Gliders in "Life"-Like Cellular Automata. 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. It evolves in time according to the second order differential equation: 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: When μ > 0, the system will enter a limit cycle. Popular culture[edit] See also[edit] 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] 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. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics. 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] . Solution in some cases[edit] . Hausdorff dimension. In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This means, for example, the Hausdorff dimension of a point is zero, the Hausdorff dimension of a line is one, and the Hausdorff dimension of the plane is two. There are, however, many irregular sets that have noninteger Hausdorff dimension. The concept was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch. The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
Sierpinski triangle. Intuition[edit] Formal definitions[edit] . Theorem. If. 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.