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We treat the logistic map here. It's known as a simple model which yields chaos, written as x_{n+1}=ax_{n}(1x_{n}) (n=0,1,2...), where 0<=a<=4. This is a discretetime map which receives a real number between 0 and 1, and returns a real number in [0,1] again. n denotes a discrete time step and x_{n} denotes a data at n. For example, x_{n} can be regarded as an annual change of the number of an animal, a change of the price of a stock of every month, and so on. Such an equation which governs the changes of data for discrete time steps is called the difference equation. Although the logistic map had been known since early times, Dr. Robert May first found that this map shows very complicated behaviors in the paper entitled as "Simple mathematical models with very complicated dynamics" published in Nature in 1976. First we take x_{0} as the initial value. Using the above equation, we can calculate x_{1}, x_{2}, x_{3}. In these procedures, we have no statistical features. So time series x_{n} (n=0,1,2...) is called deterministic. Using the above simulator, we can get these sequences x_{0}, x_{1}, x_{2}, x_{3}... diagrammatically. Two graphs are drawn in the upper field. y=4x(1x) and y=x. Please click your mouse on the upper side of the simulator, then its xcoordinate is set as the initial value x_{0} and a red point is plotted. This simulator draws a vertical line from the red point to the parabola. The ycoordinate of the intersecting point is x_{1}. This value is returned to xaxis using the line y=x. And we can get x_{2}, x_{3}... with the same procedure. This simulator calculates up to x_{20}. You can choose three initial conditions, that is, x_{0}, x'_{0}, x''_{0}. These series are drawn in different colors. In the lower field of the simulator, the derived time series x_{n} (n=1..20) is plotted in a line. Even if you choose nearby initial points, these sequences will be separated as time passes. From this result, we can see that chaos is sensitive to the initial condition. It's also called butterfly effect. With the scrollbar, you can change the parameter a from 0 to 4. In some parameter value, there is no chaos and only periodic orbits appear. The logistic map cannot predict the number of animals nor the price of stocks because the logistic map is not a quantitative model, but a qualitative one. However, the logistic map is thought to be important in the research of chaos because it is simple and has universal properties. The parabola of the logistic map shows the stretching and folding processes of chaos. Such structure of chaos is also observed in many nonlinear models, such as Lorenz attractor which is derived from a model of the atmospheric convection, and the synchronous behaviors in a neuronal network. Such similarities are observed in many models because the logistic map is universal. Reference
