Editing Logistic map (section)

=== Topological conjugate mapping ===

Let the symbol ∘ denote the composition of maps . In general, for  a topological space X, Y, two maps f  : X → X and g  : Y → Y are composed by a homeomorphism h : X → Y.

{{NumBlk|:|<math>{\displaystyle h\circ f=g\circ h}</math>|{{EquationRef|4-4}}}}

f and g are said to be phase conjugates if they satisfy the relation <!--[ 261 ]-->. The concept of phase conjugation plays an important role in the study of dynamical systems <!--[ 262 ]-->. Phase conjugate f and g exhibit essentially identical behavior, and if the behavior of f is periodic, then g is also periodic, and if the behavior of f is chaotic, then g is also chaotic <!--[ 262 ]-->.

In particular, if a homeomorphism h is linear, then f and g are said to be linearly conjugate . <!--[ 263 ]--> Every quadratic function is linearly conjugate with every other quadratic function . <!--[ 264 ]--> Hence,

{{NumBlk|:|<math>{\displaystyle x_{n+1}=x_{n}^{2}+b}</math>|{{EquationRef|4-5}}}}

{{NumBlk|:|<math>{\displaystyle x_{n+1}=1-cx_{n}^{2}}</math>|{{EquationRef|4-6}}}}

{{NumBlk|:|<math>{\displaystyle x_{n+1}=d-x_{n}^{2}}</math>|{{EquationRef|4-7}}}}

are linearly conjugates of the logistic map for any parameter a <!--[ 265 ]-->. Equations ( 4-6 ) and ( 4-7 ) are also called logistic maps <!--[ 266 ]-->. In particular, the form ( 4-7 ) is suitable for time-consuming numerical calculations, since it requires less computational effort <!--[ 134 ]-->.

[[File:Tent map cobweb diagram, example of parameter 2.png|class=skin-invert-image|thumb|Orbital view of the tent map ( 4-8 ). It has a topological conjugate relationship with the a = 4 logistic map.]]

Moreover, the logistic map <math>f_{a=4}</math> for <math>r = 4</math> is topologically conjugate to the following tent map T  ( x ) and Bernoulli shift map B  ( x ) <!--[ 267 ]-->.

{{NumBlk|:|<math>{\displaystyle T(x_{n})={\begin{cases}2x_{n}&\left(0\leq x_{n}\leq {\frac {1}{2}}\right)\\2(1-x_{n})&\left({\frac {1}{2}}\leq x_{n}\leq 1\right)\end{cases}}}</math>|{{EquationRef|4-8}}}}

{{NumBlk|:|<math>{\displaystyle B(x_{n})={\begin{cases}2x_{n}&\left(0\leq x_{n}<{\frac {1}{2}}\right)\\2x_{n}-1&\left({\frac {1}{2}}\leq x_{n}\leq 1\right)\end{cases}}}</math>|{{EquationRef|4-9}}}}

These phase conjugate relations can be used to prove that the logistic map <math>f_{a=4}</math> is strictly chaotic and to derive the exact solution ( 3-19 ) of <math>f_{r=4}</math> <!--[ 268 ]-->.

Alternatively, introducing the concept of symbolic dynamical systems, consider the following shift map σ defined on the symbolic string space consisting of strings of 0s and 1s as introduced above <!--[ 269 ]-->:

{{NumBlk|:|<math>{\displaystyle \sigma (s_{0}s_{1}s_{2}\cdots )=(s_{1}s_{2}\cdots )}</math>|{{EquationRef|4-10}}}}

Here, <math>s_i</math> is 0 or 1. On the set <math>\Lambda</math> introduced in equation ( 3-18 ), the logistic map <math>f_{r>4}</math> is [[topologically conjugate]] to the shift map, so we can use this to derive that <math>f_{r>4}</math> on <math>\Lambda</math> is chaotic <!--[ 270 ]-->.