Finding Orbits of Functions Using Suffridge Polynomials

  • Dmitriy Dmitrishin
  • Paul HagelsteinEmail author
  • Alex Stokolos
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this paper we indicate how Suffridge polynomials may be used to find orbits of functions. In particular, we describe a control mechanism that, given a function \(f: \mathbb {R}^n \rightarrow \mathbb {R}^n\) and a positive integer T, yields a dynamical system \(G: \mathbb {R}^{Tn} \rightarrow \mathbb {R}^{Tn}\) that under quantifiable conditions has (x, …, x) as an attractor provided x lies on a T-cycle of f. An explicit example of this control mechanism is provided using a logistic function.


Control theory Stability 

2010 Mathematics Subject Classification

Primary 93B52 42A05 



We wish to thank the referee for a careful reading of the paper.

P. H. is partially supported by a grant from the Simons Foundation (#521719 to Paul Hagelstein).


  1. 1.
    D. Dmitrishin, A. Khamitova, Methods of harmonic analysis in nonlinear dynamics. C. R. Acad. Sci. Paris 351, 367–370 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    D. Dmitrishin, P. Hagelstein, A. Khamitova, A. Stokolos, On the stability of cycles by delayed feedback control. Linear Multilinear A. 64, 924–946 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    D. Dmitrishin, P. Hagelstein, A. Khamitova, A. Stokolos, Limitations of robust stability of a linear delayed feedback control. SIAM J. Control Optim. 56, 148–157 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D. Dmitrishin, P. Hagelstein, A. Khamitova, A. Korenovskyi, A. Stokolos, Fejér polynomials and control of nonlinear discrete systems. Constr. Approx. (accepted for publication). arXiv: 1804.04537Google Scholar
  5. 5.
    D.V. Dmitrishin, A.M. Stokolos, I.M. Skrynnik, E.D. Franzheva, Generalization of nonlinear control for nonlinear discrete systems. Bull. NTU “KhPI” (Series: System analysis, control and information technology) 28(1250), 3–18 (2017). ISSN 2079-0023Google Scholar
  6. 6.
    P. Duren, Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259 (Springer, Berlin, 1983)Google Scholar
  7. 7.
    Ö. Morgül, On the stability of delayed feedback controllers. Phys. Lett. A 314, 278–285 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dmitriy Dmitrishin
    • 1
  • Paul Hagelstein
    • 2
    Email author
  • Alex Stokolos
    • 3
  1. 1.Odessa National Polytechnic UniversityOdessaUkraine
  2. 2.Department of MathematicsBaylor UniversityWacoUSA
  3. 3.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

Personalised recommendations