The nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well -Rayleigh oscillator combined to parametric excitations is studied in this paper. The main attention is focused on the dynamical properties of local bifurcations as well as global bifurcations including homoclinic and heteroclinic bifurcations. The original oscillator is transformed to averaged equations using the method of harmonic balance to obtain periodic solutions. The response curves show the saddle-node bifurcation and the multi-stability phenomena. Based on the Melnikov’s method, horseshoe chaos is found and its control is made by introducing an external periodic perturbation.
Issue Section:
Research Papers
1.
Wei
, Z.
, and Pei
, Y.
, 2001, “Degenerate Bifurcation Analysis on Parametrically and Externally Excited Mechanical System
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 3
, pp. 689
–709
.2.
Siewe Siewe
, M.
, Moukam Kakmeni
, F. M.
, and Tchawoua
, C.
, 2004, “Resonance Oscillations and Homoclinic Bifurcation in a Φ6-Van der Pol Oscillators
,” Chaos, Solitons Fractals
0960-0779, 21
, pp. 841
–853
.3.
Siewe Siewe
, M.
, Moukam Kakmeni
, F. M.
, Tchawoua
, C.
, and Woafo
, P.
, 2005, “Bifurcations and Chaos in Periodically and Externally Driven Φ6-Van der Pol Oscillators
,” Physica A
0378-4371, 357
, pp. 383
–396
.4.
Belhaq
, M.
, and Houssni
, M.
, 2000, “Suppression of Chaos in Averaged Oscillator Driven by External and Parametric Excitations
,” Chaos, Solitons Fractals
0960-0779, 11
, pp. 1237
–1246
.5.
Cao
, H.
, Chi
, X.
, and Chen
, G.
, 2004, “Suppressing or Inducing Chaos by Weak Resonant Excitations in an Externally-Forced Froude Pendulum
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 3
, pp. 1115
–1120
.6.
Rajasekar
, S.
, 1993, “Controlling of Chaos by Weak Periodic Perturbations in Duffingvan der Pol Oscillator
,” Pramana
0304-4289, 41
, pp. 295
–309
.7.
Dabbs
, M.
, and Smith
, P.
, 1996, “Critical Forcing for Homoclinic and Heteroclinic Orbits of a Rotating Pendulum
,” J. Sound Vib.
0022-460X, 2
, pp. 231
–248
.8.
Chacón
, R.
, 2001, “Maintenance and Suppression of Chaos by Weak Harmonic Perturbations, A Unified View
,” Phys. Rev. Lett.
0031-9007, 86
, pp. 1737
–1740
.9.
Lenci
, S.
, and Rega
, G.
, 2003, “Optimal Control of Nonregular Dynamics in a Duffing Oscillator
,” Nonlinear Dyn.
0924-090X, 33
, pp. 71
–86
.10.
Lenci
, S.
, and Rega
, G.
, 2003, “Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator
,” J. Vib. Control
1077-5463, 3
, pp. 28l
–316
.11.
Lenci
, S.
, and Rega
, G.
, 2004, “Heteroclinic Bifurcations and Optimal Control in the Nonlinear Rocking Dynamics of Generic and Slender Rigid Blocks
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274 (in press).12.
Cao
, H.
, Jiang
, Y.
, and Shan
, Y.
, 2006, “Primary Resonant Optimal Control for Nested Homoclinic and Heteroclinic Bifurcations in Single-DOF Nonlinear Oscillators
,” J. Sound Vib.
0022-460X, 289
, pp. 229
–244
.13.
Lakshmanan
, M.
, and Murali
, K.
, 1996, Chaos in Nonlinear Oscillators: Controlling and Synchronization
, World Scientific Series on Nonlinear Science Series A
, World Scientific, Singapore, Vol. 13
.14.
Warminski
, J.
, 2003, “Regular, Chaotic and Hyperchaotic Vibrations of Nonlinear Systems With Self Parametric and External Excitations
,” Mech. Autom. Control Robot.
, 14
, pp. 891
–905
.15.
Verhulst
, F.
, 2001, “Parametric and Autoparametric Resonance
,” Mathematisch Instituut University of Utrecht, Utrecht, The Netherlands, April 17.16.
Szabelski
, K.
, and Warminski
, J.
, 1995, “Self-Excited System Vibrations With Parametric and External Excitations
,” J. Sound Vib.
0022-460X, 4
, pp. 595
–607
.17.
Tchoukuegno
, R.
, and Woafo
, P.
, 2002, “Dynamics and Active Control of Motion of a Particle in a ϕ6 Potential With a Parametric Forcing
,” Physica D
0167-2789, 167
, pp. 86
–100
.18.
Nayfeh
, A. H.
, and Mook
, D. T.
, 1979, Nonlinear Oscillations
, Wiley
, New York.19.
Wiggins
, S.
, 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos
, Springer-Verlag
, Berlin.20.
Trueba
, J. L.
, Baltanás
, J. P.
, and Sanjuán
, M. A. F.
, 2003, “A Generalized Perturbed Pendulum
,” Chaos, Solitons Fractals
0960-0779, l5
, pp. 911
–924
.21.
Trueba
, J. L.
, Baltanás
, J. P.
, and Sanjuán
, M. A. F.
, 2002, “Nonlinearly Damped Oscillators
,” Recent Res. Dev. Sound Vib.
, 1
, pp. 29
–61
.22.
Gradshteyn
, I.
, and Ryzhik
, I.
, 1994, Table of Integrals, Series and Products
, Academic Press
, New York.23.
Lenci
, S.
, and Rega
, G.
, 2004, “Higher-Order Melnikov Functions for Single-DOF Mechanical Oscillators: Theoretical Treatment and Applications
,” Math. Probl. Eng.
1024-123X, 2
, pp. 145
–168
.24.
Chacón
, R.
, Balibrea
, F.
, and López
, M. A.
, 2001, “Role of Nonlinear Dissipation in the Suppression of Chaotic Escape From Potential Well
,” Phys. Lett. A
0375-9601, 279
, pp. 38
–46
.25.
Lima
, R.
, and Pettini
, M.
, 1990, “Suppression of Chaos by Resonant Parametric Perturbations
,” Phys. Rev. A
1050-2947, 41
, pp. 726
–733
.Copyright © 2006
by American Society of Mechanical Engineers
You do not currently have access to this content.
