We show in this paper that the analysis of diffusion-induced instability in spatially extended models can be performed by separating local dynamics from diffusion. This is possible not only in the case studied by Turing, namely models with two interacting variables, but also in the general case of three or more variables. The advantage of this decomposition, based on the notion of potential Tur...

We study a 2-spin quantum Turing architecture, in which discrete local rotations {αm} of the Turing head spin alternate with quantum controlled NOT-operations. We show that a single chaotic parameter input {αm} leads to a chaotic dynamics in the entire Hilbert space. The instability of periodic orbits on the Turing head and ‘chaos swapping’ onto the Turing tape are demonstrated explicitly as we...

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to g...

In this paper, we investigate the emergence of a predator-prey model with Beddington-DeAngelistype functional response and reaction-diffusion. We derive the conditions for Hopf and Turing bifurcation on the spatial domain. Based on the stability and bifurcation analysis, we give the spatial pattern formation via numerical simulation, i.e., the evolution process of the model near the coexistence...

Abstract Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these from a quiescent state. A key issue that standard techniques one-dimensional proven insufficient understanding localisation higher dimensions. In this work, we present co...

The problem of explaining the rich spectrum of different patterns exhibited by mammals has a long history. In a controversial paper (1952) Turing proposed a Reaction-Diffusion (RD) model to explain at a macroscopic scale the process of pattern formation, as related to the occurrence of what he called a diffusion-driven instability. A typical Turing system consists of at least two chemical speci...

Turing instability in activator-inhibitor systems provides a paradigm of nonequilibrium pattern formation; it has been extensively investigated for biological and chemical processes. Turing pattern formation should furthermore be possible in network-organized systems, such as cellular networks in morphogenesis and ecological metapopulations with dispersal connections between habitats, but inves...