This article reviews the history and recent development of the nonlinear dynamics theory of the stochastic layer in nonlinear Hamiltonian systems. The exponentially small splitting of separatrix, standard and whisker map approaches, Chirikov overlap criterion, renormalization group technique, and incremental energy method are presented herein for analytic predictions of the onset of resonance in stochastic layers. An energy spectrum technique is also presented for the numerical prediction of such an onset, and the numerical computation of stochastic layer widths is introduced as well. Some technical problems in this area are pointed out. The objective of this review is to excite more research in nonlinear dynamic systems. There are 47 references cited in this review article.

1.
Chirikov
BV
(
1979
),
A universal instability of many-dimensional oscillator systems
,
Phys. Rep.
52
,
263
379
.
2.
Lichtenberg AJ and Lieberman MA (1992), Regular and Chaotic Dynamics, 2nd Edition, Springer-Verlag, New York.
3.
Han
RPS
and
Luo
ACJ
(
1998
),
Resonant layers in nonlinear dynamics
,
ASME J. Appl. Mech.
65
,
727
736
.
4.
Luo
ACJ
and
Han
RPS
(
1999
),
Analytical predictions of chaos in a nonlinear rod
,
J. Sound Vib.
227
(
3
),
523
544
.
5.
Poincare´
H
(
1890
),
Sur la probleme des trois corps et les e´quations de la dynamique
,
Acta Math.
13
,
1
271
.
6.
Poincare´, H (1899), Les Me´thodes Nouvelles de la Me´canique Celeste, 3 Vols, Gauthier-Zvillars, Paris.
7.
Melnikov
VK
(
1963
),
On the stability of the center for time periodic perturbations
,
Trans. Mosc. Math. Soc.
12
,
1
57
.
8.
Holmes
PJ
,
Marsden
JE
, and
Scheurle
J
(
1988
),
Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations
,
Contemp. Math.
81
,
213
244
.
9.
Lazutkin
VF
,
Schachmannski
IG
, and
Tabanov
MB
(
1989
),
Splitting of separatrices for standard and semistandard mappings
,
Physica D
40
,
235
248
.
10.
Gelfreich
VG
,
Lazutkin
VF
, and
Tabanov
MB
(
1991
),
Exponentially small splitting in Hamiltonian systems
,
Chaos
1
,
137
142
.
11.
Gelfreich
VG
,
Lazutkin
VF
, and
Svanidze
NV
(
1994
),
A refined formula for the separatrix splitting for the standard map
,
Physica D
71
,
82
101
.
12.
Treschev
DV
(
1995
),
An averaging method for Hamiltonian systems, exponentially close to integrable ones
,
Chaos
6
,
6
14
.
13.
Treschev
DV
(
1998
),
Width of stochastic layers in near-integrable two-dimensional symplectic maps
,
Physica D
116
,
21
43
.
14.
Filonenko
NN
,
Sagdeev
RZ
, and
Zaslavsky
GM
(
1967
),
Destruction of magnetic surfaces by magnetic field irregularities–Part II:
Nuclear Fusion
7
,
253
266
.
15.
Zaslavsky
GM
and
Filonenko
NN
(
1968
),
Stochastic instability of trapped particles and conditions of application of the quasi-linear approximation
,
Sov. Phys. JETP
27
,
851
857
.
16.
Luo ACJ (1995), Analytical modeling of bifurcations, chaos and multifractals in nonlinear dynamics PhD dissertation, Univ of Manitoba, Winnipeg, Manitoba, Canada.
17.
Melnikov
VK
(
1962
),
On the behavior of trajectories of systems near to autonomous Hamiltonian systems
,
Sov. Math. Dokl.
3
,
109
112
.
18.
Arnold
VI
(
1964
),
Instability of dynamical systems with several degrees of freedom
,
Sov. Math. Dokl.
5
,
581
585
.
19.
Nekhoroshev
NN
(
1977
),
An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems
,
Russ. Math. Surveys
32
(
6
),
1
65
.
20.
Neishtadt
AI
(
1984
),
The separation of motion in systems with rapidly rotating phase, (PMM)
J. Appl. Math. Mech.
48
,
133
139
.
21.
Slutskin
AA
(
1964
),
Motion of a one-dimensional nonlinear oscillator under adiabatic conditions
,
Sov. Phys. JETP
18
,
676
682
.
22.
Luo
ACJ
,
Gu
K
, and
Han
RPS
(
1999
),
Resonant-separatrix webs in stochastic layers of the twin-well Duffing oscillator
,
Nonlinear Dyn.
19
,
37
48
.
23.
Reichl LE (1992), The Transition to Chaos in Conservative Classic System: Quantum Manifestations, Springer-Verlag, New York.
24.
Vecheslavov
VV
(
1996
),
Motion in the vicinity of the separatrix of a nonlinear resonance in the presence of high-frequency excitations
,
Sov. Phys. JETP
82
,
1190
1195
.
25.
Greene
JM
(
1968
),
Two-dimensional measure-preserving mappings
,
J. Math. Phys.
9
,
760
768
.
26.
Greene
JM
(
1979
),
A method for computing the stochastic transition
,
J. Math. Phys.
20
,
1183
1201
.
27.
Rom-Kedar
V
(
1990
),
Transport rates of a class of two-dimensional maps and flow
,
Physica D
43
,
229
268
.
28.
Rom-Kedar
V
(
1994
),
Homoclinic tangles-classification and applications
,
Nonlinearity
7
,
441
473
.
29.
Rom-Kedar
V
(
1995
),
Secondary homoclinic bifurcation theorems
,
Chaos
5
,
385
401
.
30.
Zaslavsky
GM
and
Abdullaev
SS
(
1995
),
Scaling properties and anomalous transport of particles inside the stochastic layer
,
Phys. Rev.
51
,
3901
3910
.
31.
Abdullaev
SS
and
Zaslavsky
GM
(
1995
),
Self-similarity of stochastic magnetic field lines near the X-point
,
Phys. Plasmas
2
,
4533
4541
.
32.
Abdullaev
SS
and
Zaslavsky
GM
(
1996
),
Application of the separatrix map to study perturbed magnetic field lines near the separatrix
,
Phys. Plasmas
3
,
516
528
.
33.
Ahn
T
,
Kim
G
, and
Kim
S
(
1996
),
Analysis of the separatrix map in Hamiltonian systems
,
Physica D
89
,
315
328
.
34.
Iomin
A
and
Fishman
S
(
1996
),
Semiclassical quantization of a separatrix map
,
Phys. Rev.
54
,
R1–R5
R1–R5
.
35.
Luo
ACJ
(
2001
),
Resonant-overlap phenomena in stochastic layers of nonlinear Hamiltonian systems with periodical excitations
,
J. Sound Vib.
240
(
5
),
821
836
.
36.
Luo
ACJ
and
Han
RPS
(
2000
),
Investigations of stochastic layers in nonlinear dynamics
,
J. Vibr. Acoust.
122
,
36
41
.
37.
Reichl
LE
and
Zheng
WM
(
1984
),
Field-Induced barrier penetration in the quadratic potential
,
Phys. Rev. A
29A
,
2186
2193
.
38.
Reichl
LE
and
Zheng
WM
(
1984
),
Perturbed double-well system: The pendulum approximation and low-frequency effect
,
Phys. Rev. A
30A
,
1068
1077
.
39.
Escande
DF
and
Doveil
F
(
1981
),
Renormalization method for the onset of stochasticity in a Hamiltonian system
,
Phys. Lett.
83A
,
307
310
.
40.
Escande
DF
(
1985
),
Stochasticity in classic Hamiltonian systems: universal aspects
,
Phys. Rep.
121
,
165
261
.
41.
Lin
WA
and
Reichl
LE
(
1986
),
External field induced chaos in an infinite square well potential
,
Physica D
19
,
145
152
.
42.
Luo
ACJ
,
Han
RPS
, and
Xiang
YM
(
1995
),
Chaotic analysis of subharmonic resonant waves in undamped and damped strings
,
Journal of Hydrodynamics
7B
,
92
104
.
43.
Luo
ACJ
and
Han
RPS
(
2001
),
The resonance theory for stochastic layers in nonlinear dynamic systems
,
Chaos, Solitons Fractals
12
,
2493
2508
.
44.
Arnold VI (1989), Mathematical Methods of Classical Mechanics, Springer: New York.
45.
Luo
ACJ
(
2001
),
Resonance and stochastic layer in a parametrically excited pendulum
,
Nonlinear Dyn.
25
(
4
),
355
367
.
46.
Feng
K
and
Qin
MZ
(
1991
),
Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study
,
Comput. Phys. Commun.
65
,
173
187
.
47.
McLachlan
R
and
Atela
P
(
1992
),
The accuracy of symplectic integrators
,
Nonlinearity
5
,
541
562
.
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