areas are being emphasized: