Sierpinski carpet The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions (another is Cantor dust). Sierpiński demonstrated that this fractal is a universal curve, in that it has topological dimension one, and every other compact metric space of topological dimension 1 is homeomorphic to some subset of it. The technique of subdividing a shape into smaller copies of itself, removing one or more copies, and continuing recursively can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the Sierpinski triangle. Construction[edit] The process of recursively removing squares is an example of a finite subdivision rule. The area of the carpet is zero (in standard Lebesgue measure). /** * Decides if a point at a specific location is filled or not. Process[edit] Brownian motion on the Sierpinski carpet[edit]
List of fractals by Hausdorff dimension From Wikipedia, the free encyclopedia According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."[1] Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. Deterministic fractals[edit] Random and natural fractals[edit] See also[edit] Notes and references[edit] Further reading[edit] External links[edit] Van der Pol oscillator Evolution of the limit cycle in the phase plane. Notice the limit cycle begins as circle and, with varying μ, become increasingly sharp. An example of a Relaxation oscillator. In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. History[edit] Two dimensional form[edit] Liénard's Theorem can be used to prove that the system has a limit cycle. , where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form:[8] Another commonly used form based on the transformation is leading to Results for the unforced oscillator[edit] Relaxation oscillation in the Van der Pol oscillator without external forcing. Two interesting regimes for the characteristics of the unforced oscillator are:[9] When μ = 0, i.e. there is no damping function, the equation becomes: This is a form of the simple harmonic oscillator and there is always conservation of energy. When μ > 0, the system will enter a limit cycle. See also[edit]
Hausdorff measure Fractal measurement In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of is proportional to the area of the set. Let be a metric space. , let denote its diameter, that is be any subset of and a real number. where the infimum is over all countable covers of by sets satisfying Note that is monotone nonincreasing in since the larger is, the more collections of sets are permitted, making the infimum not larger. exists but may be infinite. It can be seen that -dimensional Hausdorff measure of . are [edit]
Complex quadratic polynomial A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers. Forms[edit] When the quadratic polynomial has only one variable (univariate), one can distinguish its 4 main forms: The monic and centered form has the following properties: Conjugation[edit] Between forms[edit] Since is affine conjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images of Mandelbrot, Julia and Fatou sets. When one wants change from to With doubling map[edit] There is semi-conjugacy between the dyadic transformation (here named doubling map) and the quadratic polynomial. Family[edit] The family of quadratic polynomials parametrised by is called: the Douady-Hubbard family of quadratic polynomials[6]quadratic family Map[edit] and parameter Notation[edit] Here denotes the n-th iteration of the function not exponentiation so Because of the possible confusion it is customary to write for the nth iterate of the function Critical items[edit] of
H tree The first ten levels of an H tree The first 18 levels of an H tree as an animation. The H tree (so called because its repeating pattern resembles the letter "H") is a family of fractal sets whose Hausdorff dimension is equal to 2. An alternative process that generates the same fractal set is to begin with a rectangle with sides in the ratio 1:√2, known as a "silver rectangle", and repeatedly bisect it into two smaller silver rectangles, at each stage connecting the two centroids of the two smaller rectangles by a line segment. The Mandelbrot Tree is a very closely related fractal using rectangles instead of line segments, slightly offset from the H-tree positions, in order to produce a more naturalistic appearance. Applications[edit] The planar H tree can be generalized to the three-dimensional structure via adding line segments on the direction perpendicular to the H tree plane.[7] The resultant three-dimensional H tree has Hausdorff dimension equal to 3. Notes[edit] References[edit]
Logistic map where: is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0) r is a positive number, and represents a combined rate for reproduction and starvation. This nonlinear difference equation is intended to capture two effects. However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values lead to negative population sizes. The r=4 case of the logistic map is a nonlinear transformation of both the bit shift map and the case of the tent map. Behavior dependent on r[edit] By varying the parameter r, the following behavior is observed: For any value of r there is at most one stable cycle. A bifurcation diagram summarizes this. Chaos and the logistic map[edit] Two- and three-dimensional phase diagrams show the stretching-and-folding structure of the logistic map . is given by . for . .
Herman ring The Julia set of the cubic rational function e2πitz2(z−4)/(1−4z) with t=.6151732... chosen so that the rotation number is (√5−1)/2, which has a Herman ring (shaded). Formal definition[edit] Namely if ƒ possesses a Herman ring U with period p, then there exists a conformal mapping and an irrational number , such that So the dynamics on the Herman ring is simple. Name[edit] It was introduced by, and later named after, Michael Herman (1979[2]) who first found and constructed this type of Fatou component. Function[edit] Polynomials do not have Herman rings.Rational functions can have Herman rings. Examples[edit] Herman and parabolic basin[edit] Here is an example of a rational function which possesses a Herman ring.[1] where such that the rotation number of ƒ on the unit circle is The picture shown on the right is the Julia set of ƒ: the curves in the white annulus are the orbits of some points under the iterations of ƒ while the dashed line denotes the unit circle. A rational function . Letting