FoCM 2014 conference
Workshop A1 - Computational Dynamics
December 11, 16:00 ~ 16:25 - Room B21
Computer-assisted analysis of Craik's 3D dynamical system
Tomoyuki Miyaji
Meiji University, Japan - tmiyaji@meiji.ac.jp
The following system of equations is studied: $$ \dot{x} = a y z + b z + c y, \dot{y} = d z x + e x + f y, \dot{z} = g x y + h y + k x, $$ where $x(t), y(t)$, and $z(t)$ are real-valued functions, $\dot{x}, \dot{y}$, and $\dot{z}$ are their derivatives with respect to the independent variable $t$, and the coefficients $a$ to $k$ are real constants. This system arises several contexts in mechanics and fluid mechanics. Especially, Craik has shown that the equations of the form describe a class of exact solutions of the full incompressible Navier-Stokes equations.
Most of solution orbits for the system are unbounded. We can, however, observe characteristic behaviour. A typical solution orbit draws a helical curve, which changes amplitude in a vicinity of the origin. Some solutions change only the amplitude, while some solutions change not only the amplitude but also the axis along which they go to infinity as $t \to \infty$. Craik and Okamoto have found a four-leaf structure and a periodic orbit, which play an important role in controlling the solution orbits. We prove the existence of such a periodic orbit by a method of numerical verification.