Abstract
Using the theory of complete discrimination system and the computer algebra systemMAPLE V.17, we compute the number of forts for the logistic mappingfλ(x)=λx(1-x)on[0,1]parameterized byλ∈(0,4]. We prove that if0<λ≤2then the number of forts does not increase under iteration and that ifλ>2then the number of forts is not bounded under iteration. Furthermore, we focus on the case ofλ>2and give for eachk=1,…,7some critical values ofλfor the change of numbers of forts.
Highlights
Iteration is the act of repeating a process with the aim of approaching a desired goal, target, or result
From one-dimensional case, one can notice that an iterate of a linear function of any order remains linear but the degree of a polynomial may increase drastically, which shows that the nonlinear complexity is amplified by iteration
In the one-dimensional case, the complexity of nonlinear functions is related to nonmonotonicity
Summary
Iteration is the act of repeating a process with the aim of approaching a desired goal, target, or result. For a fixed integer n ≥ 1, the nth iterate fn of a mapping f : E → E, where E is a nonempty set, is defined recursively by fk = f ∘ fk−1, f0 = id,. From one-dimensional case, one can notice that an iterate of a linear function of any order remains linear but the degree of a polynomial may increase drastically, which shows that the nonlinear complexity is amplified by iteration. It is easy to find nonlinear functions whose number of forts, regarded as the damagers of monotonicity, increases rapidly under iteration. How can we compute the number of forts for nonmonotonic functions?. Polynomials, a special class of nonmonotonic functions, possess the advantage that each fort of a polynomial of degree.
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