SHRIMP STRUCTURE AND ASSOCIATED DYNAMICS IN PARAMETRICALLY EXCITED OSCILLATORS
Abstract
We investigate the bifurcation structures in a two-dimensional parameter space (PS) of a parametrically excited system with two degrees of freedom both analytically and numerically. By means of the Rényi entropy of second order K2, which is estimated from recurrence plots, we uncover that regions of chaotic behavior are intermingled with many complex periodic windows, such as shrimp structures in the PS. A detailed numerical analysis shows that the stable solutions lose stability either via period doubling, or via intermittency when the parameters leave these shrimps in different directions, indicating different bifurcation properties of the boundaries. The shrimps of different sizes offer promising ways to control the dynamics of such a complex system.
References
- Astron. Astrophys. 426, 353 (2004). Crossref, ISI, Google Scholar
- Phys. Rev. Lett. 78, 4561 (1997). Crossref, ISI, Google Scholar
- J. Sound Vibr. 233, 557 (2000). ISI, Google Scholar
- Int. J. Bifurcation and Chaos 14, 337 (2004). Link, ISI, Google Scholar
- Doedel, E. J. & Fairgrieve, T. F. [1997] AUTO 97, Continuation and Bifurcation Software for Ordinary Differential Equations, Tech. Rept. Concordia University, Montreal . Google Scholar
- Europhys. Lett. 4, 973 (1987). Crossref, Google Scholar
- Phys. Rev. Lett. 70, 2714 (1993). Crossref, ISI, Google Scholar
- Physics A 202, 196 (1994). Crossref, ISI, Google Scholar
-
J. Guckenheimer and P. Holmes , Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 3rd edn. ( Springer-Verlag , NY , 1990 ) . Google Scholar -
H. Kantz and T. Schreiber , Nonlinear Time Series Analysis ( Cambridge University Press , Cambridge , 1997 ) . Google Scholar - Nonlin. Proc. Geophys. 1, 72 (1994). Crossref, ISI, Google Scholar
-
R. N. Madan , Chua's Circuit: A Paradigm for Chaos ( World Scientific , Singapore , 1993 ) . Link, Google Scholar - Nonl. Proc. Geophys. 9, 325 (2002). Crossref, ISI, Google Scholar
- Phys. Rev. E 66, 026702 (2002). Crossref, Google Scholar
- Marwan, N. [2003] Encounters with Neighbours-Current Developments of Concepts Based on Recurrence Plots and Their Applications, PhD thesis, Institute of Physics, University of Potsdam . Google Scholar
-
A. H. Nayfeh and D. T. Mook , Introduction to Perturbation Techniques ( John Wiley and Sons , NY , 1981 ) . Google Scholar - Europhys. Lett. 72, 176 (2005). Crossref, Google Scholar
- Phys. Lett. A 330, 214 (2004). Crossref, ISI, Google Scholar
- Europhys. Lett. 71, 466 (2005). Crossref, Google Scholar
- Appl. Nonl. Dyn. 11, 20 (2003). ISI, Google Scholar
- Thiel, M. [2004] Recurrences: Exploiting Naturally Occurring Analogues, PhD thesis, Institute of Physics, University of Potsdam . Google Scholar
- Chaos 14, 234 (2004). Crossref, ISI, Google Scholar
- Phys. Lett. A 330, 343 (2004). Crossref, ISI, Google Scholar
- J. Appl. Physiol. 76, 965 (1994). Crossref, ISI, Google Scholar
-
S. Wiggins , Introduction to Applied Nonlinear Dynamical System and Chaos ( Springer , NY , 1990 ) . Crossref, Google Scholar - Physica D 16, 285 (1985). Crossref, ISI, Google Scholar
- Phys. Lett. A 237, 131 (1998). Crossref, ISI, Google Scholar
- Med. Eng. Phys. 24, 53 (2002). Crossref, ISI, Google Scholar
| Remember to check out the Most Cited Articles! |
|---|
|
Check out our Bifurcation & Chaos |
