Scope of the phase-only sinusoidal oscillator idealization

Consider a finite population of self-sustained oscillators, meaning systems that maintain periodic motion without an externally imposed periodic drive. For each isolated oscillator suppose a stable limit cycle is specified: a periodic orbit approached by sufficiently nearby motions. A phase map assigns a position modulo one cycle to states in a chosen neighborhood of that orbit; on the orbit the phase advances uniformly in time. The phase-only description retains this angular position and discards the remaining state variables. Weak coupling means that interactions perturb the isolated oscillations only slightly; near identity means that the isolated oscillators have similar dynamics. These qualitative conditions motivate a phase description but do not by themselves specify an error bound.

Adopt the following additional idealization for a finite nonempty vertex set $V$. Vertex $x$ represents oscillator $x$, and its phase ${\theta }_x$ is an angle modulo $2\pi$. Its intrinsic frequency ${\omega }_x$ is its angular rate when coupling is absent. Let $A_{xy}=A_{yx}\ge 0$ be the strength of the pairwise influence between $x$ and $y$, with $A_{xx}=0$; zero strength means no edge. On locally chosen real angular representatives, assume ${\dot{\theta }}_x={\omega }_x+{\sum }_{y\in V}{A}_{xy}\sin ({\theta }_y-{\theta }_x)$. Thus the interaction is sinusoidal in the phase difference, symmetric in its weights, and attractive in the sense that a small positive phase difference contributes positively to ${\dot{\theta }}_x$. In the homogeneous specialization assume ${\omega }_x=\omega$ for every $x\in V$.

The phase-only description, the sinusoidal graph interaction, symmetric nonnegative weights and identical frequencies are modeling assumptions. They are not an observation, an experimental calibration, or an assertion that a generic oscillator system obeys this equation exactly.

Phasors, trigonometric identities and periodic derivatives

Strogatz, From Kuramoto to Crawford, Section 2 and Section 3.1, motivates phases through weak coupling and near identity and presents the equally weighted all-to-all sinusoidal model. The weighted graph and homogeneous-frequency specialization above are separately adopted choices; the source citation does not establish them for a physical apparatus.