PT-Symmetric Gain–Loss Dynamics in Auto-Catalytic Reaction Networks

Auto-catalytic chemical reactions are open non-equilibrium systems
characterized by feedback-driven oscillations and instabilities that are
not always transparent within classical rate-equation formalisms. In
this work, we introduce a PT-symmetric gain–loss framework for
modeling auto-catalytic reaction networks, using the Belousov–
Zhabotinsky(BZ) reaction [1-3] as a motivating example. Chemical
species are mapped onto a three-level non-Hermitian Hamiltonian in
which autocatalysis and decay are represented by balanced gain and
loss terms. Eigenvalue analysis reveals unbroken, exceptional-point,
and broken PT phases corresponding to stable oscillations, critical
thresholds, and runaway amplification in chemical dynamics. Time
evolution is computed using matrix exponentiation e −iHt , revealing
novel gain-loss induced oscillations, PT-phase transitions, and
complex dynamical behavior in open chemical systems. This approach
establishes a dynamical-systems bridge between auto-catalytic
chemistry and PT-symmetric non-Hermitian quantum physics.