Antispiral Waves as Sources in Oscillatory Reaction−Diffusion Media

Abstract

Spiral and antispiral waves are studied numerically in two examples of oscillatory reaction-diffusion media and analytically in the corresponding complex Ginzburg-Landau equation (CGLE). We argue that both these structures are sources of waves in oscillatory media, which are distinguished only by the sign of the phase velocity of the emitted waves. Using known analytical results in the CGLE, we obtain a criterion for the CGLE coefficients that predicts whether antispirals or spirals will occur in the corresponding reaction-diffusion systems. We apply this criterion to the FitzHugh-Nagumo and Brusselator models by deriving the CGLE near the Hopf bifurcations of the respective equations. Numerical simulations of the full reaction-diffusion equations confirm the validity of our simple criterion near the onset of oscillations. They also reveal that antispirals often occur near the onset and turn into spirals further away from it. The transition from antispirals to spirals is characterized by a divergence in the wavelength. A tentative interpretaion of recent experimental observations of antispiral waves in the Belousov-Zhabotinsky reaction in a microemulsion is given.

Dynamical structures in binary media of potassium-driven neurons

“…These are rotating spirals or target waves which run from outward to the center called antispirals or antiwaves, respectively. Such patterns have been found in the Belousov-Zhabotinsky reaction and elsewhere and can be described by reaction-diffusion systems [43,44,45]. Note, that in our case the rotational direction of the spirals and the propagation of the waves is the same as for common waves like presented in Set #1.…”

Negative refraction in nonlinear wave systems

“…4(a). In and [18]. From this correspondence, we justify NW and AW for the left and right media of Fig.4(b), respectively, and expect to observe negative refraction.…”

“…Chemical reaction diffusion (RD) systems are the most probable candidates of this kind. Here we take a well known chemical reaction model, the 2D Brusselator, as our example [18,19] …”

Amplitude equations for reaction-diffusion systems with cross diffusion

“…For example, if c 1 = c 2 = 0, both the group and phase velocities of waves are zero [16]. If we perturb the SS locally, then a pseudo-oscillon can emerge, i.e., a localized spot that oscillates for many periods while the remainder of the system is quiescent.…”

“…This approach has been used for many reaction-diffusion models, including the Brusselator [9,14], Gray-Scott, Rössler [15], FitzHugh-Nagumo [16,17], and Lengyel-Epstein models [18], as well as a model for CO oxidation on a Pt surface [19].…”