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Nonlinear Growth Dynamics: The Fascinating World of Chaos and Universality

Explore the intricacies of nonlinear systems with a focus on the nonlinear growth of bugs, logistic map, bifurcation points, chaos, and self-similarity. Discover how Feigenbaum's work reveals universal constants in these systems. Learn about the intriguing geometric convergence and the scaling properties that underpin these phenomena. Delve into the world of nonlinear systems and their fundamental properties.

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Nonlinear Growth Dynamics: The Fascinating World of Chaos and Universality

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  1. Simple nonlinear systems

  2. Nonlinear growth of bugs and the logistic map xi+1=μxi(1-xi)

  3. The fix point

  4. Bifurcation, self-similarity, and chaos

  5. Bifurcation points converge geometrically 3.449 3.544, 3.5644, 3.5688 a constant

  6. Geometric convergence indicates that something is preserved when we change the scale(scaling property) Feigenbaum (1978) set out to calculate another iteration xi+1=μsin (xi) 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

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