Nonclassical Turing instabilities induced by superdiffusive transport in FitzHugh-Nagumo dynamics

Abstract

We investigate diffusion-driven instabilities in a FitzHugh-Nagumo reaction-diffusion system with superdiffusive transport, modeled by fractional Laplacian operators with different diffusion orders for the activator and the inhibitor. A linear stability analysis yields explicit expressions for the instability threshold and the critical wavenumber and shows that superdiffusion modifies the band of unstable modes and the characteristic spatial scale of emerging patterns. We show that the threshold depends only on the ratio of the fractional exponents and on the kinetic parameters, while the spatial scale is controlled by the diffusion orders and the domain size. When the diffusion orders differ, spatial instabilities may occur even in regimes where the activator diffuses faster than the inhibitor, due to the combined effect of diffusion rates, anomalous scaling and system size. This leads to instability mechanisms that depart from the classical activator-inhibitor framework. A weakly nonlinear analysis near threshold provides the amplitude equation governing nonlinear saturation and reveals that superdiffusion promotes subcritical behavior. We also analyze the interaction between stationary and oscillatory instabilities near Turing-Hopf codimension-two points. All analytical results are supported by numerical simulations.

Figures (11)

• Figure 1: Different dynamical regimes of the local FitzHugh--Nagumo system \ref{['kin_1']}–\ref{['kin_2']} in the $(a,\gamma)$ plane. The black, dark gray, and light gray regions correspond to the monostable, excitable, and bistable regimes, respectively. The white region indicates the parameter values for which the unique equilibrium loses stability through a Hopf bifurcation. (a) $\beta=0$, $\varepsilon=2$. (b) $\beta=0$, $\varepsilon=0.9$.
• Figure 2: (a) Critical wavenumber $k_c$ as a function of $\alpha_1=\alpha_2$, for different values of $\Gamma$. (b) Length of the instability interval $[k_{\max}-k_{\min}]$, representing the range of unstable modes, as a function of $\alpha_1=\alpha_2$ for different values of $\Gamma$. The other system parameters are fixed at $a=-0.1$, $\beta=0.3$, $\gamma=2$, $\varepsilon=1$, and $d_c=6.9525$, while $d$ is set to $d=7.0220>d_c$.
• Figure 3: Dependence of the bifurcation threshold $d_c$ on $\alpha$ for different values of $\Gamma$. The other parameters are $a=-0.1$, $\beta=0.3$, $\gamma=2.3$, and $\varepsilon=2$.
• Figure 4: Dependence of the bifurcation threshold $d_c$ on $\alpha_1$ for $\alpha_2=1.3$ (a) and $\alpha_2=1.001$ (b), and of the critical wavenumber $k_c$ on $\alpha_1$ for different values of the width parameter $\Gamma$ with $\alpha_2=1.001$ (c). The dashed line represents the curve $d=\ell^{\alpha_1-\alpha_2}$, corresponding to the condition $D_V/D_U=1$ in the original variables. The other parameters are fixed as $a=0.1$, $\beta=0.4$, $\gamma=1.8$, and $\varepsilon=1.01$.
• Figure 5: Dispersion curves for $\alpha_2=1.001$: (a) varying $\Gamma$ with $\alpha_1=2$, (b) varying $\alpha_1>\alpha_2$ with $\Gamma=5$, and (c) corresponding one-dimensional pattern for $\alpha_1=2$, $\alpha_2=1.001$, and $\Gamma=5$. The remaining parameters are fixed at $a=0.1$, $\beta=0.4$, $\gamma=1.8$, and $\varepsilon=1.01$, and $d=d_c(1+0.1^2)$ in all panels.

Theorems & Definitions (12)

• Proposition 1: Existence of a unique equilibrium
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1
• Proof 1
• Theorem 2
• Proof 2
• Remark 1
• Remark 2: Dependence of $k_c$ on $\Gamma$