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How an anomalous cusp bifurcates in a weak-noise system
1Mathematics Department, University of Arizona, Tucson, Arizona 85721 and Physics Department, University of Arizona, Tucson, Arizona 85721, USA.
Physical Review Letters
|September 6, 2000
Summary
Singularities in double well systems reveal critical phenomena. We found a scaling law governing anomalous cusp bifurcations, extending catastrophe theory for noise-perturbed systems.
Area of Science:
- Statistical physics
- Nonlinear dynamics
- Complex systems
Background:
- Symmetric double well systems exhibit complex activated trajectories.
- Absence of detailed balance can lead to singularities like cusps.
- These phenomena are analogous to optical caustics.
Purpose of the Study:
- To investigate the nature and bifurcation of singularities in activated trajectories.
- To derive a scaling law for anomalous cusp formation.
- To extend classical catastrophe theory to noise-perturbed systems.
Main Methods:
- Analysis of activated trajectories in a symmetric double well system.
- Derivation of scaling laws and nonpolynomial equations of state.
- Examination of system quasipotential to understand bifurcations.
Main Results:
- Identified anomalous cusps (coinciding with saddle points) in trajectory patterns.
- Derived a scaling law governing the bifurcation of anomalous cusps into conventional ones.
- Demonstrated that these bifurcations are reflected in the system quasipotential, akin to phase transitions.
Conclusions:
- The study reveals critical phenomena in noise-perturbed systems.
- Results extend classical catastrophe theory by describing cusp bifurcations.
- The findings offer insights into nonclassical critical points and system dynamics.
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