A system of harmonic oscillators which can exchange energy is described by a system of coupled second order linear ordinary differential equations. The classic example is a set of n masses coupled by springs in some configuration.
The complete solution of such a system consists of a sum of n special solutions called Normal Modes. A normal mode is a solution of the system in which all the masses execute sinusoidal motion at the same frequency, a normal mode frequency. That is, the solution has the form x1=A1*cos(ωt+φ1), x2=A2*cos(ωt+φ2), etc. Each individual normal mode consists of a frequency and a set of relationships between the As and φs.
Remember, n masses each moving in 1-D gives n different modes at n frequencies, some of which might be the same as each other. If the masses can move in more directions then each direction adds n more modes. For example 3 pendula allowed to swing in any direction have 6 modes, 3 masses × 2 directions.
The lowest frequency mode is always the most symmetric, with all masses moving in the same direction at once. The highest frequency mode is always the most anti-symmetric, with adjacent masses moving in opposite directions at all times.
For example the system of two identical masses, m, tied together and to a set of walls by three identical springs, k, has two normal modes.
Lower frequency mode ω1=√(k/m), masses move together, A1=A2.
Higher frequency mode ω2=√(3k/m), masses move equal and opposite, A1=-A2.
The total solution is a sum of the two modes
x1=A1cos(ω1t+ φ1) + A2cos(ω2t+ φ2) and x2=A1cos(ω1t+ φ1) - A2cos(ω2t+ φ2)
The four arbitrary constants, A1, A2, φ1, and φ2 are determined by the four initial conditions, the starting positions and velocities of the two masses.