From Turing patterns to quasiperiodic dynamics: A global bifurcation route mediated by Turing-Hopf interactions

We investigate a global bifurcation route from stationary Turing patterns to quasiperiodic spatiotemporal dynamics in two-component reaction-diffusion systems, in which a positive parameter α is introduced into the second component. In this setting, Turing instability can coexist with a Hopf instability of the spatially homogeneous zero mode for suitable values of α. This creates an interaction between Turing-type spatial instability and homogeneous temporal oscillation. Using numerical continuation and direct time integration, we show that this interaction can generate oscillatory Turing patterns, torus bifurcations, and stable quasiperiodic solutions. In the Schnakenberg model, torus bifurcations arise through the interaction between Hopf bifurcations and fold bifurcations on nonconstant stationary branches. We further examine the Brusselator, Holling-Tanner, Mimura-Murray, and Lengyel-Epstein models to clarify which aspects of this mechanism are robust and which are model-dependent. Our results show that even two-component reaction-diffusion systems can exhibit rich global bifurcation structures leading to quasiperiodic pattern dynamics.