The onset of a new resonance in the stochastic layer is predicted numerically through the maximum and minimum energy spectra when the energy jump in the spectra occurs. The incremental energy approach among all the established, analytic approaches gives the best prediction of the onset of resonance in the stochastic layer compared to numerical investigation. The stochastic layers in the periodically-driven pendulum are discussed as another example. Illustrations of stochastic layers in the twin-well Duffing oscillator and the periodically-driven pendulum are given through the Poincare´ mapping sections. [S0739-3717(00)00701-7]

Issue Section:
Technical Papers
Keywords:
nonlinear dynamical systems, classical mechanics, resonance, stochastic processes, pendulums, Poincare mapping
Topics:
Excitation, Pendulums, Renormalization (Physics), Resonance, Nonlinear dynamics
1.
Chirikov
,
B. V.
,
1979
, “
A Universal Instability of Many-Dimensional Oscillator System
,”
Phys. Rep.
,
52
, pp.
263
–
379
.
2.
Lichtenberg. A. J., and Lieberman, M. A., 1992, Regular and Chaotic Dynamics, 2nd ed., Springer-Verlag, New York.
3.
Poincare, H., 1890, Les Methods Nouvelles de la Mecanique Celests, 3 Vols, Gauthier-Villars, Paris.
4.
Melnikov
,
V. K.
,
1963
, “
On the Stability of the Center for Time Periodic Perturbations
,”
Trans. Moscow Math. Soc.
,
12
, pp.
1
–
57
.
5.
Melnikov
,
V. K.
,
1962
, “
On the Behavior of Trajectories of System near to Autonomous Hamiltonian Systems
,”
Sov. Math. Dokl.
,
3
, pp.
109
–
112
.
6.
Lazutkin
,
V. F.
,
Schachmannski
,
I. G.
, and
Tabanov
,
M. B.
,
1989
, “
Splitting of Separatrices for Standard and Semistandard Mappings
,”
Physica D
,
40
, pp.
235
–
248
.
7.
Gelfreich
,
V. G.
,
Lazutkin
,
V. F.
, and
Tabanov
,
M. B.
,
1991
, “
Exponentially Small Splitting in Hamiltonian Systems
,”
Chaos
,
1
, pp.
137
–
142
.
8.
Gelfreich
,
V. G.
,
Lazutkin
,
V. F.
, and
Svanidze
,
N. V.
,
1994
, “
A Refined Formula for the Separatrix Splitting for the Standard Map
,”
Physica D
,
71
, pp.
82
–
101
.
9.
Treschev
,
D. V.
,
1995
, “
An Averaging Method for Hamiltonian Systems, Exponentially Close to Integrable Ones
,”
Chaos
,
6
, pp.
6
–
14
.
10.
Holmes
,
P. J.
,
Marsden
,
J. E.
, and
Scheurle
,
J.
,
1988
, “
Exponentially Small Splittings of Separatrices with Applications to KAM Theory and Degenerate Bifurcations
,”
Contemp. Math.
,
81
, pp.
213
–
244
.
11.
Treschev
,
D. V.
,
1998
, “
Width of Stochastic Layers in Near-Integrable Two-Dimensional Symplectic Maps
,”
Physica D
,
116
, pp.
21
–
43
.
12.
Rom-Kedar
,
V.
,
1990
, “
Transport Rates of a Class of Two-Dimensional Maps and Flow
,”
Physica D
,
43
, pp.
229
–
268
.
13.
Rom-Kedar
,
V.
,
1994
, “
Homoclinic Tangles-Classification and Applications
,”
Nonlinearity
,
7
, pp.
441
–
473
.
14.
Rom-Kedar
,
V.
,
1995
, “
Secondary Homoclinic Bifurcation Theorems
,”
Chaos
,
5
, pp.
385
–
401
.
15.
Zaslavsky
,
G. M.
, and
Abdullaev
,
S. S.
,
1995
, “
Scaling Properties and Anomalous Transport of Particles inside the Stochastic Layer
,”
Phys. Rev.
,
51
, pp.
3901
–
3910
.
16.
Abdullaev
,
S. S.
, and
Zaslavsky
,
G. M.
,
1995
, “
Self-similarity of Stochastic Magnetic Field Lines near the X-Point
,”
Phys. Plasmas
,
2
, pp.
4533
–
4541
.
17.
Abdullaev
,
S. S.
, and
Zaslavsky
,
G. M.
,
1996
, “
Application of the Separatrix Map to Study Perturbed Magnetic Field Lines near the Separatrix
,”
Phys. Plasmas
,
3
, pp.
516
–
528
.
18.
Ahn
,
T.
,
Kim
,
G.
, and
Kim
,
S.
,
1996
, “
Analysis of the Separatrix Map in Hamiltonian Systems
,”
Physica D
,
89
, pp.
315
–
328
.
19.
Iomin
,
A.
, and
Fishman
,
S.
,
1996
, “
Semiclassical Quantization of a Separatrix Map
,”
Phys. Rev.
,
54
, pp.
R1–R5
R1–R5
.
20.
Luo, A. C. J., 1995, Analytical Modeling of Bifurcations, Chaos, and Multifractals in Nonlincar Dynamics, Ph.D. dissertation, University of Manitoba, Winnipeg, Manitoba, Canada.
21.
Luo, A. C. J., 1999, “Resonant-Overlap Phenomena in Stochastic Layers of 1.5 Degrees of Freedom Nonlinear Systems,” J. Sound Vib., submitted.
22.
Reichl
,
L. E.
, and
Zheng
,
W. M.
,
1984a
, “
Field-Induced Barrier Penetration in the Quadratic Potential
,”
Phys. Rev.
,
29A
, pp.
2186
–
2193
.
23.
Reichl
,
L. E.
, and
Zheng
,
W. M.
,
1984b
, “
Perturbed Double-Well System: The Pendulum Approximation and Low-Frequency Effect
,”
Phys. Rev.
,
30A
, pp.
1068
–
1077
.
24.
Escande
,
D. F.
, and
Doveil
,
F.
,
1981
, “
Renormalization Method for the Onset of Stochasticity in a Hamiltonian System
,”
Phys. Lett. A
,
83
, pp.
307
–
310
.
25.
Escande
,
D. F.
,
1985
, “
Stochasticity in Classic Hamiltonian Systems: Universal Aspects
,”
Phys. Rep.
,
121
, pp.
165
–
261
.
26.
Luo, A. C. J., and Han, R. P. S., 1999a, “Analytical Predictions of Chaos in a Nonlinear Rod,” J. Sound Vib., in press.
27.
Luo
,
A. C. J.
,
Han
,
R. P. S.
, and
Xiang
,
Y. M.
,
1995
, “
Chaotic Analysis of Subharmonic, Resonant Waves in Undamped and Damped Strings
,”
J. Hydrodyn.
,
7B
, pp.
92
–
104
.
28.
Luo
,
A. C. J.
,
Gu
,
K.
, and
Han
,
R. P. S.
,
1999
, “
Resonant Separatrix Webs in the Stochastic Layers of the Twin-Well Duffing Oscillator
,”
Nonlinear Dyn.
,
19
, pp.
37
–
48
.
29.
Luo, A. C. J. and Han, R. P. S., 1999b, “The Resonant Mechanism of Stochastic Layers in Nonlinear Dynamic Systems with 1.5 Degrees of Freedom,” Int. J. Nonlinear Mech., submitted.
30.
Zaslavsky
,
G. M.
, and
Filonenko
,
N. N.
,
1968
, “
Stochastic Instability of Trapped Particles and Conditions of Application of the Quasi-Linear Approximation
,”
Sov. Phys. JETP
,
27
, pp.
851
–
857
.
31.
Reichl, L. E., 1992, The Transition to Chaos in Conservative Classic System: Quantum Manifestations, Springer-Verlag, New York.
32.
Han
,
Ray P.S.
, and
Luo
,
Albert C.J.
,
1998
, “
Resonant Layers in Nonlinear Dynamics
,”
ASME J. Appl. Mech.
,
65
, pp.
727
–
736
.
33.
Luo, A. C. J., and Han, R. P. S., 1997, “Stochastic and Resonant Layers in a Periodically Driven Pendulum,” International Mechanical Engineering Congress and Exposition, Dallas, Texas, USA, DE-Vol. 95/AMD-Vol. 223, pp. 207–215.
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