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An animated cobweb diagram of the logistic map = (), showing chaotic behaviour for most values of >. A cobweb plot , known also as Lémeray Diagram or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions , such as the logistic map .
A cobweb diagram of the logistic map, showing chaotic behaviour for most values of r > 3.57 Logistic function f (blue) and its iterated versions f 2, f 3, f 4 and f 5 for r = 3.5. For example, for any initial value on the horizontal axis, f 4 gives the value of the iterate four iterations later.
Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic. If μ is less than 1 the point x = 0 is an attractive fixed point of the system for all initial values of x i.e. the system will converge towards x = 0 from any initial value of x. If μ is 1 all values of x less than or ...
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).
A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2]
Extending to 3 dimensions the physically impossible Riemann surfaces used to classify all closed orientable 2-manifolds, Heegaard's 1898 thesis "looked at" similar structures for functions of two complex variables, taking an imaginary 4-dimensional surface in Euclidean 6-space (corresponding to the function f=x^2-y^3) and projecting it ...
Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.. In a dynamical system such as ¨ + (;) + =, which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case = is the symmetric pitchfork bifurcation.
Given a function :, the canonical surjection of f onto its image () = {()} is the function from X to f(X) that maps x to f(x). For every subset A of a set X , the inclusion map of A into X is the injective (see below) function that maps every element of A to itself.