A rigorous lower bound for the stability regions of the quadratic map. (English) Zbl 1197.37060
The authors take the quadratic map \(Q_a(x) = ax(1 - x)\), where \(Q_a: [0,1] \rightarrow [0,1]\), and prove that the set of regular parameters for the quadratic map \(Q_a(x)\) satisfies the lower bound \(|\mathbb{R} \cap [2,4]|\geq 1.61394824439594656781\), where \(\operatorname{Re}\) denotes the set of regular parameters.
Partitioning the parameter plane \(A =[2,4]\) into subintervals \(A_i\) the authors prove that for each subinterval \(A_i\) the quadratic map \(Q_a(x)\) has a unique stable periodic orbit which persists for all \(a \in A_i\).
The authors prove the existence of period-doubling bifurcations. From these results they obtain a non-trivial upper bound on the set of stochastic parameters for \(Q_a(x)\).
Partitioning the parameter plane \(A =[2,4]\) into subintervals \(A_i\) the authors prove that for each subinterval \(A_i\) the quadratic map \(Q_a(x)\) has a unique stable periodic orbit which persists for all \(a \in A_i\).
The authors prove the existence of period-doubling bifurcations. From these results they obtain a non-trivial upper bound on the set of stochastic parameters for \(Q_a(x)\).
Reviewer: Patricia Dominguez (Puebla)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
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