Global stability of the Lengyel-Epstein systems

TL;DR

This work analyzes the global dynamics of the Lengyel-Epstein (Belousov-Zhabotinsky) two-parameter cubic system. It uses Poincaré compactification, invariant sets, and blow-up techniques to characterize infinite equilibria and establish when the unique equilibrium is not globally attracting, while deriving explicit parameter regions ($A$, $B$) yielding local or regional stability and describing the basin of attraction. The authors also characterize Hopf bifurcations, compute Lyapunov coefficients, identify a Bautin point, and reveal parameter zones with two coexisting limit cycles, elucidating the rich bifurcation structure of the model. The results enhance understanding of global stability, basins of attraction, and bifurcation-driven transitions in Belousov-Zhabotinsky-type chemical dynamics, with implications for predicting multi-stability and oscillatory behavior in reaction-diffusion systems.

Abstract

We study the global (asymptotic) stability of the Lengyel-Epstein differential systems, sometimes called Belousov-Zhabotinsky differential systems. Such systems are topologically equivalent to a two-parameter family of cubic systems in the plane. We show that for each pair of admissible parameters the unique equilibrium point of the corresponding system is not globally (asymptotically) stable. On the other hand, we provide explicit conditions for this unique equilibrium point to be asymptotically stable and we study its basin of attraction. We also study the generic and degenerate Hopf bifurcations and highlight a subset of the set of admissible parameters for which the phase portraits of the systems have two limit cycles.

Figures (2)

• Figure 1: (a) Desingularization of systems \ref{['u2eq:01']} using polar blow-ups where $\theta_1=0$, $\theta_2=\pi/2$, $\theta_3=\pi$, $\theta_4=\pi+\arcsin(\sqrt{5} - 2)$, $\theta_5=3\pi/2$ and $\theta_6=2\pi - \arcsin(\sqrt{5} - 2)$. (b) Topological local phase portraits at the origin of systems \ref{['u2eq:01']}. (c) Dynamics near infinity of systems \ref{['eq:02']} in the Poincaré disk. The equilibrium point $I_4$ has a parabolic sector that intersects the interior of the Poincaré disk.
• Figure 2: (a) The set of admissible parameters: $A$ is defined in \ref{['eq:05']}, $H$ is the Hopf curve, $a_1 = 3 \sqrt{3}$, $a_2 = 5 \sqrt{5/3}$, $I = (5 \sqrt{5}, 2 \sqrt{5})$ is the intersection point of the graphs $b = b_H$ (blue) and $b = b_a$ (red). (b) Phase portrait of \ref{['eq:02']} for $(a, b) = (24.712, 13.85) \in D$.

Theorems & Definitions (5)

• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 1
• Lemma 2