1. Simple nonlinear systems

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

3. The fix point

5. Bifurcation, self-similarity, and chaos

6. Bifurcation points converge geometrically 3.4493.544, 3.5644, 3.5688 a constant

7. 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: 4. 669201609102990671853203820466201617258185577475768632745651 343004134330211314737138689744023948013817165984855189815134 408627142027932522312442988890890859944935463236713411532481 714219947455644365823793202009561058330575458617652222070385 410646749494284981453391726200568755665952339875603825637225