Stochastic dynamics in a two-dimensional oscillator near a saddle-node bifurcation

Area of Science:

• Nonlinear Dynamics
• Quantum Electronics
• Computational Neuroscience

Background:

• Nonlinear amplifiers, like the two-junction superconducting quantum interference device (SQUID), exhibit complex dynamics.
• These systems can transition between stable states and oscillatory "running states" via bifurcations.
• Enhanced sensitivity to weak magnetic signals is observed near the onset of spontaneous oscillations.

Purpose of the Study:

• To analytically approximate the oscillatory behavior of nonlinear amplifiers past a saddle-node bifurcation.
• To compute the oscillation period and validate scaling laws.
• To model the system's dynamics using an integrate-fire model and compare with numerical simulations.

Main Methods:

• Center manifold technique applied to the oscillator equations.
• Analytical approximation of oscillatory behavior near bifurcation.
• Renewal theory used for calculating interspike interval and power spectral density.
• Comparison with results from numerical simulations.

Main Results:

• Analytical approximation of oscillation period obeying standard scaling laws.
• Successful representation of dynamics using an integrate-fire model.
• High agreement between theoretical predictions (interspike interval, power spectral density) and numerical simulations.
• Observed noise-lowering effects like injection locking and heterodyning when driven by sinusoids.

Conclusions:

• The center manifold technique provides an effective analytical tool for understanding nonlinear amplifier dynamics near bifurcations.
• The integrate-fire model accurately captures the oscillatory behavior and statistical properties of the system.
• The study demonstrates a link between nonlinear dynamics in physical systems and computational neuroscience models.
• Exploitation of these dynamics offers potential for enhanced magnetic signal detection and signal processing.