Transitions in weakly-coupled nonlinear oscillators

Paul H. Steen, Stephen H. Davis

Research output: Contribution to journalArticlepeer-review


Two oscillators having natural frequencies δ and γ and damping coefficients proportional to -λ are weakly coupled through cubic nonlinearities. As λ is increased through λc ≡ 0, nonlinear solutions of amplitude ɛ bifurcate. To leading order in ɛ such systems (including the case of coupled van der Pol oscillators) have behaviors governed by the single parameter ρ ∼ 2(δ-γ)/(λ-λc). ρ has a critical value ρc determined by a Josephson-junction equation. For |ρ| > ρ the oscillations are quasiperiodic; the modulations die out in the far-from-resonance case |ρ| → ∞. For |ρ| < ρ there are phase-locked oscillations and competing stable states can occur. An increase in λ, causing |ρ| to cross ρ, results in a type of reverse bifurcation (quasiperiodic to periodic). No chaotic behavior is found.

Original languageEnglish (US)
Pages (from-to)1-5
Number of pages5
JournalNorth-Holland Mathematics Studies
Issue numberC
StatePublished - Jan 1 1983
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics


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