Chaos in some planar nonautonomous polynomial differential equation

• Autorzy: Klaudiusz Wójcik
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

periodic solutions   fixed point index   Lefschetz number   chaos  

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 73
• Numer: 2
• Strony: 159-168

Daty

wydano

2000

otrzymano

1999-05-31

poprawiono

1999-10-20

Twórcy

autor

Klaudiusz Wójcik

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

Bibliografia

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