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BifurcationDiagramQuadraticFamily

BifurcationDiagramQuadraticFamily - For instance why are...

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Bifurcation Diagram for the Quadratic Family The diagram is the bifurcation diagram for the quadratic family, f μ ( x ) = μ ! x ! (1 " x ) . The graph is for μ in the range 7 2 ! μ ! 4 and 0 ! x ! 1 . The diagram represents the long-term behavior of the function f μ ( x ) . It is created by taking a particular value of μ and starting with the point x 0 = 1 2 . Iterate the function for 500 times without plotting. Then plot the points μ , f μ n ( x 0 ) ( ) n = 501 ! 1000 { } . Do the same thing for a large number of equidistributed values of μ in the interval 7 2 ! μ ! 4 , say for 1000 points. The result of the plot is the diagram above. In class we will show what is happening in the diagram.
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Unformatted text preview: For instance, why are there spectral lines in the diagram? These blank spaces are where there are attracting periodic points. What are the dark regions? These are values of for which there are no attracting periodic points. Some of the most prestigious and talented mathematicians of our time have devoted a major portion of their careers to studying this diagram. It is just one of many major discoveries in mathematics in our time. ! axis x ! axis 3.5 4 1 f ( x ) = ! x ! (1 " x )...
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