An Efficient Adjoint-Free Melnikov–Lyapunov Diagnostic for Transition Detection in Slow–Fast Chemical Reaction Networks

Chemical reaction networks (CRNs) with pronounced time-scale separation exhibit complex dynamical behaviours including oscillations, bistability, and abrupt regime transitions. Detecting parameter thresholds at which these qualitative changes occur is essential for understanding biochemical systems but poses significant computational challenges, particularly for medium-to-high-dimensional stiff systems with singular perturbation parameter ε < 10⁻³.This work develops a computationally efficient adaptation of Melnikov’s method tailored for nonlinear slow–fast CRNs, validated on systems of dimension n = 2–10. The principal innovation is an adjoint-free formulation: the full variational adjoint system is replaced by an exponentially weighted surrogate w(t) = e⁻ˡᵗ derived from the dominant stable eigenvalue of the fast subsystem Jacobian. This substitution reduces per-evaluation complexity from O(n²Tsteps) to O(nTsteps) and eliminates the need for backward integration of potentially unstable adjoint equations.For extreme stiffness regimes (ε < 10⁻³) we introduce a Melnikov–Lyapunov surrogate that replaces geometric normal information with Lyapunov energy-gradient information, stabilising computation while preserving transition-detection capability. Rigorous theoretical justification under geometric singular perturbation theory (GSPT) proves that the simplified functional approximates the classical Melnikov integral to order O(ε).Validation on seven representative biochemical models—including a substrate inhibition oscillator (2D), enzymatic feedback network (3D), glycolysis oscillator (7D), MAPK cascade (8D), and circadian rhythm network (10D)—demonstrates relative errors of 2.8–8.1% compared with MATCONT continuation, with evaluation times below 0.8 s versus minutes-to-hours for full continuation. The method achieves a 235–612× speedup (average 401×), enabling parametric design studies requiring 100+ evaluations that were previously computationally prohibitive.