Global bifurcations at the onset of pulse self-replication

Area of Science:

• * Reaction-diffusion systems
• * Nonlinear dynamics
• * Computational physics

Background:

• * Pulse self-replication is a recently discovered phenomenon in reaction-diffusion systems.
• * Understanding the stability and bifurcations of solutions is key to explaining this behavior.
• * The one-dimensional Gray-Scott model provides a framework for studying these dynamics.

Purpose of the Study:

• * To numerically explore the global dynamics of pulse self-replication.
• * To analyze the stability and bifurcations of singular homoclinic stationary solutions.
• * To understand the implications for self-replicating pulse phenomena.

Main Methods:

• * Extensive numerical exploration of the one-dimensional Gray-Scott model.
• * Stability analysis of homoclinic stationary solutions.
• * Investigation of bifurcation scenarios and global bifurcation diagrams.

Main Results:

• * Identified a codimension-2 global bifurcation organizing solutions.
• * Revealed a hierarchy structure of folding bifurcation branches related to self-replication.
• * Found Bogdanov-Takens points and critical points are central to bifurcations and chaotic dynamics.
• * Observed modulating two-pulse or multipulse solutions accompanying self-replication.

Conclusions:

• * The global dynamics of pulse self-replication are organized by a complex bifurcation structure.
• * Homoclinic orbits and their bifurcations are critical for understanding solitary pulse solutions.
• * The findings provide insights into the mechanisms driving self-replication and associated chaotic behaviors in reaction-diffusion systems.