Melnikov's criteria and chaos in systems with fractional order deflection

Journal of Sound and Vibration

Volumn 326, Issue 3-5, 2009, Pages 768-779

  Cveticanin, L ^a   Zukovic, M ^a  

^a UNIVERSITY OF NOVI SAD   (Serbia)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION DIAGRAM; CLOSED ORBITS; CONTROL OF CHAOS; CRITICAL PARAMETER VALUES; DYNAMIC SYSTEMS; FIXED POINTS; FRACTIONAL ORDER; HOMOCLINIC ORBITS; LYAPUNOV EXPONENT; PERIOD DOUBLING; PERIODIC MOTION; PHASE PLANE; POINCARE MAP; QUALITATIVE ANALYSIS; SECOND-ORDER DIFFERENTIAL EQUATION; TRANSITION TO CHAOS; UNSTABLE MANIFOLD;

EQUATIONS OF MOTION; QUALITY CONTROL;

CHAOTIC SYSTEMS;

EID: 68049148360     PISSN: 0022460X     EISSN: 10958568     Source Type: Journal    
DOI: 10.1016/j.jsv.2009.05.012     Document Type: Article

References (52)

1. Cveticanin L. Oscillator with fraction order restoring force. Journal of Sound and Vibration 320 (2008) 1064-1077
2. Guckenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer, New York
3. Wiggins S. Global Bifurcations and Chaos: Analytical Methods (1988), Springer, New York
4. Tseng W.Y., and Dugundji J. Nonlinear vibrations of a buckled beam under harmonic excitation. Journal of Applied Mechanics 38 (1971) 467-476
5. Holmes P. A nonlinear oscillator with a strange attractor. Philosophical Transactions of the Royal Society of London Series A 292 (1979) 419-448
6. Holmes P., and Marsden J. A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Archives for Rational Mechanics and Analysis 76 (1981) 135-166
7. Prathap G., and Varadan T.K. The inelastic large deformation of beams. Journal of Applied Mechanics 43 (1976) 689-690
8. Lo C.C., and Gupta S.D. Bending of a nonlinear rectangular beam in large deflection. Journal of Applied Mechanics 45 (1978) 213-215
9. Lewis G., and Monasa F. Large deflections of cantilever beams of non-linear material of the Ludwick type subjected to an end moment. International Journal of Non-Linear Mechanics 17 (1982) 1-6
10. Haslach H.W. Post-buckling behavior of columns with non-linear constitutive equations. International Journal of Non-Linear Mechanics 20 (1982) 53-67
11. Haslach H.W. Influence of adsorbed moisture on the elastic post-buckling behavior of columns made of non-linear hydrophilic polymers. International Journal of Non-Linear Mechanics 27 (1992) 527-546
12. Chen W.-H., and Gibson R.F. Property distribution for nonuniform composite beams from vibration response measurements and Galerkin's method. Journal of Applied Mechanics 65 (1988) 127-133
13. Duffing G. Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung, Druck und Verlag vor Friedr (1918), Vieweg & Sohn, Braunschweig
14. Russell D., and Rossing T. Testing the nonlinearity of piano hammers using residual shock spectra. Acustica-Acta Acustica 84 (1998) 967-975
15. Zhu Q., and Ishitoby M. Chaos and bifurcations in a nonlinear vehicle model. Journal of Sound and Vibration 275 (2004) 1136-1146
16. Patten W.N., Sha S., and Mo C. A vibration model of open celled polyurethane foam automative seat cushions. Journal of Sound and Vibration 217 (1998) 145-161
17. C.V. Jutte, Generalized Synthesis Methodology of Nonlinear Springs for Prescribed Load-Displacement Functions, PhD Dissertation, Mechanical Engineering, The University of Michigan, 2008.
18. Rhoads J.F. Generalized parametric resonance in electrostatically actuated micromechanical oscillators. Journal of Sound and Vibration 296 (2006) 797-829
19. Rhoads J.F. Tunable micromechanical filters that exploit parametric resonance. Journal of Vibration and Acoustics 127 (2005) 423-430
20. C. Cortopassi, O. Englander, Nonlinear Springs for Increasing the Maximum Stable Deflection of MEMS Electrostatic Gap Closing Actuators, UC Berkeley, 〈http://robotics.eecs.berkeley.edu/∼pister/245/project/Cortopas siEnglander〉, March 10, 2009.
21. Melnikov V.K. On the stability of the center for time periodic perturbations. Transactions of the Moscow Mathematical Society 12 (1963) 1-57
22. Holmes P. A nonlinear oscillator with a strange attractor. Philosophical Transactions of the Royal Society of London, Series A, Mathematical and Physical Sciences 292 (1979) 419-448
23. Chacon R. General result on chaos suppression for biharmonically driven dissipative systems. Physics Letters A 257 (1999) 293-300
24. Cveticanin L. Extension of Melnikov criterion for the differential equation with complex function. Nonlinear Dynamics 4 (1993) 139-152
25. Wang R., Deng J., and Jing Z. Chaos, control in Duffing system. Chaos, Solitons and Fractals 23 (2006) 249-257
26. J. Yang, Z. Jing, Control of chaos in a three-well duffing system, Chaos, Solitons and Fractals, in press (doi:10.1016/j.chaos.2008.05018).
27. Pyragas K. Continuous control of chaos by self controlling feedback. Physics Letters A 170 (1992) 421-428
28. Pyragas K. Continuous control of chaos by self-controlling feedback. Controlling Chaos (1996), Academic Press, San Diego 118-123
29. Ushio T. Limitation of delayed feedback control in nonlinear discrete-time systems. IEEE Transactions on Circuits and Systems I 43 (1996) 815-816
30. Chen G.R., and Yu X.H. On time-delayed feedback control of chaotic systems. IEEE Transactions on Circuits and Systems I 46 (1999) 767-772
31. Pyragas K. Control of chaos via an unstable delayed feedback controller. Physical Review Letters 86 (2001) 2265-2268
32. Pyragas K. Delayed feedback control of chaos. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364 (2006) 2309-2334
33. Zhu J., and Tian Y.-P. Necessary and sufficient conditions for stabilizability of discrete-time systems via delayed feedback control. Physics Letters A 343 (2005) 95-107
34. Dudnik E.N., Kuznetsov Yu.I., Minakova I.I., and Romanovskiii Yu.M. Synchronization in systems with strange attractors. Moscow University Physics Bulletin Series 3 24 (1983) 84-87
35. Chacon R. Maintenance and suppression of chaos by weak harmonic perturbations: a unified view. Physical Review Letters 86 (2001) 1737-1740
36. Lima R., and Pettini M. Suppression of chaos by resonant parametric perturbations. Physical Review A 41 (1990) 726-733
37. Lima R., and Pettini M. Parametric resonant control of chaos. International Journal of Bifurcation and Chaos 8 (1998) 1675-1684
38. Zhalnin A.Y. Control of chaos in nonautonomous systems with quasiperiodic excitation. Technical Physics Letters 25 (1999) 662-664
39. Mirus K.A., Sprott J.C., and 1999. Controlling chaos in low-and high-dimensional systems with periodic parametric perturbations. Physical Review E 59 (1999) 5313-5324
40. Belhaque M., and Houssni M. Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dynamics 18 (1999) 1-24
41. Pisarchik A.N., and Corbalan R. Parametric nonfeedback resonance in period doubling systems. Physical Review E 59 (1999) 1669-1674
42. Yang J., and Jing Z. Inhibition of chaos in a pendulum equation. Chaos, Solitons and Fractals 33 (2008) 726-737
43. Pyragas K., Lange F., Letz T., Parisi J., and Kittel A. Stabilization of an unstable steady state in intracavity frequency-doubled lasers. Physical Review E-Statistical Physicas, Plasmas, Fluids, and Related Interdisciplinary Topics 61 4A (2000) 3721-3731
44. Pyragas K., Lange F., Letz T., Parisi J., and Kittel A. Dynamics and control of a multimode laser: reduction of space-dependent rate equations to a low dimensional system. Physical Review E-Statistical, Nonlinear and Soft Matter Physics 63 (2001) 1-12
45. Hai W.H., Duan Y.W., and Pan L.X. An analytical study for controlling unstable periodic motion in magneto-elastic chaos. Physics Letters A 234 (1997) 198-204
46. Hikihara T., Touno M., and Kawagoshi T. Experimental stabilization of unstable periodic orbit in magneto-elastic chaos by delayed feedback control. International Journal of Bifurcation and Chaos 7 (1997) 2837-2846
47. Fradkov A.L., Evans R.J., and Andrievsky B.R. Control of chaos: methods and applications in mechanics. Philosophical Transactions of the Royal Society A 364 (2006) 2279-2307
48. V. Pyragas, K. Pyragas, Delayed feedback control of the Lorenz system: an analytical treatment at a subcritical Hopf bifurcation, Physical Review E-Statistical, Nonlinear and Soft Matter Physics 73 (036215) (2006) 1-10.
49. Zukovic M., and Cveticanin L. Chaotic response in a stable Duffing system of non-ideal type. Journal of Vibration and Control 13 (2007) 751-767
50. M. Zukovic, L. Cveticanin, Chaos in non-ideal mechanical system with clearance, Journal of Vibration and Control (2009) (doi:10.1177/1075546308091216).
51. I.S. Gradstein, I.M. Rjizhik, Tablici integralov, summ, rjadov i proizvedenij, Nauka, Moscow, 1971.
52. Sandri M. Numerical calculation of Lyapunov exponents. The Mathematical Journal 6 (1996) 78-84