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Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence. In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.
The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes. There are several related notions: oscillation of a sequence of real numbers , oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set ).
In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation (,, ′, …, ()) = [, +)is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating.
Quantity (common name/s) (Common) symbol/s SI units Dimension Number of wave cycles N: dimensionless dimensionless (Oscillatory) displacement Symbol of any quantity which varies periodically, such as h, x, y (mechanical waves), x, s, η (longitudinal waves) I, V, E, B, H, D (electromagnetism), u, U (luminal waves), ψ, Ψ, Φ (quantum mechanics).
The equation + + = is known as the universal oscillator equation, since all second-order linear oscillatory systems can be reduced to this form. [ citation needed ] This is done through nondimensionalization .
This definition of periodicity can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic tessellations of the plane. A sequence can also be viewed as a function defined on the natural numbers , and for a periodic sequence these notions are defined accordingly.
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function of some real variable (such as time) is an angle-like quantity representing the fraction of the cycle covered up to .
In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative ...