Local Stability and Hopf Bifurcation in a Three-Dimensional Photocatalytic Microplastic Reactor Model with Adaptive Gain
SYMMETRY-BASEL, vol.18, no.8, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 18 Issue: 8
- Publication Date: 2026
- Doi Number: 10.3390/sym18081390
- Journal Name: SYMMETRY-BASEL
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, zbMATH, Academic Search Ultimate (EBSCO), Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Acibadem Mehmet Ali Aydinlar University Affiliated: Yes
Abstract
Adaptive feedback can destabilize a loop that would be stable under any fixed gain, so the speed at which the gain adapts is itself a design parameter. We study this effect in a minimal three-dimensional model motivated by the photocatalytic degradation of microplastics: a pollutant concentration is driven toward a setpoint by an ultraviolet (UV) actuator whose gain adapts online. The model has a single bilinear nonlinearity, so the local analysis can be carried out in closed form. Under an explicit feasibility condition, the system has a unique positive equilibrium. The Routh-Hurwitz criterion shows that this equilibrium is locally asymptotically stable below an explicit critical adaptation speed kappa c and unstable above it. At kappa=kappa c , a purely imaginary eigenvalue pair crosses the imaginary axis transversally, and a Hopf bifurcation occurs, with an explicit onset frequency. The first Lyapunov coefficient is computed in closed form; it separates a supercritical onset, for well-damped actuators, from a subcritical onset with hysteresis, for weakly damped actuators. Numerical experiments confirm the predicted limit cycle and the classification. All the stability results established here are local.