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

Roy, R. V. , 1997, “ Global Stability Analysis of Nonlinear Dynamical Systems,” Ser. Stab., Vib. Control Syst. Ser. B, 9, pp. 261–295.
Roy, R. V. , 1994, “ Averaging Method for Strongly Nonlinear Oscillators With Periodic Excitations,” Int. J. Non-Linear Mech., 29(5), pp. 737–753. [CrossRef]
Roy, R. V. , 1995, “ Noise-Induces Transitions in Weakly-Nonlinear Oscillators Near Resonance,” ASME J. Appl. Mech., 62(2), pp. 496–504. [CrossRef]
Roy, R. V. , 1993, “ Noise Perturbations of Nonlinear Dynamical Systems,” Computational Stochastic Mechanics, Elsevier, Amsterdam, The Netherlands, pp. 125–147.
Roy, R. V. , 1995, “ Large Deviation Theory, Weak-Noise Asymptotics, and First-Passage Problems: Review and Results,” Applications of Statistics and Probability, M. Lemaire , J.-L. Favre , and A. Mebarki , eds., A.A. Balkema, Rotterdam, The Netherlands, pp. 1129–1135.
Roy, R. V. , 1996, “ Probabilistic Analysis of a Nonlinear Pendulum,” Acta Mech., 115(1), pp. 87–101. [CrossRef]
Roy, R. V. , 1994, “ Noise Perturbations of a Non-Linear System With Multiple Steady States,” Int. J. Non-Linear Mech., 29(5), pp. 755–773. [CrossRef]
Roy, R. V. , 1996, “ Asymptotic Analysis of First-Passage Problems,” Int. J. Non-Linear Mech., 32(1), pp. 173–186. [CrossRef]
Roy, R. V. , and Nauman, E. , 1995, “ Noise-Induced Effects on a Non-Linear Oscillator,” J. Sound Vib., 183(2), pp. 269–295. [CrossRef]
Chen, L. C. , and Zhu, W. Q. , 2010, “ First Passage Failure of Quasi-Partial Integrable Generalized Hamiltonian Systems,” Int. J. Non-Linear Mech., 45(1), pp. 56–62. [CrossRef]
Zhu, W. Q. , and Wu, Y. J. , 2003, “ First-Passage Time of Doffing Oscillator Under Combined Harmonic and White-Noise Excitation,” Nonlinear Dyn., 32(3), pp. 291–305. [CrossRef]
Chen, L. C. , Deng, M. L. , and Zhu, W. Q. , 2009, “ First Passage Failure of Quasi Integrable-Hamiltonian Systems Under Combined Harmonic and White Noise Excitations,” Acta Mech., 206(3), pp. 133–148. [CrossRef]
Huang, Z. L. , Zhu, W. Q. , and Suzuki, Y. , 2000, “ Stochastic Averaging of Strongly Non-Linear Oscillators Under Combined Harmonic and White-Noise Excitations,” J. Sound Vib., 238(2), pp. 233–256. [CrossRef]
Wu, Y. J. , Luo, M. , and Zhu, W. Q. , 2000, “ First-Passage Failure of Strongly Nonlinear Oscillators Under Combined Harmonic and Real Noise Excitations,” Arch. Appl. Mech., 78(7), pp. 501–515. [CrossRef]
Rodrigues, C. S. , Grebogi, C. , and de Moura, A. P. S. , 2010, “ Escape From Attracting Sets in Randomly Perturbed Systems,” Phys. Rev. E, 82(4), p. 046217. [CrossRef]
Kifer, Y. , 1981, “ The Exit Problem for Small Random Perturbations of Dynamical Systems With a Hyperbolic Fixed Point,” Isr. J. Math., 40(1), pp. 74–96. [CrossRef]
Guckenheimer, J. , 1982, “ Noise in Chaotic Systems,” Nature, 298(5872), pp. 358–361. [CrossRef]
Djeundam, S. R. D. , Yamapi, R. , Kofane, T. C. , and Aziz-Alaoui, M. A. , 2012, “ Deterministic and Stochastic Bifurcations in the Hindmarsh-Rose Neuronal Model,” Chaos, 23(3), p. 033125. [CrossRef]
Crutchfield, J. P. , Farmer, J. D. , and Huberman, B. A. , 1982, “ Fluctuations and Simple Chaotic Dynamics,” Phys. Rep. (Rev. Sec. Phys. Lett.), 92(2), pp. 45–82.
Crutchfield, J. P. , and Huberman, B. A. , 2013, “ Fluctuations and the Onset of Chaos,” Phys. Lett. A, 77(6), pp. 407–410. [CrossRef]
He, T. , and Habib, S. , “ Chaos and Noise,” Chaos, 23(3), p. 033123. [CrossRef] [PubMed]
Deissler, R. J. , and Farmer, J. D. , 1992, “ Deterministic Noise Amplifiers,” Phys. D, 55(1–2), pp. 155–165. [CrossRef]
Ellner, S. , and Turchin, P. , 1995, “ Chaos in a Noisy World: New Methods and Evidence From Time-Series Analysis,” Am. Nat., 145(3), pp. 343–375. [CrossRef]
Soskin, S. M. , Mannella, R. , Arrayás, M. , and Silchenko, A. N. , 2001, “ Strong Enhancement of Noise-Induced Escape by Nonadiabatic Periodic Driving Due to Transient Chaos,” Phys. Rev. E, 63(5), p. 051111. [CrossRef]
Tél, T. , Lai, Y. C. , and Gruiz, M. , 2008, “ Noise-Induced Chaos: A Consequence of Long Deterministic Transients,” Int. J. Bifurcation Chaos, 18(02), pp. 509–520. [CrossRef]
Billings, L. , and Schwartz, I. B. , 2002, “ Exciting Chaos With Noise: Unexpected Dynamics in Epidemic Outbreaks,” J. Math. Biol., 44(1), pp. 31–48. [CrossRef] [PubMed]
Lai, Y. C. , Liu, Z. , Billings, L. , and Schwartz, I. B. , 2003, “ Noise-Induced Unstable Dimension Variability and Transition to Chaos in Random Dynamical Systems,” Phys. Rev. E, 67(2), p. 026210. [CrossRef]
Armbruster, D. , Stone, E. , and Kirk, V. , 2003, “ Noisy Heteroclinic Networks,” Chaos, 13(1), pp. 71–79. [CrossRef]
Kirk, V. , and Silber, M. , 1994, “ A Competition Between Heteroclinic Cycles,” Nonlinearity, 7(6), pp. 1605–1621. [CrossRef]
Stone, E. , and Armbruster, D. , 1999, “ Noise and O(1) Amplitude Effects on Heteroclinic Cycles,” Chaos, 9(2), pp. 499–506. [CrossRef] [PubMed]
Stone, E. , and Holmes, P. , 1990, “ Random Perturbation of Heteroclinic Attractors,” SIAM J. Appl. Math., 50(3), pp. 726–743. [CrossRef]
Zhou, C. S. , Kurths, J. , Allaria, E. , Boccaletti, S. , Meucci, R. , and Arecchi, F. T. , 2003, “ Constructive Effects of Noise on Homoclinic Chaotic Systems,” Phys. Rev. E, 67(6), p. 066220. [CrossRef]
Tél, T. , and Lai, Y. C. , 2010, “ Quasipotential Approach to Critical Scaling in Noise-Induced Chaos,” Phys. Rev. E, 81(5), p. 056208. [CrossRef]
Kraut, S. , and Feudel, U. , 2003, “ Noise-Induced Escape Through a Chaotic Saddle: Lowering of the Activation Energy,” Phys. D, 181(3–4), pp. 222–234. [CrossRef]
Kraut, S. , and Feudel, U. , 2003, “ Enhancement of Noise-Induced Escape Through the Existence of a Chaotic Saddle,” Phys. Rev. E, 67(1), p. 015204. [CrossRef]
Ibrahim, R. A. , 2008, “ Noise-Induced Transition in Ship Roll Oscillations,” Adv. Stud. Theor. Phys., 2(2), pp. 51–69.
Hunt, B. R. , Ott, E. , and Yorke, J. A. , 1996, “ Fractal Dimensions of Chaotic Saddles of Dynamical Systems,” Phys. Rev. E, 54(5), pp. 4819–4823. [CrossRef]
Kraut, S. , and Feudel, U. , 2002, “ Multistability, Noise, and Attractor Hopping: The Crucial Role of Chaotic Saddles,” Phys. Rev. E, 66, p. 015207. [CrossRef]
Hong, L. , and Xu, J. , 1999, “ Crises and Chaotic Transients Studied by the Generalized Cell Mapping Digraph Method,” Phys. Lett. A, 262(4–5), pp. 361–375. [CrossRef]
Hong, L. , and Xu, J. , 2003, “ Chaotic Saddles in Wada Basin Boundaries and Their Bifurcations by the Generalized Cell-Mapping Digraph (GCMD) Method,” Nonlinear Dyn., 32(4), pp. 371–385. [CrossRef]
Hong, L. , and Xu, J. , 2004, “ A Chaotic Crisis Between Chaotic Saddle and Attractor in Forced Duffing Oscillators,” Commun. Nonlinear Sci. Numer. Simul., 9(3), pp. 313–329. [CrossRef]
Hsu, C. S. , 1995, “ Global Analysis of Dynamical Systems Using Posets and Digraphs,” Int. J. Bifurcation Chaos, 5(4), pp. 1085–1118. [CrossRef]
Ling, H. , 2010, “ A Fuzzy Crisis in a Duffing-Van Der Pol System,” Chin. Phys. B, 19(3), p. 030513. [CrossRef]
Hong, L. , and Sun, J.-Q. , 2006, “ Bifurcations of Fuzzy Nonlinear Dynamical Systems,” Commun. Nonlinear Sci. Numer. Simul., 11(1), pp. 1–12. [CrossRef]
Hong, L. , Zhang, Y. , and Jiang, J. , 2010, “ A Hyperchaotic Crisis,” Int. J. Bifurcation Chaos, 20(4), pp. 1193–1200. [CrossRef]
Xu, W. , He, Q. , Fang, T. , and Rong, H. , 2005, “ Global Analysis of Crisis in Twin-Well Duffing System Under Harmonic Excitation in Presence of Noise,” Chaos, Solitons Fractals, 23(1), pp. 141–150. [CrossRef]
Han, Q. , Xu, W. , Yue, X. , and Zhang, Y. , 2015, “ First-Passage Time Statistics in a Bistable System Subject to Poisson White Noise by the Generalized Cell Mapping Method,” Commun. Nonlinear Sci. Numer. Simul., 23(1–3), pp. 220–228. [CrossRef]
Han, Q. , Xu, W. , and Yue, X. , 2014, “ Global Bifurcation Analysis of a Duffing-Van Der Pol Oscillator With Parametric Excitation,” Int. J. Bifurcation Chaos, 23(4), p. 1450051. [CrossRef]
Yue, X. , Xu, W. , and Zhang, Y. , 2012, “ Global Bifurcation Analysis of Rayleigh-Duffing Oscillator Through the Composite Cell Coordinate System Method,” Nonlinear Dyn., 69(1), pp. 437–457. [CrossRef]
Hailin, Z. , and JianXue, X. , 2009, “ Improved Generalized Cell Mapping for Global Analysis of Dynamical Systems,” Sci. China Ser. E: Technol. Sci., 52(3), pp. 787–800. [CrossRef]
Hsu, C. S. , 1987, Cell-to-Cell Mapping, Springer-Verlag, New York.
Sun, J. Q. , and Hsu, C. S. , 1988, “ First-Passage Time Probability of Non-Linear Stochastic Systems by Generalized Cell Mapping Method,” J. Sound Vib., 124(2), pp. 233–248. [CrossRef]
Hsu, C. S. , and Chiu, H. M. , 1986, “ A Cell Mapping Method for Nonlinear Deterministic and Stochastic Systems-Part I: The Method of Analysis,” ASME J. Appl. Mech., 53(3), pp. 695–701. [CrossRef]
Chiu, H. M. , and Hsu, C. S. , 1986, “ A Cell Mapping Method for Nonlinear Deterministic and Stochastic Systems-Part II: Examples of Application,” ASME J. Appl. Mech., 53(3), pp. 702–710. [CrossRef]
Xu, J. , 2009, “ Some Advances on Global Analysis of Nonlinear Systems,” Chaos, Solitons Fractals, 39(4), pp. 1839–1848. [CrossRef]
Kong, C. , and Liu, X. B. , 2014, “ Research for Attracting Region and Exit Problem of a Piecewise Linear System Under Periodic and White Noise Excitations,” Chin. J. Theor. Appl. Mech., 46(3), pp. 447–456 (in Chinese).

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