# Concept derivation guides

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Concept guide



## [Orbits, fixed points and periodic points: classifying all six states of T(x)=(2x+1) mod 6](/en/derivations/derivation-mko-orbit-fixed-periodic-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-orbit-fixed-periodic)

On the state space X={0,1,2,3,4,5} take the self-map T(x)=(2x+1) mod 6. Find all fixed points and all periodic points of least period 2; prove that for a point of least period p, T^n(x)=x exactly when p divides n, and use this to explain why T^4(1)=1 does not make 4 the least period of 1; finally prove that 0, 2 and 4 are not periodic, although their orbits enter a cycle after one step.

Orbits, fixed points and periodic points

derivation-mko-orbit-fixed-periodic-01





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