Singular Singularly Perturbed Systems

A Singular Singular Perturbation Problem Arising From a Class of Biomolecular Feedback Controllers

“…In particular, if one regards the concentrations of the two controller species as fast variables, the Jacobian of the boundary layer (BL) is singular everywhere in the state space. In SP literatures, such a system is called a singular singularly perturbed (SSP) system [10], [11], [12], [13]. While existing results have provided conditions to reduce the dimensionality of SSP systems, we find that the FSFs we consider do not satisfy these conditions.…”

“…This property arises from the annihilation reaction, c 1 + c 2 → ∅, which affects both fast variables (c 1 and c 2 ) in an identical fashion (−θc 1 c 2 / ). SSP problems have been studied in [10], [11], [12], [13]. For these results to be applicable to (2), it is necessary for the Jacobian matrix of the fast timescale system (3) evaluated at = 0:…”

“…to have a zero eigenvalue with same algebraic and geometric multiplicities. Assuming that this condition is satisfied, system (2) can be transformed into standard SP form under additional technical conditions [10], [11], [12], [13]. However, the zero eigenvalue of (5) has algebraic multiplicity 2 and geometric multiplicity 1 for any positive c 1 , c 2 and parameters α and θ.…”

“…When A 0 22 has a zero eigenvalue, we are faced with an SSP problem. For a subclass of SSP problems, where the multiplicities of the zero eigenvalue satisfy µ = λ, existing results can be applied for model reduction [10], [11], [12], [13]. However, the type of singularity in Assumption 1, where µ = q + 1 = q = λ, cannot benefit from these previous results.…”

“…In this section, we analyze the error dynamics between the full system (9) and the candidate reduced system (12) to demonstrate that their trajectories are close to each other.…”