| GENERAL |
|
|
|
|
An Effective Numerical Procedure to Determine Saddle-Type Unstable Invariant Limit Sets in Nonlinear Systems |
| JIANG Jun** |
| State Key Laboratory of Strength and Vibration, Xi'an Jiaotong University, Xi'an 710049 |
|
| Cite this article: |
|
JIANG Jun 2012 Chin. Phys. Lett. 29 050503 |
|
|
|
|
Abstract A new technique that can efficiently approximate the attracting set of a nonlinear dynamical system is proposed under the framework of point mapping with the cell reference method. With the aid of the approximated attracting set, the difficulties encountered by the PIM-triple method and bisection procedure in finding trajectories on the stable manifolds of chaotic saddles in basins of attraction and on basin boundaries can be overcome well. On the basis of this development, an effective method to determine saddle-type invariant limit sets of nonlinear dynamical systems can be devised. Examples are presented for the purposes of illustration and to demonstrate the capabilities of the proposed method.
|
|
|
Received: 21 November 2011
Published: 30 April 2012
|
|
| PACS: |
05.45.-a
|
(Nonlinear dynamics and chaos)
|
| |
02.40.Vh
|
(Global analysis and analysis on manifolds)
|
|
|
|
|
|
[1] Pomeau Y and Manneville P 1980 Commun. Math. Phys. 74 189
[2] Grebogi C, Ott E and York J A 1982 Phys. Rev. Lett. 48 1507
[3] Grebogi C, Ott E and York J A 1983 Physica D 7 181
[4] Grebogi C, Ott E and York J A 1986 Phys. Rev. Lett. 56 1011
[5] Grebogi C, Ott E and York J A 1987 Physica D 24 243
[6] Nusse H E and Yorke J A 1989 Physica D 36 137
[7] Nusse H E and Yorke J A 1991 Nonlinearity 4 1183
[8] Jiang J 2011 Theor. Appl. Mech. Lett. 1 063001
[9] Jiang J and Xu J X 1994 Phys. Lett. A 188 137
[10] Hsu C S 1995 Int. J. Bifurcation Chaos 5 1085
[11] Hong L and Xu J X 1999 Phys. Lett. A 262 361
[12] He Q, Xu W, Li S and Xiao Y Z 2008 Acta Phys. Sin. 57 743 (in Chinese)
[13] Zou H L and Xu J X 2009 Sci. Chin. Ser. E: Technolog. Sci. 52 787
[14] Hsu C S 1987 Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems (Berlin: Springer)
[15] Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Berlin: Springer)
[16] Dellnitz M and Hohmann A 1997 Numer. Math. 75 293
[17] Jiang J and Xu J X 1996 J. Sound Vib. 194 605
[18] Jiang J and Xu J X 1998 Nonlinear Dyn. 15 103
[19] Cao Y Y, Chung K W and Xu J 2011 Nonlinear Dyn. 64 221
[20] Chen S H, Chen Y Y and Sze K Y 2010 Sci. Chin. Ser. E: Technol. Sci. 53 692
[21] Manucharyan G V and Mikhlin Y V 2005 J. Appl. Math. Mech. 69 39 |
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
