Hopf bifurcation in addition-shattering kinetics

Area of Science:

• Physical Chemistry
• Chemical Kinetics
• Nonlinear Dynamics

Background:

• Aggregation-fragmentation processes typically reach a steady state, implying an attractive fixed point.
• Asymptotically periodic motion is the next simplest dynamic behavior.
• Never-ending oscillations have been numerically observed but not rigorously established in these systems.

Purpose of the Study:

• To provide convincing numerical evidence for never-ending oscillations in a class of addition-shattering processes.
• To investigate the conditions under which these oscillations emerge.
• To explore the role of bifurcations in generating complex dynamics.

Main Methods:

• Numerical simulations of addition-shattering processes.
• Analysis of coupled ordinary differential equations.
• Investigation of parameter space regions for dynamic transitions.
• Identification of Hopf bifurcations.

Main Results:

• Convincing numerical evidence for never-ending oscillations was found in a specific parameter region (U).
• The fixed point of the system becomes unstable within this region U.
• Never-ending oscillations emerge through a Hopf bifurcation.

Conclusions:

• Addition-shattering processes can exhibit never-ending oscillations, challenging the typical steady-state assumption.
• Hopf bifurcations are a key mechanism driving the emergence of these oscillatory dynamics.
• The findings expand the understanding of complex dynamics in aggregation-fragmentation systems.