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Qué (quién) es averaging method - definición

Method of Averaging

$Figure 2: A simple harmonic oscillator with small periodic damping term given by <math>\ddot{z} + 4 \varepsilon \cos^2{(t)} \dot{z} + z = 0, ~ z(0) = 0,~ \dot{z}(0) = 1;~\varepsilon = 0.05</math>.The numerical simulation of the original equation (blue solid line) is compared with averaging system (orange dashed line) and the crude averaged system (green dash-dotted line). The left plot displays the solution evolved in time and the right plot represents on the phase space. We note that the crude averaging disagrees with the expected solution.$
$Figure 1: Solution to perturbed logistic growth equation <math>\dot {x} = \varepsilon (x (1 - x) + \sin{t}) ~ x \in \R, ~\varepsilon = 0.05</math> (blue solid line) and the averaged equation <math>\dot {y} = \varepsilon y (1 - y),~ y \in \R</math> (orange solid line).$
$Figure 4: The plot depicts two fundamental quantities the average technique is based on: the bounded and connected region <math>D</math> of the phase space and how long (defined by the constant <math>c</math>) the averaged solution is valid. For this case, <math display="inline">\ddot{z} + z = 8 \varepsilon \cos{(t)} \dot{z}^2 , ~ z(0) = 0,~ \dot{z}(0) = 1;~ 8 \varepsilon = \frac{2}{15} </math>. Note that both solutions blow up in finite time.  Hence, <math>D</math> has been chosen accordingly in order to maintain the boundedness of the solution and the time interval of validity of the approximation is <math>0 \leq \varepsilon t < L < \frac{1}{3}</math>.$
$Figure 3: Phase space of a Van der Pol oscillator with <math>\varepsilon = 0.1</math>. The stable limit cycle (orange solid line) in the system is captured correctly by the qualitative analysis of the averaged system. For two different initial conditions ( black dots ) we observe the trajectories.(dashed blue line) converging to the periodic orbit.$

Method of averaging

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 behavior from the resulting dynamics.

https://en.wikipedia.org/wiki/Method_of_averaging

Krylov–Bogoliubov averaging method

The Krylov–Bogolyubov averaging method (Krylov–Bogolyubov method of averaging) is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics.Krylov–Bogolyubov method of averaging at Encyclopedia of Mathematics The method is based on the averaging principle when the exact differential equation of the motion is replaced by its averaged version.

https://en.wikipedia.org/wiki/Krylov–Bogoliubov_averaging_method

Highest averages method

METHOD TO ALLOCATE SEATS PROPORTIONALLY FOR REPRESENTATIVE ASSEMBLIES WITH PARTY LIST VOTING SYSTEMS

Highest average method; Method of smallest divisors; Imperiali method; Danish method; Adams's method; Divisor method; Adams apportionment method; Rank-index apportionment method

A highest-averages method, also called a divisor method, is a class of methods for allocating seats in a parliament among agents such as political parties or federal states. A divisor method is an iterative method: at each iteration, the number of votes of each party is divided by its divisor, which is a function of the number of seats (initially 0) currently allocated to that party.

https://en.wikipedia.org/wiki/Highest_averages_method

Wikipedia

Method of averaging

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 behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale. It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution.

More precisely, the system has the following form

of a phase space variable

x.

The fast oscillation is given by

f

versus a slow drift of

{\dot {x}}

. The averaging method yields an autonomous dynamical system which approximates the solution curves of

{\dot {x}}

inside a connected and compact region of the phase space and over time of

1/\varepsilon

.

Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for $y$ . In this way, qualitative methods for autonomous dynamical systems may be employed to analyze the equilibria and more complex structures, such as slow manifold and invariant manifolds, as well as their stability in the phase space of the averaged system.

In addition, in a physical application it might be reasonable or natural to replace a mathematical model, which is given in the form of the differential equation for ${\dot {x}}$ , with the corresponding averaged system ${\dot {y}}$ , in order to use the averaged system to make a prediction and then test the prediction against the results of a physical experiment.

The averaging method has a long history, which is deeply rooted in perturbation problems that arose in celestial mechanics (see, for example in ).

https://en.wikipedia.org/wiki/Method of averaging