0
Research Papers

On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise

[+] Author and Article Information
Chen Kong

State Key Laboratory of Mechanics
and Control of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: kongchen_bill@126.com

Xue Gao

State Key Laboratory of Mechanics
and Control of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: xgao.detec@nuaa.edu.cn

Xianbin Liu

State Key Laboratory of Mechanics
and Control of Mechanical Structures,
Nanjing University of Aeronautics
and Astronautics,
Nanjing 210016, China
e-mail: xbliu@nuaa.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 30, 2015; final manuscript received May 18, 2016; published online June 9, 2016. Assoc. Editor: Corina Sandu.

J. Comput. Nonlinear Dynam 11(5), 051029 (Jun 09, 2016) (10 pages) Paper No: CND-15-1404; doi: 10.1115/1.4033687 History: Received November 30, 2015; Revised May 18, 2016

The global analysis is very important for a nonlinear dynamical system which possesses a chaotic saddle and a nonchaotic attractor, especially for the one that is driven by a noise. For a random dynamical system, within which, chaotic saddles exist, it is found that if the noise intensity exceeds a critical value, the so called “noise-induced chaos” is observed. Meanwhile, the exit behavior is also found to be influenced significantly by the existence of chaotic saddles. In the present paper, based on the generalized cell-mapping digraph (GCMD) method, the global dynamical behaviors of a piecewise linear system, wherein a chaotic saddle exists and consists of subharmonic solutions in a wide frequency range, are investigated numerically. Further, in order to simplify the system that is driven by a Gaussian white noise excitation, the stochastic averaging method is applied and through which, a five-dimensional Itô system is obtained. Some of the global dynamical behaviors of the original system are retained in the averaged one and then are analyzed. The researches in this paper show that GCMD method is a good numerical tool to investigate the global behaviors of a nonlinear random dynamical system, and the stochastic averaging method is effective for solving the global problems.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

Bifurcation diagram of system (1) for ζ=0.01,f0=f1=0.25,D=0. The x -component of the Poincaré points on Eq. (4) at steady-state is shown versus the external forcing frequency ω. T in these two pictures stands for the period of the external excitation T=2π/ω.

Grahic Jump Location
Fig. 3

(a) The phase diagram projected on Poincaré section (4) by using GCMD method at ω=0.47. (b) The phase diagram obtained on Poincaré section (4) around the period-1 solution at ω=0.49.

Grahic Jump Location
Fig. 4

The phase diagram projected on Poincaré section (4) by using GCMD method at (a) and (b) ω=0.68, (c) and (d) ω=0.70, (e) and (f) ω=0.75, (g) and (h) ω=0.80, (i) and (j) ω=0.89

Grahic Jump Location
Fig. 5

The phase diagram projected on Poincaré section (4) by using GCMD method at (a) ω=0.47, (b) ω=0.475, (c) ω=0.48, and (d) ω=0.49

Grahic Jump Location
Fig. 6

The phenomenon of noise-induced chaos at ω=0.89,D=6e−5. The time history of y -component of the Poincaré points on Eq. (4) is shown versus time T which stands for the period of the external excitation T=2π/ω.

Grahic Jump Location
Fig. 7

The phase diagram projected on the (a,b) plane by using GCMD method at (a) ω=0.80, (b) ω=0.89

Grahic Jump Location
Fig. 8

(a) The phase diagram projected on Poincaré section (4) by using GCMD method at ω=0.89. (b) The two-dimensional phase trajectory projection on (x,y) plane corresponding to the period-3 saddle in Fig. 8(a).

Grahic Jump Location
Fig. 9

The two-dimensional phase trajectories projection on (x, y) plane included three harmonic solutions which are all elliptical orbits, and a subharmonic solution with order of two-third which is a self-intersected orbit in (a) the original system (1) (b) the new averaged system (16) and (18)

Grahic Jump Location
Fig. 10

The phenomenon of intermittency at ω=0.89,D=5e−4. The time history of d2+e2 of the new averaged system is shown versus time 3T which stands for the triple periods of the external excitation T=2π/ω.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In