    October  1998, 4(4): 691-708. doi: 10.3934/dcds.1998.4.691

## Torsion numbers, a tool for the examination of symmetric reaction-diffusion systems related to oscillation numbers

 1 Mathematisches Institut, Justus-Liebig-Universität Gießen, Beskidenstr. 9, D-35398 Gießen, Germany

Received  November 1997 Revised  February 1998 Published  July 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 $(u_1(t,x), u_2(t,x))\ne (0,0)$. 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.
Citation: Matthias Büger. Torsion numbers, a tool for the examination of symmetric reaction-diffusion systems related to oscillation numbers. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 691-708. doi: 10.3934/dcds.1998.4.691
  Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515  Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304  A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65  Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49  Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189  C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872  Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191  Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21  Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815  Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415  Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523  Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179  Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69  Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891  Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669  Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301  Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001  Ana Carpio, Gema Duro. Explosive behavior in spatially discrete reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 693-711. doi: 10.3934/dcdsb.2009.12.693  Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105  Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020053

2018 Impact Factor: 1.143