Simple nonlinear systems. Nonlinear growth of bugs and the logistic map x i+1 =μx i (1-x i )

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Presentation transcript:

Simple nonlinear systems

Nonlinear growth of bugs and the logistic map x i+1 =μx i (1-x i )

The fix point

Bifurcation, self-similarity, and chaos

Bifurcation points converge geometrically , , a constant

Geometric convergence indicates that something is preserved when we change the scale (scaling property) Feigenbaum (1978) set out to calculate another iteration x i+1 =μsin (x i ) and got the same constant (4.6692…)! So are other 1-dim maps that have bifurcations! He discovered “universality” in nonlinear systems Note: Keith Briggs from the Mathematics Department of the University of Melbourne in Australia computed what he believes to be the world-record for the number of digits for the Feigenbaum number: