Nonlinear effects on Turing patterns: Time oscillations and chaos

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, the patterns oscillate in time. When varying a single parameter, a series of bifurcations leads to period doubling, quasiperiodic, and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examine the Turing conditions for obtaining a diffusion-driven instability and show that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. These results demonstrate the limitations of the linear analysis for reaction-diffusion systems.

Figure 1

The bold line is the saddle-node bifurcation curve (2) of system (1) without diffusion. (Only the region $C<0$ is shown.) In the shaded region, $\left(0,0\right)$ is the only equilibrium point, which is linearly stable. The dashed curve is the locus of a Hopf bifurcation calculated numerically from Eq. (1) as described in Sec. 3.

Figure 2

Numerical simulations of the system (1) in one dimension for $H=3$ and $D=0.08$. The left column shows the space-time plot of $u$ for three values of $C$. (The plots for $v$ are similar.) On the right, the time series registered at the central point $u\left(25\right)$ of the corresponding space-time plots are shown. (a) $C=-0.6$, (b) $C=-1$, and (c) $C=-1.5$.

Figure 3

(a) Numerically calculated bifurcation diagram of system (1) using auto. Supercritical Hopf (HB), period doubling (PD), and torus (TR) bifurcations are detected at $C=-0.7546515$, $-1.200656$, and $-1.281831$, respectively. (b) Numerically calculated maximal Liapunov exponents.

Figure 4

Phase portraits of the central points for different values of $C$ after 50 000 time iterations ($t=500$). (a) $C=-1.15$, (b) $C=-1.21$, (c) $C=-1.285$, and (d) $C=-1.5$.

Figure 5

Power spectrum for $C=-1.285$ after 500 000 iterations ($t=5000$). The frequency ${\omega }_1=0.3382$ and its integer multiples appear after the Hopf bifurcation. The period doubling bifurcation introduces the frequency ${\omega }_1/2$ and the torus bifurcation ${\omega }_2=0.2732$. Inset: Power spectrum for $C=-1.455$ showing a continuous band of frequencies, typical of a chaotic oscillation.