[Paper Review] Reliability of Coupled Oscillators I: Two-Oscillator Systems

Kevin Lin, Eric Shea‐Brown+1 more|ArXiv.org|Aug 22, 2007

Nonlinear Dynamics and Pattern Formation46 references3 citations

TL;DR

This paper investigates reliability in two coupled phase oscillators driven by noisy inputs, showing that reliability depends critically on the balance between feedforward and feedback coupling. Using stochastic dynamical systems theory, it identifies shear-induced chaos near phase-locking as a geometric mechanism causing unreliable responses when coupling strengths are comparable, especially at low input amplitudes.

ABSTRACT

This paper concerns the reliability of a pair of coupled oscillators in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations regardless of the network's initial condition. Our main result is that both reliable and unreliable behaviors occur in this network for broad ranges of coupling strengths, even though individual oscillators are always reliable when uncoupled. A new finding is that at low input amplitudes, the system is highly susceptible to unreliable responses when the feedforward and feedback couplings are roughly comparable. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed.

Motivation & Objective

• To understand how network architecture, particularly coupling geometry, affects reliability in coupled oscillator systems.
• To determine whether reliable responses can emerge in two-oscillator networks despite internal dynamics that may disrupt reproducibility.
• To identify geometric and dynamical mechanisms—particularly shear-induced chaos—that lead to unreliable behavior in such systems.
• To establish a mathematical framework linking Lyapunov exponents and reliability in stochastic oscillator networks.
• To explore the role of input amplitude and coupling strength balance in determining response reproducibility.

Proposed method

• Models the system as a stochastic differential equation with white noise input, treating the oscillator dynamics as phase oscillators on the circle.
• Uses top Lyapunov exponent (λ_max) as a quantitative measure of reliability, based on random dynamical systems theory.
• Applies Gronwall’s lemma to compare trajectories under varying feedback coupling (Δafb), estimating sensitivity to parameter changes.
• Employs rescaling techniques (δ → b²δ, z → z/b²) to analyze dynamics in the limit of small b, enabling asymptotic analysis near phase-locking regions.
• Compares trajectories of Systems A (baseline), B (no feedback), and C (with feedback) over finite time intervals to assess response divergence.
• Analyzes the role of the z-function (representing coupling strength) and its monotonicity in determining trajectory separation during transit through critical regions.

Experimental results

Research questions

• RQ1 Under what conditions does a two-oscillator system with feedforward and feedback coupling produce reliable, reproducible responses to repeated inputs?
• RQ2 How does the balance between feedforward and feedback coupling strengths affect the system’s reliability?
• RQ3 What geometric or dynamical mechanisms underlie unreliable behavior in coupled oscillator networks?
• RQ4 Why is the system particularly susceptible to unreliable responses when input amplitudes are low and coupling strengths are comparable?
• RQ5 Can Lyapunov exponents serve as a reliable indicator of response reproducibility in stochastic oscillator networks?

Key findings

• Reliability is not guaranteed even in simple two-oscillator systems; both reliable and unreliable behaviors coexist depending on coupling parameters.
• At low input amplitudes, the system exhibits high susceptibility to unreliable responses when feedforward and feedback coupling strengths are comparable.
• Shear-induced chaos at the onset of phase-locking is identified as the primary geometric mechanism driving unreliable dynamics.
• The top Lyapunov exponent (λ_max) serves as a valid and measurable indicator of reliability in the stochastic system.
• Trajectory divergence due to feedback coupling is quantitatively bounded: |a′ − a| < k₂b²Δafb, showing sensitivity to feedback strength at small b.
• For sufficiently large Δafb (> CΔω), the system exhibits persistent divergence (T(x₁) > x₁), indicating instability and unreliability, especially when Δafb > Kb⁻²Δω.