Intro to Chaos and Fractals: Summary

Iteration

• Start with a seed, repeatedly apply a function. This is an example of a feedback loop.

• In homework and in class, we iterated functions numerically (using a calculator or by hand) and by making graphically iterating via cobweb diagrams.

• Vocabulary to describe the results of iteration:
  • Orbit: The sequence of values obtained while iterating. Also known as an itinerary.

  • Fixed Point: A number which, when the function is applied to it, remains unchanged. The fixed point equation is f(x) = x.

  • Periodic Point: A number which, when the function is applied to it, returns to itself after N iterations. The number N is known as the periodicity.

  • A fixed point is attracting if nearby points are drawn closer to it under iteration.

  • A fixed point is repelling if nearby points are pushed away from it under iteration.

  • Attracting fixed points or periodic cycles are also called stable.

  • Repelling fixed points or periodic cycles are also called unstable.

  • If successive numbers in the orbit get larger and larger, then we say that the orbit tends to infinity.

    
    

• Simplest model of population growth that doesn't assume unlimited growth and an unbounded population.

• Iterated the logistic equation, f(x) = rx(1-x). The variable r is viewed as a parameter. The population x is a number between 0 and 1; think of it as a fraction of the maximum possible population.

• Experimented with different values of the parameter. We saw that bifurcations occurred as r was varied. A bifurcation is a sudden change in the fate of an orbit as a parameter is varied.

• We summarized the behavior of the logistic equation by using a final state diagram. This type of diagram is also known as a bifurcation diagram.

• The bifurcation diagram showed an intricate structure. For different values of r there were all sorts of periodic behavior as well as lots of values for which r was chaotic.

  
  

• Orbit Diagram
• Graphical Iteration aka Cobweb Diagram
• Parameter Plane
• Final-State Diagram aka Bifurcation Diagram

Chaos

• A system is chaotic if:
  • The orbit is bounded (i.e. it doesn't fly off to infinity) and it never repeats.

  • The system displays Sensitive Dependence on Initial Conditions.

• The above two criteria are equivalent, in the sense that one implies the other, and vice versa.

• A system has sensitive dependence on initial conditions -- also known as the butterfly effect -- if small differences in initial conditions are quickly magnified.

  
  

• A simple deterministic system can produce unpredictable results.

• A simple deterministic system can produce complicated, ordered behavior, e.g. period 32 or period 17 cycles.

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