Complex mode-locking oscillations and Stern-Brocot derivation tree in a CSTR reaction with impulsive perturbations

Area of Science:

• Chemical Engineering
• Nonlinear Dynamics
• Complex Systems

Background:

• Continuous stirred tank reactors (CSTRs) are fundamental in chemical processes.
• Understanding complex dynamics like mode-locking is crucial for reactor control.
• Impulsive perturbations can significantly alter reactor behavior.

Purpose of the Study:

• To identify and characterize the mode-locking topology in a CSTR model under impulsive perturbations.
• To elucidate the global structure and interconnections of mode-locking oscillations.
• To provide a framework for classifying mode-locking phenomena in impulsive systems.

Main Methods:

• High-resolution stability diagrams were employed to analyze the system.
• Bi-parametric planes were used to visualize mode-locking structures.
• Analysis focused on the relationship between control parameters and oscillation complexity.

Main Results:

• A complex mode-locking (entrainment) topology was identified in the CSTR model.
• The structure of mode-locking oscillations was found to be organized by the Stern-Brocot sum tree.
• Mode-locking organization is governed by an invariant torus initiated from Hopf bifurcations.

Conclusions:

• The study provides a novel description and classification of mode-locking oscillations for impulsive CSTR systems.
• The findings demonstrate a precise agreement between mode-locking order and the reciprocal of the Stern-Brocot sum tree.
• This research contributes to a deeper understanding of complex dynamics in chemical reactors.