LettersNo Access

CHAOS BEHAVIOR IN THE DISCRETE BVP OSCILLATOR

Department of Mathematics, Hunan Normal University, Changsha, 410081, Hunan Province, China

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Search for more papers by this author

ZHIYUAN JIA

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Beijing, 100080, China

Search for more papers by this author

, and

RUIQI WANG

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Beijing, 100080, China

Search for more papers by this author

https://doi.org/10.1142/S0218127402004577Cited by:19 (Source: Crossref)

Abstract

The discrete BVP oscillator obtained through the Euler method is investigated, and also first proved that there exist chaotic phenomena in the sense of Marotto's definition of chaos and two-period cycles. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits in Marotto's chaos and intermitten's chaos. The computations of Lyapunov exponents confirm the existence of dynamical behaviors.

This work was supported by the National Key Basic Research Special Found (No. G1998020307).

Keywords:

References

B. Braaksma and J. Grasman, Physica D 68, 265 (1993). Crossref, Web of Science, Google Scholar
D. S. Chernvskiiet al., Biosystems 9, 187 (1977). Crossref, Web of Science, Google Scholar
R. FitzHugh, Biophysical J. 1, 445 (1961). Crossref, Web of Science, Google Scholar
A. Goldbeter and L. A. Segel, Proc. Natl. Acad. Sci. USA 74, 1543 (1977). Crossref, Web of Science, Google Scholar
A. L. Hodgin and A. F. Huxley, J. Physiol. 117, 500 (1952). Crossref, Web of Science, Google Scholar
K. Kaumann and U. Staude, Equadiff 82, Lecture Notes in Mathematics 1017, eds. H. W. Knobloch and K. Schmitt (Springer-Verlag, NY, 1983) pp. 313–321. Crossref, Google Scholar
F. R. Marotto, J. Math. Anal. Appl. 63, 199 (1978). Crossref, Web of Science, Google Scholar
C. Mira et al. , Chaotic Dynamics in Two-Dimensional Noninvertible Maps ( World Scientific , Singapore , 1996 ) . Link, Google Scholar
S. Rajasekar and M. Lakshmanan, Physica D 67, 282 (1993). Crossref, Web of Science, Google Scholar
S. Rajasekar, S. Parthasarathy and M. Lakshmanan, Chaos Solit. Fract. 2, 271 (1992). Crossref, Google Scholar
J. Sugie, Quart. Appl. Maths. 49, 543 (1991). Crossref, Web of Science, Google Scholar
S. A. Treskov and E. P. Volokitin, Quart. Appl. Math. 54, 601 (1996). Crossref, Web of Science, Google Scholar
B. van der Pol and J. van der Mark, Philos. Mag. 6, 763 (1928). Crossref, Google Scholar
S. Wiggins , An Introduction to Applied Nonlinear Dynamics and Chaos ( Springer , NY , 1990 ) . Crossref, Google Scholar