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An initial value problem is a differential equation ′ = (, ()) with : where is an open set of , together with a point in the domain of (,),called the initial condition.. A solution to an initial value problem is a function that is a solution to the differential equation and satisfies
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1. The minimum value of x is ...
Use of a user-defined function sq(x) in Microsoft Excel. The named variables x & y are identified in the Name Manager. The function sq is introduced using the Visual Basic editor supplied with Excel. Subroutine in Excel calculates the square of named column variable x read from the spreadsheet, and writes it into the named column variable y.
In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai( x ) and the related function Bi( x ) , are linearly independent solutions to the differential equation d 2 y d x 2 − x y = 0 , {\displaystyle {\frac ...
The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
Whereas the (global) Lyapunov exponent gives a measure for the total predictability of a system, it is sometimes of interest to estimate the local predictability around a point x 0 in phase space. This may be done through the eigenvalues of the Jacobian matrix J 0 (x 0). These eigenvalues are also called local Lyapunov exponents. [22]
The Poincaré map is constructed by plotting the value of the function every time it passes through a set plane in a specific direction. An example would be plotting the y , z {\displaystyle y,z} value every time it passes through the x = 0 {\displaystyle x=0} plane where x {\displaystyle x} is changing from negative to positive, commonly done ...