One-Dimensional Dynamical Systems

Parameter dependent dynamics

• Qualitative change of dynamics
  
• Bifurcation points
  
• Orbit diagram
  

As the parameter of the Logistic map increases, a series of period-doubling bifurcations occurs. The Logistic map becomes chaotic for even larger (certainly for 4). We say that the Logistic map becomes chaotic as it undergoes a series of period-doubling bifurcations. In fact, this is a typical route to chaos, which is not at all restricted to the Logistic map alone. We never gave a definition of chaotic behaviour, and we are not going to, but you can think of a dynamical system being chaotic if it has periodic points of any period, which are all repelling.

Before a map can possibly have infinitely many periodic points with distinct periods, it must have periodic points with all periods of the form 2ⁿ. If for a particular parameter value a periodic point of period 5 appears, you can imagine how periodic points of periods 10, 20, 40, etc. are born as the parameter is increased. In fact, there is a very strict, and universal, ordering of the periods described in a famous paper by Li and Yorke called Period 3 implies Chaos. You can read more about this in the Geometry Forum article Sharkovskii's Theorem.