Resonance & Synchronization
Many systems have a natural rhythm. Put rhythms near one another and new behavior appears: beats, normal modes, energy exchange, phase pulling and synchronization. Change the parameters yourself and watch the collective motion emerge.
Beats are interference in time.
For two nearby frequencies, the sum repeatedly grows and fades. The fast oscillation is wrapped in a slower envelope. For equal amplitudes, the envelope repeats at the difference frequency |f₁ − f₂|.
The phenomenon does not require the oscillators to exchange energy; simple superposition is enough.
Coupling creates collective normal modes.
Two identical oscillators connected together no longer have only their individual natural motion. The system has an in-phase mode and an out-of-phase mode. A localized initial displacement is a mixture of both, so energy can move back and forth between the two masses.
For this dimensionless model: ω₊ = ω₀ and ω₋ = √(ω₀² + 2κ).
Synchronization is an emergent order parameter.
Each oscillator keeps its own natural frequency, but coupling pulls phases toward one another. The Kuramoto order parameter r measures phase coherence: r ≈ 0 means phases are widely dispersed; r ≈ 1 means they are tightly aligned.
This finite simulation shows the mechanism, not a universal critical coupling. The location of any transition depends on the frequency distribution, network, finite size and model assumptions.