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Hopf bifurcation in addition-shattering kinetics.

S S Budzinskiy1,2, S A Matveev1,2, P L Krapivsky3,4

  • 1Faculty of Computational Mathematics and Cybernetics, Lomonosov MSU, 119991 Moscow, Russia.

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Summary
This summary is machine-generated.

This study provides numerical evidence for never-ending oscillations in addition-shattering processes. These oscillations emerge via a Hopf bifurcation when a fixed point becomes unstable in a specific parameter region.

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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.