It has been proved by experiments that the fitness function value of random numbers generated by chaotic maps is significantly improved, and better results can be obtained by replacing conventional uniformly distributed random number generators with chaotic maps, especially when there are many local solutions in the search space. It is easier to search for the global optimal solution. Using chaotic sequences to perform operations such as population initialization, selection, crossover, and mutation will affect the entire process of the algorithm, and often achieve better results than pseudo-random numbers.
Chaos mainly has the following two properties:
- Orbital instability, a property that under appropriate constraints leads toMovement instability and bifurcation
- Highly dependent on the initial value, that is, two similar initial values will get completely different random number sequences
A typical example of describing chaotic motion is the nonlinear Logistic map, which is a one-dimensional nonlinear function:
where μ is the control parameter, and μ>1; the value range of xn is 0≤xn≤1.
This blog takes logistic chaotic mapping as an example to learn the properties of chaotic mapping.
This paper considers the mapping of formula (1) when the control parameter μ is in the range of 1<μ≤4.
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When μ=2.8, x0=0.63, the solution after 20 iterations is x20=0.642 7 and x21=0.643 0, which gradually tends to a fixed point between 0.642 7 and 0.643 0.
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When μ=3.14, x0=0.66, the solution after 30 iterations is stable around two fixed points x30=0.538 1 and x31=0.780 4.
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When μ=3.45 and x0=0.68, the solution after 68 iterations is stable around four fixed points x68=0.429 4, x69=0.845 3, x70=0.451 1 and x71=0.854 2.
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When μ=4.0, x0=0.80, the result of iterative calculation no longer tends to one or several fixed points, but completely enters the chaotic state, and the value in this case is random.
What needs to be explained here is that the initial value x0 can start from 0, and the initial value here is only for the convenience of the reader's verification.
From the analysis of the above four points, it can be seen that the chaotic motion in the nonlinear dynamics described by the Logistic map shows a high degree of order under certain conditions. For example, when μ=2.8, the solution after 20 iterations tends to a fixed point , no longer changing, this is a regular and orderly movement. If the value of μ is increased, the number of stable fixed points in this orderly movement will increase continuously. The increase in the number can still be regarded as orderly, and this situation of increasing fixed points is called"Fork phenomenon". With the continuous increase of the number of bifurcation fixed points, the orderly motion gradually loses balance and enters a chaotic state, that is, the situation where μ≥4.0. Therefore, a set of random numbers can be obtained by substituting μ=4.0 into formula (1): 
The larger the value of n, the larger the value of xn+1 obtained. By giving the corresponding random numbers obtained from different initial values, it can be known thatThese random numbers are highly dependent on the initial value, that is, two similar initial values will get completely different random number sequences, which is the property of the chaotic map, and also the theoretical basis for citing it. This property will bring great benefits to optimize computation.
Some chaotic sequences commonly used in the field of swarm intelligence are given below:
References:
Application of Chaos Maps in Optimal Calculation
Several Chaos Maps