Chaos in some planar nonautonomous polynomial differential equation

Klaudiusz Wójcik

Annales Polonici Mathematici (2000)

  • Volume: 73, Issue: 2, page 159-168
  • ISSN: 0066-2216

Abstract

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We show that under some assumptions on the function f the system ż = z ̅ ( f ( z ) e i ϕ t + e i 2 ϕ t ) 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.

How to cite

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Wójcik, Klaudiusz. "Chaos in some planar nonautonomous polynomial differential equation." Annales Polonici Mathematici 73.2 (2000): 159-168. <http://eudml.org/doc/262600>.

@article{Wójcik2000,
abstract = {We show that under some assumptions on the function f the system $ż = z̅(f(z) e^\{iϕt\} + e^\{i2ϕt\})$ 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.},
author = {Wójcik, Klaudiusz},
journal = {Annales Polonici Mathematici},
keywords = {periodic solutions; fixed point index; Lefschetz number; chaos; chaotic dynamics; Lefschetz fixed point theorem; Ważewski retract theorem; isolating segment; Poincaré map},
language = {eng},
number = {2},
pages = {159-168},
title = {Chaos in some planar nonautonomous polynomial differential equation},
url = {http://eudml.org/doc/262600},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Wójcik, Klaudiusz
TI - Chaos in some planar nonautonomous polynomial differential equation
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 2
SP - 159
EP - 168
AB - We show that under some assumptions on the function f the system $ż = z̅(f(z) e^{iϕt} + e^{i2ϕt})$ 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.
LA - eng
KW - periodic solutions; fixed point index; Lefschetz number; chaos; chaotic dynamics; Lefschetz fixed point theorem; Ważewski retract theorem; isolating segment; Poincaré map
UR - http://eudml.org/doc/262600
ER -

References

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  1. [Ma] J. Mawhin, Periodic solutions of some planar non-autonomous polynomial differential equations, J. Differential Integral Equations 7 (1994), 1055-1061. Zbl0802.34045
  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. Zbl0848.34027
  3. [MM] K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. 32 (1995), 66-72. Zbl0820.58042
  4. [S] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22 (1994), 707-737. Zbl0801.34041
  5. [S1] R. Srzednicki, On periodic solutions of planar differential equations with periodic coefficients, J. Differential Equations 114 (1994), 77-100. Zbl0811.34031
  6. [SW] R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics, ibid. 135 (1997), 66-82. Zbl0873.58049
  7. [W] S. Wiggins, Global Bifurcation and Chaos. Analytical Methods, Springer, New York, 1988. Zbl0661.58001
  8. [W1] K. Wójcik, Isolating segments and symbolic dynamics, Nonlinear Anal. 33 (1998), 575-591. Zbl0955.37005
  9. [W2] K. Wójcik, On detecting periodic solutions and chaos in ODE's, ibid., to appear. 

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