In the theory of dynamical systems, one may encounter many different kinds of oscillators, with many different formulations. Still, reading "through" all these seemingly different equations. one can always detect the presence of a similar structure or relationship between the variables involved in the oscillation (usually two). This common structure descends directly from the "godfather" of all oscillators: the harmonic oscillator. Following Hooke's law the harmonic oscillator can be written in the form on the right.
or in the form of ordinary (i.e. first order) differential equations:
Generalising the above formulation, one can see that any system of two differential equations in two variables, of the form nad the right with α and β positive numbers, will produce an oscillatory behaviour. This can be readily seen when looking at the phase space of this system.
The phase space of a particular dynamical system, is the space of all possible states that system can have. Moreover, each point in this space is associated to a vector: the whole space is covered with a vector field. One can image this field as a sort of flow (one speaks of phase flow) that "pushes" each point or state to the next according to the laws of evolution of the system, its differential equations. Fixing a point and following its evolution in time will draw a path in this space which is called a phase trajectory.
For systems as the one above the phase spaces look very similar: in any case, given a starting point, under the influence of the phase flow (depicted as the arrows in the figures on the side) the phase trajectory followed by that point will result in a closed trajectory. In dependence of the relationship between the parameters α and β, it might be a bit stretched in one direction, but the fundamental behaviour, that of returning to the same state from which the observation has begun, is always given. Projecting the motion of the the state on one axis, one always gets a sinusoidal function.
In terms of attractors in this case one speaks of a centre attractor. To note: the only stable point in this phase space, that is a point that remains static for all time, is the origin.
The same behaviour is also given in the case of non-linear equations, like these:
Also in this case, the phase trajectories will result in closed loops around the origin, eve of their form will be a bit more "squarish".
So, the fundamental relationship between variables from which oscillatory behaviours are generated is a relationship that relates the change of one variable to the state of other, with opposite signs for each variable.