Diffusive instabilities in hyperbolic reaction-diffusion equations

We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.

Figure 1

Wave number versus control parameter $k-B$ diagram for the hyperbolic Brusselator with fixed $A=5$ and (a) $D_u=3,D_v=1$ and (b) $D_u=1,D_v=3$. The curves represent the null-cline equation $h(k,B)=0$.

Figure 2

Turing instability. The dispersion curves for real eigenvalues ${\lambda }_1$ (thick line) and ${\gamma }_1$ (thin line) for the hyperbolic and parabolic Brusselators, respectively, are shown for various values of the control parameter $B$: (i) before the Turing bifurcation point at $B=14.6$, (ii) at the Turing bifurcation point with the critical value $B_{\text{T}}=15.106836$ and the critical wave number $k_{\text{T}}=1.6990442$, and (iii) beyond the Turing instability at $B=16.5$. The model parameters are fixed at $A=5, D_u=1, D_v=3$, i.e., the case of Fig. 1, and $\tau =1$.

Figure 3

Criteria for the onset of the wave instability depicted by thick lines (i) $\mathrm{Re}\left({\lambda }_{1,3}\right)=-1/\left(2\tau \right)+y=0$ and (ii) $d\mathrm{Re}\left({\lambda }_{1,3}\right)/dk=dy/dk=0$ in the wave number versus control parameter $k-B$ plane for the hyperbolic Brusselator at fixed $A=5, D_u=3, D_v=1$, i.e., the case of Fig. 1, and $\tau =1$. The intersection of (i) and (ii) curves corresponds to the wave bifurcation point with the critical values $B_{\text{w}}=1.4693878$ and $k_{\text{w}}=3.4255939$. The thin lines represent the null-cline equation $h=0$; see Fig. 1.

Figure 4

Wave instability. The dispersion curves for real $\mathrm{Re}\left({\lambda }_{1,3}\right)=-1/\left(2\tau \right)+y$ (thick line) and imaginary $\mathrm{Im}\left({\lambda }_{1,3}\right)=z$ (thin line) parts of the eigenvalues ${\lambda }_{1,3}$ for the hyperbolic Brusselator at various values of the control parameter $B$: (i) below the wave bifurcation point at $B=0.1$, (ii) at the bifurcation point with the critical value $B_{\text{w}}=1.4693878$ and the critical wave number $k_{\text{w}}=3.4255939$, and (iii) beyond the wave instability point at $B=10$. The model parameters are fixed at $A=5, D_u=3, D_v=1$, i.e., the case of Fig. 1, and $\tau =1$.

Figure 5

Wave instability. Critical values $B_{\text{w}}$ (circles) and $k_{\text{w}}$ (diamonds) as functions of the inertial time $\tau$ for $A=5, D_u=3$, and $D_v=1$, i.e., the case of Fig. 1, for the hyperbolic Brusselator.

Figure 6

Turing instability. The dispersion curves for real eigenvalues ${\lambda }_1$ (thick line) and ${\gamma }_1$ (thin line) for the hyperbolic and parabolic Gierer-Meinhardt models, respectively, at various values of the control parameter $b$: (i) before the Turing bifurcation point at $b=8$, (ii) at the Turing bifurcation point with the critical value $b_{\text{T}}=5.8359213$ and the critical wave number $k_{\text{T}}=0.58450046$, and (iii) beyond the Turing bifurcation point at $b=4$. The model parameters are fixed at $a=9, D_u=1, D_v=50$, and $\tau =1$.

Figure 7

Wave instability. The dispersion curves for (a) real $\mathrm{Re}\left({\lambda }_{1,3}\right)=-1/\left(2\tau \right)+y$ and (b) imaginary $\mathrm{Im}\left({\lambda }_{1,3}\right)=z$ parts of the eigenvalues ${\lambda }_{1,3}$ for the hyperbolic Gierer-Meinhardt model at various values of the control parameter $b$: (i) before the wave bifurcation point at $b=10$, (ii) at the bifurcation point with the critical value $b_{\text{w}}=5.4166667$ and the critical wave number $k_{\text{w}}=1.6583124$, and beyond the wave bifurcation point at (iii) $b=4.5$, (iv) $b=4.03$, and (v) $b=3$. (The critical value for the Hopf bifurcation is $b_{\text{H}}=4.0275875$.) The model parameters are fixed at $a=2, D_u=5, D_v=4$, and $\tau =3$.

Figure 8

Wave instability. Critical values $b_{\text{w}}$ (circles) and $k_{\text{w}}$ (diamonds) as functions of the inertial time $\tau$ for $a=2, D_u=5$, and $D_v=4$ for the hyperbolic Gierer-Meinhardt model.