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Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so...
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Related Experiment Video

Updated: Sep 26, 2025

Kinematic History of a Salient-recess Junction Explored through a Combined Approach of Field Data and Analog Sandbox Modeling
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Self-organized quantization and oscillations on continuous fixed-energy sandpiles.

Jakob Niehues1, Gorm Gruner Jensen1, Jan O Haerter1,2,3

  • 1Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark.

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|April 16, 2022
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Summary

This study models local activation-inhibition dynamics, revealing complex checkerboard patterns and phase transitions in energy redistribution. The findings highlight robust energy levels within the system, even with noise.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Statistical Physics

Background:

  • Self-organization and activator-inhibitor dynamics are observed in atmospheric and biological systems.
  • Checkerboard-like spatiotemporal patterns emerge from such processes.

Purpose of the Study:

  • To investigate a simple model of local activation-inhibition processes.
  • To analyze the emergent spatiotemporal organization and phase transitions.

Main Methods:

  • A continuous-energy, non-Abelian sandpile model was employed.
  • Lattice sites redistribute energy exceeding a threshold to nearest neighbors.
  • Analysis of phase diagrams based on mean energy (μ) and spatial variance (σ).

Main Results:

  • Low mean energy leads to cessation of dynamics.
  • High mean energy results in diffusion-like behavior, removing spatial variance.
  • Intermediate mean energy exhibits checkerboard phases and complex, long-period phases.
  • Phase transitions are identified by discontinuous jumps in spatial variance.

Conclusions:

  • The model exhibits rich phase diagrams with transitions akin to first- and higher-order phase transitions.
  • A few dominant energy levels form sharp, noise-robust spikes in the density of states.