Geometrical approach for description of the mixed state in multi-well potentials
We use the so-called geometrical approach [1] in description of transition from regular motion to chaotic one in Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems is coexistence of different types of dynamics (regular or chaotic) in...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2007 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/110968 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Geometrical approach for description of the mixed state in multi-well potentials / V.P. Berezovoj, Yu.L. Bolotin, G.I. Ivashkevych // Вопросы атомной науки и техники. — 2007. — № 3. — С. 249-254. — Бібліогр.: 8 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859826669701824512 |
|---|---|
| author | Berezovoj, V.P. Bolotin, Yu.L. Ivashkevych, G.I. |
| author_facet | Berezovoj, V.P. Bolotin, Yu.L. Ivashkevych, G.I. |
| citation_txt | Geometrical approach for description of the mixed state in multi-well potentials / V.P. Berezovoj, Yu.L. Bolotin, G.I. Ivashkevych // Вопросы атомной науки и техники. — 2007. — № 3. — С. 249-254. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | We use the so-called geometrical approach [1] in description of transition from regular motion to chaotic one in Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems is coexistence of different types of dynamics (regular or chaotic) in different wells at the same energy [2]. Application of traditional criteria for transition to chaos (resonance overlap criterion, negative curvature criterion and stochastic layer destruction criterion) is inefficient in case of potentials with complex topology. Geometrical approach allows considering only configuration space but not phase space when investigating the stability. In this approach all information about chaos and regularity is contained in potential function. The aim of this work is to determine what details of geometry of potential lead to chaos in Hamiltonian systems using geometrical approach. Numerical calculations are executed for potentials that are relevant with lowest umbilical catastrophes.
Ми використовуємо так званий геометричний підхід [1] в описі переходу від регулярного руху до хаотичного в гамільтонових системах, у яких поверхня потенційної енергії має кілька локальних мінімумів. Відмітна риса таких систем – співіснування різних типів динаміки (регулярного або хаотичного) у різних потенційних ямах при тій же самій енергії [2]. Застосування традиційних критеріїв для переходу до хаосу (критерій перекриття резонансів, критерій негативної кривизни й критерій руйнування стохастичного шару) неефективно у випадку потенціалів із комплексною топологією. Геометричний підхід при дослідженні стабільності дозволяє розглядати тільки простір конфігурацій, але не фазовий простір. У цьому підході вся інформація щодо хаосу й регулярності міститься в потенційній функції. Ціль даної роботи полягає в тому, щоб, використовуючи геометричний підхід, визначити які деталі геометрії потенціалу приводять до хаосу в гамільтонових системах. Чисельні розрахунки виконані для потенціалів, які відповідають найнижчим омбілічним катастрофам.
Мы используем так называемый геометрический подход [1] в описании перехода от регулярного движения к хаотическому в гамильтоновых системах, в которых поверхность потенциальной энергии имеет несколько локальных минимумов. Отличительная черта таких систем – сосуществование различных типов динамики (регулярного или хаотического) в разных потенциальных ямах при той же самой энергии [2]. Применение традиционных критериев для перехода к хаосу (критерий перекрытия резонансов, критерий отрицательнoй кривизны и критерий разрушения стохастического слоя) неэффективно в случае потенциалов с комплексной топологией. Геометрический подход при исследовании устойчивости позволяет рассматривать только пространство конфигураций, но не фазовое пространство. В этом подходе вся информация относительно хаоса и регулярности содержится в потенциальной функции. Цель настоящей работы состоит в том, чтобы, используя геометрический подход, определить какие детали геометрии потенциала приводят к хаосу в гамильтоновых системах. Численные расчеты выполнены для потенциалов, которые соответствуют самым низким омбилическим катастрофам.
|
| first_indexed | 2025-12-07T15:29:24Z |
| format | Article |
| fulltext |
GEOMETRICAL APPROACH FOR DESCRIPTION
OF THE MIXED STATE IN MULTI-WELL POTENTIALS
V.P. Berezovoj1, Yu.L. Bolotin1, and G.I. Ivashkevych2
1National Science Center “Kharkov Institute of Physics and Technology”,
1, Akademicheskaya str., 61108, Kharkov, Ukraine;
e-mail: bolotin@kipt.kharkov.ua;
2Kharkov National University, Department of Physics and Technology,
4, Svobody sqr., 61077, Kharkov, Ukraine
We use the so-called geometrical approach [1] in description of transition from regular motion to chaotic one in
Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems
is coexistence of different types of dynamics (regular or chaotic) in different wells at the same energy [2]. Applica-
tion of traditional criteria for transition to chaos (resonance overlap criterion, negative curvature criterion and sto-
chastic layer destruction criterion) is inefficient in case of potentials with complex topology. Geometrical approach
allows considering only configuration space but not phase space when investigating the stability. In this approach all
information about chaos and regularity is contained in potential function. The aim of this work is to determine what
details of geometry of potential lead to chaos in Hamiltonian systems using geometrical approach. Numerical calcu-
lations are executed for potentials that are relevant with lowest umbilical catastrophes.
PACS: 01.30.Cc, 45.50.-j, 05.45.-a.
1. MIXED STATE. PHENOMENOLOGICAL
DESCRIPTION
Hamiltonian system with multi-well potential energy
surface (PES) represents a realistic model, describing
the dynamics of transition between different equilibrium
states, including such important cases as chemical reac-
tions, nuclear fission and phase transitions.
It became known in 80-th that existence of mixed
state is an important feature of such systems [2]. Mixed
state means that there are different dynamical regimes in
different local minima at the same energy, either regu-
lar, or chaotic. For example let’s demonstrate the exis-
tence of mixed state for nuclear quadrupole oscillations
Hamiltonian.
It can be shown that using only transformation prop-
erties of the interaction the deformation potential of
surface quadrupole oscillations of nuclei takes on the
form [4]:
2 2 2 2
0 2 0 2 0 2 0
,
( , ) ( 2 ) (6 )m n n
mn
m n
U a a C a a a a a= + −∑ , (1)
where a0 and a2 are internal coordinates of the nuclar
surface during the quadrupole oscillations:
0 0 2,0
2 2,2 2, 2
( , ) {1 ( , )
[ ( , ) ( , )]}.
R R a Y
a Y Y
θ ϕ θ ϕ
θ ϕ θ ϕ−
= +
+ +
(2)
Constants can be considered as phenomenol-
ogical parameters. Restricting with the terms of the
fourth degree in the deformation and assuming the
equality of mass parameters for two independent direc-
tions, we get C
mnC
3v-symmetric Hamiltonian:
2 2( ) / 2 ( , ; ,x y QOH p p m U x y a b c= + + , ) , (3)
where
2 2 2 3 2 2 2
2 0 10 01 20
( , ; , , )
1( ) ( ) (
2 3
2 , , 2 , 3 , .
QOU x y a b c
a x y b x y y c x y
x a y a a C b C c C
= + + − + +
= = = = =
) , (4)
Hamiltonian (3) and corresponding equations of mo-
tion depend only on parameter W=b2/ac, the unique
dimensionless quantity we can build from parameters
a,b,c. The same parameter determines the geometry of
PES. Interval 0<W≤16 includes potentials with single
extremum – minimum in the origin that corresponds to
spherical symmetric shape of the nucleus. In the interval
W>16 PES U contains seven extrema: four minima
(central, placed in the origin and three peripheral, which
correspond to deformed states of nuclei) and three sad-
dles, which separate peripheral minima from central
one. The distinctive feature of transition from regularity
to chaos in such a potential lies in the fact that energy of
transition is not the same in different local minima.
Thus, E
QO
cr ~ Es/2 (Es – energy in the saddles) for the cen-
tral minimum and Ecr~Es for peripheral. Due to this in
the interval Es/2<E<Es classical dynamics is mainly
chaotic in the central minimum and remains regular in
peripheral minima (Fig. 1). Term “mixed state” is used
for designation of such specific dynamics.
Mixed state is natural for multi-well potentials. This
statement is illustrated by Fig.1, which represents level
lines and Poincaré sections in different energies for
multi-well potentials from family of umbilical catastro-
phes D5 and D7:
7
5
2 2 2 4
2 2 2 4
3 12 ,
8 2
12 .
4
D
D
U y x xy x
U y x xy x
= + + − +
= − + +
61
6
x
(5)
One can see that there exists chaos in wells with
three saddles, while in other wells motion is regular.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 249-254. 249
Fig. 1. Level lines and Poincaré sections for (left), (center) and U (right). Sections are presented at
energies
5D 7D QO
4sE , 2sE and sE
Let’s note the distinction of sections structure in dif-
ferent wells. At the lowest energy there exists a hyper-
bolic point in the section for wells with chaotic motion.
At the same time there is no such a point in the regular
wells and structure of sections is similar at the different
energies.
2. IMPORTANCE OF THE MIXED STATE
FOR QUANTUM CHAOS
The mixed state represents optimal object for inves-
tigation of quantum manifestations of classical stochas-
ticity (QMCS) in wave function structure. Indeed, usual
procedure of search for QMCS in wave functions im-
plies distinction in its structure below and above classi-
250
cal critical energy (or other parameters of regularity-
chaos transition). However, such procedure meets diffi-
culties connected with necessity to separate QMCS
from modifications of wave functions structure due to
trivial changes of its quantum numbers. Wave functions
of the mixed state allow finding QMCS in comparison
not different eigenfunctions, but different parts of the
same eigenfunction, situated in different regions of con-
figuration space (different local minima of the poten-
tial).
Fig. 2. Wave function structure in potential 5D
For example, comparing the structure of the eigen-
functions in central and peripheral minima of the QO
potential or in left and right minima of the D5, it is evi-
dent that nodal structure of the regular and chaotic parts
is clearly different, but correlating with the character of
the classical motion (see Fig. 2).
3. STOCHASTIC CRITERIA
FOR THE MIXED STATE
As is well known [5], stochasticity is understood as
a rise of statistical properties in purely deterministic
system due to local instability. According to this idea
values of parameters of dynamical system, under which
local instability arises, are identified as regularity-chaos
transition values. However, stochasticity criteria of such
a type are not sufficient (their necessity offers a separate
and complicated question), since loss of stability could
lead to transformation of one kind of regular motion to
another one. Regardless this serious limitation, stochas-
tic criteria in combination with numerical experiments
facilitate the analysis of motion and essentially extend
efficiency of numerical calculations.
The first among widely used stochasticity criteria is
nonlinear resonances overlap criterion presented by
Chirikov [6]. According to this criterion rise of local
instability is generated by contact of separatrices of
neighboring nonlinear resonances. In this approach the
scenario of stochasticity is the following. The averaged
motion of the system in the neighborhood of the isolated
nonlinear resonance on the plane of the action-angle
variables is similar to the particle behavior in the poten-
tial well. Several resonances correspond to several po-
tential wells. The overlap of the resonances is responsi-
ble for the possibility of the random walk of particle
between these wells. This method could be modified for
the systems with unique resonance [7]. In this case the
origin of the large-scale stochasticity is connected with
the destruction of the stochastic layer near the separatrix
of the isolated resonance.
Application of these criteria in presence of strong
nonlinearity (which is inevitable when considering
multi-well potentials) encounters an obstacle: action-
angle variables effectively work only in neighborhood
of local minimum. Because of this, the interest to meth-
ods, based on direct estimation of trajectories diver-
gence speed, arises. The criterion of such a type is so-
called negative curvature criterion (NCC) [8]. This cri-
terion connects stochastisation of motion with getting to
part of configuration space, where Gaussian curvature
of PES is negative when energy increases (while in
neighborhood of minima curvature is always positive).
Then energy of transition to chaos is close to minimal
energy on the zero-curvature line. However, when pass-
ing on to the multi-well potentials, NCC fails to work
correct. In particular, for above mentioned potentials
(D5 and D7), structure of Gaussian curvature is similar
in different wells. For example, for D5 potential accord-
ing to NCC we get the same value of critical energy for
both minima: -5/9, but chaotic motion is observed only
in the left well (see Fig. 1). A natural question immedi-
ately arises: is it possible to formulate, using only geo-
metrical properties of PES but not solving numerically
equations of motion, the algorithm for finding the criti-
cal energy for single local minima in multi-well poten-
tial? We’ll try to answer this question below in the
framework of geometrical approach.
4. GEOMETRICAL APPROACH
TO HAMILTONIAN MECHANICS
We will use so-called geometrical approach in con-
sideration of mixed state [1]. Let’s recall the basics of
this method.
It is known that Hamiltonian dynamics could be
formulated in the terms of Riemannian geometry. In this
approach trajectories of the system are considered as
geodesics of some manifold. Grounds for such consid-
eration lie in variational base of Hamiltonian mechanics.
Geodesics are determined by condition:
0
L
dsδ =∫ . (6)
At the same time trajectories of dynamical system are
determined according to the Maupertuis principle:
2Tdt
γ
δ =∫ 0 (7)
(γ are all isoenergetic paths connecting end points) or to
the Hamilton’s principle:
2
1
0
t
t
Ldtδ =∫ . (8)
Once chosen a suitable metric action could be re-
wrote as a length of the curve on the manifold. Then
251
trajectories will be geodesics on this manifold. This
approach has an evident advantage: potential energy
function includes all information about the system, so
one needs to consider only configuration space but not
phase space.
Equations of motion in this case take on the form:
2
2 0
i j k
i
jk
d q dq dq
ds dsds
+ Γ = . (9) K q
• •
Christoffel symbols in this approach play role of coun-
terparts of forces in ordinary mechanics.
The most natural metric is the Jacobi one. It has the
form:
2[ ( )]ij ijg E V q= − δ . (10)
By means of this metric Maupertuis principle could be
rewritten in the form equivalent to condition for ge-
odesics.
Let’s consider local instability in the framework of
above mentioned geometrical approach. Let q and q` be
two trajectories, close at t=0:
' ( ) ( ) (i i iq s q s J s= + ) . (11)
Separation vector then satisfy the Jacobi-Levi-Civita
equation:
2
2 0
i j l
i k
jkl
d J dq dqR J
ds dsds
+ = . (12)
It can be shown that dynamics of the deviation is de-
termined only by Riemannian curvature of the manifold.
For two-degrees-of-freedom systems Riemannian curva-
ture has the form:
2
2
1 [2( ) 2 ]
4( )
R E V V
E V
= − ∆ +
−
V∇
.
. (13)
Laplacian of V is positive for considered potentials
so Riemannian curvature is positive too. Due to this we
couldn’t connect divergence of trajectories with nega-
tive Riemannian curvature.
One way to solve this problem consists in introduc-
tion of higher-dimensional (than N) metrics. Let’s
examine this question closer
It can be shown that equation for separation vector J
could be reformulated in the form, which doesn’t de-
pend on dimensionality of manifold:
22 22(2)
2
1 ( , ) 0
2
d J
K J v J J
dsds
∇
+ − = , (14)
where K(2) is a sectional curvature in two-dimensional
direction:
(2) ( , )
i k l m
iklm
J dq J dqK J v R
ds dsJ J
= (15)
and , 0J v = .
Note that the point where K(2)<0 is unstable. Since
there are more than one sectional curvature for the case
N>2, we could connect instability with negative sign of
some of them.
One of the enlarged metrics is the Eisenhart metric.
Eisenhart metric is N+2-dimentional and contains two
additional coordinates. One of these coordinates coin-
cides with time and second is connected with action.
Using Eisenhart metric, quantity K(2) could be rewritten
in the form:
2 22 2
(2)
2 12 2
1 2
2
1 2
1 2
1( , ) (
2( )
2 ).
V Vq q q
E V q q
V q q
q q
• •
∂ ∂
= +
− ∂ ∂
∂
−
∂ ∂
(16)
•
Now, investigation the K(2)-structure on the consid-
ered manifold could be used for studying the chaotic
regimes and, in particular, the mixed state.
Let’s briefly summarize the basics of geometrical
approach to Hamiltonian mechanics:
dynamics ~ geometry
t (time) ~ s (arc-length)
V (potential energy) ~ g (metric)
∂V (forces) ~ Γ (Christoffel symbols)
∂2V, (∂V)2 (curvature of potential) ~
R (curvature of potential)
5. INVESTIGATION OF THE MIXED STATE
IN THE FRAMEWORK OF GEOMETRICAL
APPROACH
As mentioned above, negative sign of K(2) is a condi-
tion for rise of local instability. It is necessary to clarify
whether this condition is sufficient for development of
chaoticity or not, clearly speaking, one needs to answer
the question: does the presence of negative curvature
parts on CM always lead to chaos? Potentials with
mixed state represent a very convenient model for
investigation of this question, since there exist both re-
gimes of motion.
So, we need to study, how differs the structure of
K(2) in different wells. For that we calculate the fraction
of phase space with negative curvature as a function of
energy, i.e. a volume of phase space where K(2) <0 re-
ferred to the total volume:
(2)( ) ( ( , )
( )
( ( , ) )
dqdp K H q p E
E
dqdp H q p E
Θ − δ −
µ =
δ −
∫
∫
)
. (17)
An advantage of this approach consists in necessity to
calculate only geometrical properties of system without
solving equations of motion.
We carried out calculations for two potentials: D5
and D7.
Calculations of µ (Fig. 3) show that there are re-
gions, where K(2) <0, in all wells, but nevertheless chaos
exists only in one well. Moreover, for the well with
chaotic motion function µ(E) gives correct value of
critical energy (in the sense, specified in part 3). At this
energy µ becomes positive.
252
Fig. 3. Function µ(E) for (a) and (b) poten-
tials. Data for chaotic wells are represented by circles,
for regular – by triangles
5D 7D
Fig.4. Function µg (E) for (a) and (b) 5D 7D
Situation with regular wells is more complicated.
Although the fraction of phase space, where K(2) <0, is
nonzero, chaos in the well doesn’t exist. This can be
seen on the Poincaré sections. For comparison the frac-
tion of CS with negative Gaussian curvature is shown in
Fig. 4. One can see that structure of negative Gaussian
curvature is similar to the K(2)-structure. To understand
this similarity let’s introduce polar coordinates in space
of momenta. K(2) then becomes:
2 2
(2) 2 2
2 2
1 2
2
1 2
( , ) (sin ) (cos )
2 cos sin ,
V VK q
q q
V
q q
ϕ ϕ
ϕ ϕ
∂ ∂
= +
∂ ∂
∂
−
∂ ∂
ϕ
(18)
where is the polar angle. Evidently Kϕ (2) could be
negative only if Gaussian curvature is negative.
6. CONCLUSIONS
Investigation of curvature of manifold, as one can
see from the cited above data, doesn’t give a plain
method for identification of chaos in any minimum,
especially if there exist both regular and chaotic regimes
of motion. It is impossible to determine a priori whether
chaos exists in the system without using the dynamical
description (in our case that are Poincaré sections).
Nevertheless, one can efficiently use geometrical meth-
ods for investigation of chaos in multi-well potentials.
In above considered potentials chaos exists only in
wells, which have two details: non-zero fraction of
negative curvature on the manifold and at least one hy-
perbolic point in the Poincaré section. According to this,
one can use the following method for identification of
chaos and calculation of critical energy. At the first step
the Poincaré section in low energy is drown for the well
and the presence of hyperbolic point is determined. If
so, the quantity µ must be calculated (or the fraction of
CS with negative Gaussian curvature). Value of energy,
in which µ becomes positive, could be associated with
critical energy. If there is no hyperbolic point in the
section than chaos doesn’t exist in the well.
Consequently, geometrical methods could be effi-
ciently used for determination of critical energy in com-
plicated potentials and identification of chaos in general.
However, one must carefully use these methods and
combine them with qualitative methods, such as Poinca-
ré sectioning method.
REFERENCES
1. M. Cerruti-Sola, M. Pettini //Phys.Rev. E. 1996,
v. 53, p. 179.
2. Yu.L. Bolotin, V.Yu. Gonchar, E.V. Inopin. Chaos
and catastrophes in quadrupole oscillations of nuclei
//Yad. Fiz.1987, v. 45, p. 351-356.
3. V.P. Berezovoj, Yu.L. Bolotin, V.A. Cherkaskiy.
Signatures of quantum chaos in wave function struc-
ture for multi-well 2D potentials //Phys. Lett. A
2004, v. 323, p. 218-223.
4. V. Mosel, W. Greiner. Exploration of collective sur-
face of the potential energy //Z.Phys. 1968, v. 217,
p. 256-281.
5. G.M. Zaslavsky. Chaos in Dynamical Systems.
N.Y.:Harwood, 1985.
253
6. B.V. Chirikov. Resonance processes in magnetic
traps //Atom.Energy. 1959, v. 6, p. 630-638 (in Rus-
sian).
7. P. Doviel, D. Escande, J. Codacconi. Stochasticity
threshold for Hamiltonians with zero and one pri-
mary resonances //Phys.Rev.Lett. 1983, v. 49,
p. 1879-1883.
8. M. Toda. Instability of trajectories of lattice with
cubic nonlinearity //Phys.Lett. A. 1974, v. 48,
p. 335-336.
ГЕОМЕТРИЧЕСКИЙ ПОДХОД К ОПИСАНИЮ СМЕШАННОГО СОСТОЯНИЯ
В МНОГОЯМНЫХ ПОТЕНЦИАЛАХ
В.П. Березовой, Ю.Л. Болотин, Г.И. Ивашкевич
Мы используем так называемый геометрический подход [1] в описании перехода от регулярного движе-
ния к хаотическому в гамильтоновых системах, в которых поверхность потенциальной энергии имеет не-
сколько локальных минимумов. Отличительная черта таких систем – сосуществование различных типов
динамики (регулярного или хаотического) в разных потенциальных ямах при той же самой энергии [2].
Применение традиционных критериев для перехода к хаосу (критерий перекрытия резонансов, критерий
отрицательнoй кривизны и критерий разрушения стохастического слоя) неэффективно в случае потенциалов
с комплексной топологией. Геометрический подход при исследовании устойчивости позволяет рассматри-
вать только пространство конфигураций, но не фазовое пространство. В этом подходе вся информация от-
носительно хаоса и регулярности содержится в потенциальной функции. Цель настоящей работы состоит в
том, чтобы, используя геометрический подход, определить какие детали геометрии потенциала приводят к
хаосу в гамильтоновых системах. Численные расчеты выполнены для потенциалов, которые соответствуют
самым низким омбилическим катастрофам.
ГЕОМЕТРИЧНИЙ ПІДХІД ДО ОПИСУ ЗМІШАНОГО СТАНУ У БАГАТОЯМНИХ ПОТЕНЦІАЛАХ
В.П. Березовий, Ю.Л. Болотін, Г.І. Івашкевич
Ми використовуємо так званий геометричний підхід [1] в описі переходу від регулярного руху до хаоти-
чного в гамільтонових системах, у яких поверхня потенційної енергії має кілька локальних мінімумів. Від-
мітна риса таких систем – співіснування різних типів динаміки (регулярного або хаотичного) у різних поте-
нційних ямах при тій же самій енергії [2]. Застосування традиційних критеріїв для переходу до хаосу (кри-
терій перекриття резонансів, критерій негативної кривизни й критерій руйнування стохастичного шару) не-
ефективно у випадку потенціалів із комплексною топологією. Геометричний підхід при дослідженні стабі-
льності дозволяє розглядати тільки простір конфігурацій, але не фазовий простір. У цьому підході вся інфо-
рмація щодо хаосу й регулярності міститься в потенційній функції. Ціль даної роботи полягає в тому, щоб,
використовуючи геометричний підхід, визначити які деталі геометрії потенціалу приводять до хаосу в гамі-
льтонових системах. Чисельні розрахунки виконані для потенціалів, які відповідають найнижчим омбіліч-
ним катастрофам.
254
|
| id | nasplib_isofts_kiev_ua-123456789-110968 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:29:24Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Berezovoj, V.P. Bolotin, Yu.L. Ivashkevych, G.I. 2017-01-07T15:20:14Z 2017-01-07T15:20:14Z 2007 Geometrical approach for description of the mixed state in multi-well potentials / V.P. Berezovoj, Yu.L. Bolotin, G.I. Ivashkevych // Вопросы атомной науки и техники. — 2007. — № 3. — С. 249-254. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 01.30.Cc, 45.50.-j, 05.45.-a. https://nasplib.isofts.kiev.ua/handle/123456789/110968 We use the so-called geometrical approach [1] in description of transition from regular motion to chaotic one in Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems is coexistence of different types of dynamics (regular or chaotic) in different wells at the same energy [2]. Application of traditional criteria for transition to chaos (resonance overlap criterion, negative curvature criterion and stochastic layer destruction criterion) is inefficient in case of potentials with complex topology. Geometrical approach allows considering only configuration space but not phase space when investigating the stability. In this approach all information about chaos and regularity is contained in potential function. The aim of this work is to determine what details of geometry of potential lead to chaos in Hamiltonian systems using geometrical approach. Numerical calculations are executed for potentials that are relevant with lowest umbilical catastrophes. Ми використовуємо так званий геометричний підхід [1] в описі переходу від регулярного руху до хаотичного в гамільтонових системах, у яких поверхня потенційної енергії має кілька локальних мінімумів. Відмітна риса таких систем – співіснування різних типів динаміки (регулярного або хаотичного) у різних потенційних ямах при тій же самій енергії [2]. Застосування традиційних критеріїв для переходу до хаосу (критерій перекриття резонансів, критерій негативної кривизни й критерій руйнування стохастичного шару) неефективно у випадку потенціалів із комплексною топологією. Геометричний підхід при дослідженні стабільності дозволяє розглядати тільки простір конфігурацій, але не фазовий простір. У цьому підході вся інформація щодо хаосу й регулярності міститься в потенційній функції. Ціль даної роботи полягає в тому, щоб, використовуючи геометричний підхід, визначити які деталі геометрії потенціалу приводять до хаосу в гамільтонових системах. Чисельні розрахунки виконані для потенціалів, які відповідають найнижчим омбілічним катастрофам. Мы используем так называемый геометрический подход [1] в описании перехода от регулярного движения к хаотическому в гамильтоновых системах, в которых поверхность потенциальной энергии имеет несколько локальных минимумов. Отличительная черта таких систем – сосуществование различных типов динамики (регулярного или хаотического) в разных потенциальных ямах при той же самой энергии [2]. Применение традиционных критериев для перехода к хаосу (критерий перекрытия резонансов, критерий отрицательнoй кривизны и критерий разрушения стохастического слоя) неэффективно в случае потенциалов с комплексной топологией. Геометрический подход при исследовании устойчивости позволяет рассматривать только пространство конфигураций, но не фазовое пространство. В этом подходе вся информация относительно хаоса и регулярности содержится в потенциальной функции. Цель настоящей работы состоит в том, чтобы, используя геометрический подход, определить какие детали геометрии потенциала приводят к хаосу в гамильтоновых системах. Численные расчеты выполнены для потенциалов, которые соответствуют самым низким омбилическим катастрофам. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Nonlinear dynamics Geometrical approach for description of the mixed state in multi-well potentials Геометричний підхід до опису змішаного стану у багатоямних потенціалах Геометрический подход к описанию смешанного состояния в многоямных потенциалах Article published earlier |
| spellingShingle | Geometrical approach for description of the mixed state in multi-well potentials Berezovoj, V.P. Bolotin, Yu.L. Ivashkevych, G.I. Nonlinear dynamics |
| title | Geometrical approach for description of the mixed state in multi-well potentials |
| title_alt | Геометричний підхід до опису змішаного стану у багатоямних потенціалах Геометрический подход к описанию смешанного состояния в многоямных потенциалах |
| title_full | Geometrical approach for description of the mixed state in multi-well potentials |
| title_fullStr | Geometrical approach for description of the mixed state in multi-well potentials |
| title_full_unstemmed | Geometrical approach for description of the mixed state in multi-well potentials |
| title_short | Geometrical approach for description of the mixed state in multi-well potentials |
| title_sort | geometrical approach for description of the mixed state in multi-well potentials |
| topic | Nonlinear dynamics |
| topic_facet | Nonlinear dynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110968 |
| work_keys_str_mv | AT berezovojvp geometricalapproachfordescriptionofthemixedstateinmultiwellpotentials AT bolotinyul geometricalapproachfordescriptionofthemixedstateinmultiwellpotentials AT ivashkevychgi geometricalapproachfordescriptionofthemixedstateinmultiwellpotentials AT berezovojvp geometričniipídhíddoopisuzmíšanogostanuubagatoâmnihpotencíalah AT bolotinyul geometričniipídhíddoopisuzmíšanogostanuubagatoâmnihpotencíalah AT ivashkevychgi geometričniipídhíddoopisuzmíšanogostanuubagatoâmnihpotencíalah AT berezovojvp geometričeskiipodhodkopisaniûsmešannogosostoâniâvmnogoâmnyhpotencialah AT bolotinyul geometričeskiipodhodkopisaniûsmešannogosostoâniâvmnogoâmnyhpotencialah AT ivashkevychgi geometričeskiipodhodkopisaniûsmešannogosostoâniâvmnogoâmnyhpotencialah |