Turing-like instabilities from a limit cycle

The Turing instability is a paradigmatic route to pattern formation in reaction-diffusion systems. Following a diffusion-driven instability, homogeneous fixed points can become unstable when subject to external perturbation. As a consequence, the system evolves towards a stationary, nonhomogeneous attractor. Stable patterns can be also obtained via oscillation quenching of an initially synchronous state of diffusively coupled oscillators. In the literature this is known as the oscillation death phenomenon. Here, we show that oscillation death is nothing but a Turing instability for the first return map of the system in its synchronous periodic state. In particular, we obtain a set of approximated closed conditions for identifying the domain in the parameter space that yields the instability. This is a natural generalization of the original Turing relations, to the case where the homogeneous solution of the examined system is a periodic function of time. The obtained framework applies to systems embedded in continuum space, as well as those defined on a networklike support. The predictive ability of the theory is tested numerically, using different reaction schemes.

Figure 1

The extended region of the Turing instability for the Brusselator model, as parameters $b$ and $c$ are varied. The diffusion coefficients were $D_{\varphi }=0.07, D_{\psi }=0.5$. The solid line shows the Hopf bifurcation for the aspatial model: above the line the aspatial system converges toward a stable fixed point. Below the line, a stable homogeneous limit-cycle solution is instead found. The circular symbols show results from the Floquet approach: the larger symbols indicating an instability, the smaller ones indicating that the homogeneous system is stable. The shaded region identifies the region of parameter space where the instability is predicted to occur, following Eq. (42).

Figure 2

Dispersion relations for the Brusselator model for three parameter choices, calculated from the Floquet analysis. Fixing $b=2.5$, we used $c=1.3$ (purple squares), $c=1.2$ (red triangles), and $c=1.1$ (orange circles). The diffusion coefficients were $D_{\varphi }=0.07, D_{\psi }=0.5$.

Figure 3

The late time evolution for species $\varphi$ (left) and $\psi$ (right) for the Brusselator model inside the extended region of Turing-like order. The initial homogeneous limit-cycle state is disturbed by a small nonhomogeneous perturbation. The synchrony of the spatially coupled oscillators is lost and the system evolves towards a stationary stable configuration. The patterns resemble (indeed, under the Fourier lens, are identical to) the patterns obtained inside the classical Turing region, i.e., above the Hopf transition line. In other words, it looks like the same Turing attractor can be reached following two alternative dynamical pathways. Parameters are $b=2.4$; $c=1.2$; $D_{\varphi }=0.07, D_{\psi }=0.5$. The simulations are carried our over a square box of linear size $L=10$ partitioned in 64 mesh points.

Figure 4

Stationary pattern attained by the Brusselator model, defined on a network of the Watts-Strogatz type (number of node $N=100$ and probability of rewiring $p=0.8$). In the main panel, the asymptotic concentration of species ${\varphi }_i$ is plotted as function of the nodes index $i$. In the inset, the evolution of the concentration on a particular node is shown, in order to appreciate the transition from the initial oscillatory regime to the final stationary state. The parameters are set as in Fig. 3.

Figure 5

The extended region of the Turing instability for the Schnakenberg model, as parameters $\alpha$ and $\beta$ are varied. The diffusion coefficients were $D_{\varphi }=0.01, D_{\psi }=1$. The solid line shows the Hopf bifurcation for the aspatial model: above the line the aspatial system converges toward a stable fixed point. Below the line, a stable homogeneous limit-cycle solution is instead found. The circular symbols show results from the Floquet approach: the larger symbols indicating an instability, the smaller ones indicating that the homogeneous system is stable. The shading delimits the region where the instability is predicted to occur by Eqs. (42).

Figure 6

Dispersion relations for the Schnakenberg model for three parameter choices, calculated from the Floquet analysis. Fixing $\alpha =0.36$, we used $\beta =0.56$ (purple squares), $\beta =0.52$ (red triangles), and $\beta =0.48$ (orange circles). The diffusion coefficients were $D_{\varphi }=0.01, D_{\psi }=1$.

Figure 7

The final stationary state obtained for species $\varphi$ by initializing the Schnakenberg model inside the region where the homogeneous (hence aspatial) limit cycle is stable, and imposing a small perturbation to the initially synchronous oscillations. As already remarked in the caption of Fig. 3, the patterns are practically indistinguishable from those obtained inside the classical Turing region. Here also, it seems plausible to hypothesize that the same Turing attractor can be reached following alternative dynamical paths. Parameters are $a=0.125, b=0.475$ (or $\alpha =0.35$ and $\beta =0.6$), $D_{\varphi }=0.01, D_{\psi }=1$. The simulations are carried our over a square box of linear size $L=10$ partitioned in 64 mesh points.

Figure 8

Stationary pattern attained by the Schnakenberg model, defined on a network of the Watts-Strogatz type (number of nodes $N=100$ and probability of rewiring $p=0.8$). In the main panel, the asymptotic concentration of species ${\varphi }_i$ is plotted as function of the nodes index $i$. In the inset, the time evolution of the concentration on one of the nodes of the network is shown. The transition from the initial oscillation to the final stationary state is clearly displayed. The parameters are set as in Fig. 7.

Figure 9

Comparison of the matrix operator $\mathrm{\Gamma }$ used in Eqs. (26) (orange diamonds) and (36) (red circles). Results were obtained from the Brusselator model with reaction parameters $c=1.2$ and $b=2.25$.

Figure 10

Comparison of the matrix operator $\mathrm{\Gamma }$ used in Eqs. (26) (orange diamonds) and (36) (red circles). Results were obtained from the Brusselator model with reaction parameters $c=1.2$ and $b=2.35$.