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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 Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. [2] Van der Pol found stable oscillations, [3] which he subsequently called relaxation-oscillations [4] and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes.
Stable limit cycle (shown in bold) and two other trajectories spiraling into it Stable limit cycle (shown in bold) for the Van der Pol oscillator. In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as ...
In the continuum limit, a → 0, N → ∞, while Na is held fixed. The canonical coordinates Q k devolve to the decoupled momentum modes of a scalar field, ϕ k {\displaystyle \phi _{k}} , whilst the location index i ( not the displacement dynamical variable ) becomes the parameter x argument of the scalar field, ϕ ( x , t ) {\displaystyle ...
Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly. Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero ...
is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution. The number of roots carries also information on the spectrum of associated boundary value problems .
In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source. In this case the oscillation is said to be driven . The simplest example of this is a spring-mass system with a sinusoidal driving force.
The amplitude of the oscillation of an unstable system grows exponentially with time (i.e., small oscillations are negatively damped), until nonlinearities become important and limit the amplitude. This can produce a steady and sustained oscillation.