Physica D 36 (1989) 167-172 North-Holland, Amsterdam EVOLUTION OF TWIST OF AN AUTOWAVE VORTEX V.N. BIKTASHEV Research Computing Centre, 142292, Pushchino, Moscow Region, USSR Received 3 October 1988 Revised manuscript received 21 December 1988 Communicated by A.V. Holden An equation is derived for the evolution of an inhomogeneousiy twisted scroll wave in a general reaction-diffusion system, with twist regarded as an order parameter and not assumed to be small. The equation describes the interaction of dispersion and diffusion of the twist. A criterion is given of twist alignment in the absence of twisting forces. 1. Introduction Reaction-diffusion systems with autowave properties may support sources of autowaves. The best studied sources for a 2-dimensional homoge- neous autowave medium are steadily rotating spi- ral waves, also called vortices, rotors, reverbera- tors or leading circles (e.g., see [1]). Their proper- ties only depend on the medium parameters. A 3-dimensional medium admits a much greater va- riety of sources of the vortex type [2]. The simplest of them is the simple scroll wave in which the state of the medium at each instant of time is independent of the coordinate along a certain fixed direction, and every perpendicular cross section shews a reverberator. A :nore compficate,9.~ source is the twisted sex ~11, in which every cross secfien shows a reverbero¢or- like re,me, wLth the phase of ~e reverbezator changing linearly fiom one cross s~fion t,~ an- other. This is a stationary re, me whose pr¢ger- ties, beside the medium parameters, depeed on ,~he rate of the phase shift along the scroll a.vds. V~ call this rate "twist" and denote it by w. The deflrfition of twist might have been done more precisely for vortices with non-planar filaments, but here we are interested in the simplest case of straight filament only, so that ambiglfity does not appear. For the model of the FitzHugh-Nagumo type with piece-linear fight-hand sides sucla effects were studied in [3, 4]. Panfi!ov et al. [3] proposed a method for numerical investigation of twisted scroU, and described results of some numerical experhnents; Mikhailov et al. [4] developed an analytical theory for uniformly twisted scroll. Later on, the regimes were investigated by Br~hnik et al. [5] with a phenomenological model of rare waves in exdtable medium. All Authors conc~.uded that the twisted scroll has a higher frequency than the simple one with the same medium parameter values. If the twist varies rather slowly along the scroll axis, then for small enough time and space scales the vortex will behave much the same way as the case of constant twist. And hew does it behave 0167-2789/89/$03.50 © Elsevier Science Publishe~z 3.V. (North-HoUand Physics Pubfishing Division)