Torsion numbers, a tool for the examination of symmetric reaction-diffusion systems related to oscillation numbers
Matthias Büger · Discrete and Continuous Dynamical Systems · 1998
Given a solution of a symmetric reaction-diffusion system of the form $\frac d{ dt} u_k = \lambda \frac{d^2}{dx^2} u_k + u_k \hat{g} (t,x,u_1,u_2,\frac d{dx}u_1,\frac d{dx}u_2, \frac{d^2}{ dx^2}u_1,\frac{d^2}{ dx^2}u_2,\sqrt{u_1^2+u_2^2} )$, $k=1,2$, with Dirichlet boundary conditions on the interval $(0,1)$, we introduce a non-negative number called torsion number which vanishes iff the solution is planar, where we call the solution $(u_1,u_2)$ planar, if the curves $\gamma_t: x\mapsto (u_1(t,x),u_2(t,x))\in\mathbb{R}^2$, for $x \in [0,1]$ and $t>0$, are contained in a space $\{\xi(\cos\a,\sin\a):\xi\in\mathbb{R}\}\subset\mathbb{R}^2$, for some $\a\in [0,2\pi)$ and all $t>0$. Loosely speaking, the torsion number measures the torsion of the curve $x\mapsto (x,u_1(t,x),u_2(t,x))\in\mathbb{R}^3$, for $x \in [0,1]$. We introduce a function called angle function $\varphi(t,x)$ which is a continuous and coincides with the polar angle $\arctan $ $u_2(t,x)$/$u_1(t,x))$ wherever $(u_1(t,x), u_2(t,x)) e (0,0)$. Then the torsion number is given by the difference between the supremum and the infimum of $\varphi(t,\cdot)$. Under certain conditions, which are, in particular, satisfied if the underlying solution is stationary, we show that this torsion number is either strictly decreasing in time or it vanishes identically. Torsion numbers are designed to play a role in the investigation of reaction-diffusion systems. Their role is comparable to the role of oscillation numbers which are a useful tool for the examination of solutions of one single reaction-diffusion equation.