Applying dynamical systems approaches, our group studies complicated phenomena such as chaos and bifurcations in various systems appearing in sciences, engineering and other disciplines and develops novel engineering technologies. For this purpose, we not only use standard approaches but also develop new innovative theories in dynamical systems. We also utilize numerical approaches such as verifiable computation and large-scale numerical simulation, and tackle the following topics:
1. Development and application of dynamical systems theory in infinite-dimensional systems
We develop new theories for bifurcation and stability of pulse solutions and pattern formations in partial differential equations and lattice systems such as infinite-degree-of-freedom Hamiltonian systems, and establish verified numerical methods for bifurcation analysis of these systems.
2. Understanding of diverse bifurcation structures in various systems of differential equations
We consider various systems of differential equations including Hamiltonian systems, reversible systems, piecewise smooth systems and random dynamical systems, and develop new theories to reveal bifurcation structures in these systems.
3. Equilibrium/Nonequilibrium statistical mechanics and dynamics in many-body Hamiltonian systems
Temporal evolution of the system is described by a distribution function instead of considering each particle. The kinetic theory reveals collective motion of the system, which are nonequilibrium phase transitions, dependence of relaxation time on the number of particles, and existence of algebraic slow relaxation. This method can be applied to the linear response theory by considering the responses to applied external forces.
4. Applications in natural science, engineering and social science
We establish dynamical systems approaches for design of spacecraft transfer trajectories. We also develop chaos control and optimal control methods based on dynamical systems theory and apply them to various problems appearing from natural science and engineering to social science.
Last modified: Wed Apr 2 20:19:29 JST 2014