coupledoscillator

A coupled oscillator is a system in which two or more individual oscillators interact through couplings that allow energy and information to be exchanged. The interaction leads to collective dynamics that differ from the behavior of isolated components. Coupled oscillators appear in mechanical, electrical, optical, chemical, and biological contexts. They can exhibit synchronized motion, phase locking, amplitude modulation, or complex chaotic behavior depending on the strength and form of the coupling and the intrinsic properties of the components.

Mathematically, N coupled oscillators can be described by a set of differential equations for their generalized coordinates x_i(t), i = 1,...,N. A common form is m_i x_i'' + gamma_i x_i' + k_i x_i + sum_j K_ij (x_i - x_j) = F_i(t), where m_i are masses, gamma_i damping, k_i stiffnesses, and K_ij the coupling strengths between oscillators i and j. The coupling matrix K is often symmetric with zeros on the diagonal. In the linear regime, the system can be decomposed into normal modes via eigenvalue analysis.

In nonlinear cases, resonance and synchronization phenomena arise. As coupling increases, oscillators can lock their phases and evolve with a common frequency, even if natural frequencies differ. The Kuramoto model is a widely studied phase-only description for weakly coupled limit-cycle oscillators, highlighting the transition to synchronization. Other behaviors include amplitude death, chimera states, and pattern formation in networks.

Classic demonstrations include coupled pendulums linked by a shared support, metronomes on a moving cart, and arrays of LC circuits connected by capacitive or inductive couplings. In biology and neuroscience, coupled oscillators model circadian rhythms, cardiac cells, and neural synchronization. Mathematics and physics tools such as normal-mode analysis, perturbation theory, and bifurcation theory are used to study these systems.

Related topics include synchronization, normal modes, and network dynamics. See also Kuramoto model, coupled systems, and oscillator networks.