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A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8 / 3 . The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz.
Portal:Systems science/Picture/1 The Lorenz attractor is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow , noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior.Maps may be parameterized by a discrete-time or a continuous-time parameter.
A plot of Lorenz' strange attractor for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternative initial conditions on the attractor, the iterated points will become arbitrarily spread out from each other.
In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Specifically, [22] the invariant measure is ().
If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get
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A Lorenz curve always starts at (0,0) and ends at (1,1). The Lorenz curve is not defined if the mean of the probability distribution is zero or infinite. The Lorenz curve for a probability distribution is a continuous function. However, Lorenz curves representing discontinuous functions can be constructed as the limit of Lorenz curves of ...