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Fejér Polynomials and Chaos

  • Dmitriy Dmitrishin
  • Anna Khamitova
  • Alexander M. Stokolos
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 108)

Abstract

We show that given any μ > 1, an equilibrium x of a dynamic system
$$\displaystyle{ x_{n+1} = f(x_{n}) }$$
(1)
can be robustly stabilized by a nonlinear control
$$\displaystyle{ u = -\sum _{j=1}^{N-1}\varepsilon _{ j}\left (f\left (x_{n-j+1}\right ) - f\left (x_{n-j}\right )\right ),\,\vert \varepsilon _{j}\vert < 1,\;j = 1,\ldots,N - 1, }$$
(2)
for f (x) ∈ (−μ, 1). The magnitude of the minimal value N is of order \(\sqrt{\mu }.\) The optimal explicit strength coefficients are found using extremal nonnegative Fejér polynomials. The case of a cycle as well as numeric examples and applications to mathematical biology are considered.

Keywords

Chaotic Attractor Robust Stability Population Development Unstable Periodic Orbit Strong Numeric Evidence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Dmitriy Dmitrishin
    • 1
  • Anna Khamitova
    • 2
  • Alexander M. Stokolos
    • 2
  1. 1.Odessa National Polytechnic UniversityOdessaUkraine
  2. 2.Georgia Southern UniversityStatesboroUSA

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