Download PDF - Chaos in some planar nonautonomous polynomial differential equationAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Chaos in some planar nonautonomous polynomial differential equation

Authors 1

Affiliations

  1. Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Abstract

We show that under some assumptions on the function f the system ż=z̅(f(z)eiϕt+ei2ϕt) generates chaotic dynamics for sufficiently small parameter ϕ. We use the topological method based on the Lefschetz fixed point theorem and the Ważewski retract theorem.

Keywords

ENG
periodic solutions, fixed point index, Lefschetz number, chaos

Bibliography

  1. [Ma] J. Mawhin, Periodic solutions of some planar non-autonomous polynomial differential equations, J. Differential Integral Equations 7 (1994), 1055-1061.
  2. [MMZ] R. Manásevich, J. Mawhin and F. Zanolin, Periodic solutions of complex-valued differential equations and systems with periodic coefficients, J. Differential Equations 126 (1996), 355-373.
  3. [MM] K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. 32 (1995), 66-72.
  4. [S] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22 (1994), 707-737.
  5. [S1] R. Srzednicki, On periodic solutions of planar differential equations with periodic coefficients, J. Differential Equations 114 (1994), 77-100.
  6. [SW] R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics, ibid. 135 (1997), 66-82.
  7. [W] S. Wiggins, Global Bifurcation and Chaos. Analytical Methods, Springer, New York, 1988.
  8. [W1] K. Wójcik, Isolating segments and symbolic dynamics, Nonlinear Anal. 33 (1998), 575-591.
  9. [W2] K. Wójcik, On detecting periodic solutions and chaos in ODE's, ibid., to appear.
Annales Polonici Mathematici
2000/73/2
Export to PBN, Opens in new tab
Pages:
159-168
Main language of publication
English
Received
1999-05-31
Accepted
1999-10-20
Published
2000
Exact and natural sciences