Chaos is an irregular and complicated solution of deterministic equations. It is often found in the solutions to nonlinear equations describing oscillators, fluid motions, electric circuits, and so on. It is not only interesting as the complicated solution to a simple equation, but also important as the manifestation of an irregular response of a system to regular forcing, such as the irregular motion of a nonlinear oscillator due to the regular forcing. In our Section, the chaotic behavior of fluid systems and other dynamical systems is studied theoretically. Particularly, we are interested in the route to chaos, characterization of chaos, and chaos in the system of many degrees of freedom. Also, as one of the applications of chaos, the mixing of fluids by the chaotic motion of fluid particles in a simple velocity field, called Lagrangian chaos, is examined.
Some of the behavior of water waves or internal waves ( waves in density-stratified fluid ) cannot be explained by the linear theory. This behavior has been examined using a variety of theories on nonlinear waves. The most famous one among them is the theory of solitons, solitary waves which do not lose their identity through their collision with each other. In our Section, generation of nonlinear waves due to the external forcing such as the oscillation of the container or the movement of a submerged object, pattern formation in the generated waves, resonant interaction among waves, and three-dimensional nonlinear waves are studied, using the soliton theory, the theory of nonlinear dynamical systems, and singular perturbation theory.
Vortices are not only commonly observed in the flow through a rigid body, but are also used as a basic mode in the description of complicated fluid motion. In our Section, the interaction of vortices, which is strongly nonlinear and complicated, and the formation of a new vortex structure accompanying it, are examined using a variety of mathematical methods. Also, the chaotic behavior of vortices themselves, or of the fluid particles near vortices, is studied.
Fluid motion is generated by the external forcing such as heating or the movement of an object or a rigid boundary. With the increase in the strength of the forcing, a simple fluid motion usually becomes unstable and changes to another more complicated motion. In our Section, the formation of a regular spatial pattern, and the generation of a different fluid motion associated with this instability, are examined using the theories of hydrodynamic instability and pattern formation.