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This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country.
The usual values of interest for the parameter r are those in the interval [0, 4], so that x n remains bounded on [0, 1]. The r = 4 case of the logistic map is a nonlinear transformation of both the bit-shift map and the μ = 2 case of the tent map. If r > 4, this leads to negative population sizes.
Map Time domain Space domain Number of space dimensions Number of parameters Also known as 3-cells CNN system: continuous: real: 3: 2D Lorenz system [1] discrete: real: 2: 1: Euler method approximation to (non-chaotic) ODE. 2D Rational chaotic map [2] discrete: rational: 2: 2: ACT chaotic attractor [3] continuous: real: 3: Aizawa chaotic ...
One can also define degree modulo 2 (deg 2 (f)) the same way as before but taking the fundamental class in Z 2 homology. In this case deg 2 ( f ) is an element of Z 2 (the field with two elements ), the manifolds need not be orientable and if n is the number of preimages of p as before then deg 2 ( f ) is n modulo 2.
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
A plot of 100,000 iterations of the Kaplan-Yorke map with α=0.2. The initial value (x 0,y 0) was (128873/350377,0.667751). The Kaplan–Yorke map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. The Kaplan–Yorke map takes a point (x n, y n ) in the plane and maps it to a new point ...
An RR tachograph is a graph of the numerical value of the RR-interval versus time. In the context of RR tachography, a Poincaré plot is a graph of RR(n) on the x-axis versus RR(n + 1) (the succeeding RR interval) on the y-axis, i.e. one takes a sequence of intervals and plots each interval against the following interval. [3]
If μ is between 1 and the square root of 2 the system maps a set of intervals between μ − μ 2 /2 and μ/2 to themselves. This set of intervals is the Julia set of the map – that is, it is the smallest invariant subset of the real line under this map. If μ is greater than the square root of 2, these intervals merge, and the Julia set is ...