Instabilities, induced by a noise
The dynamics of systems, parameters of which are subject to casual forces are considered. The casual action can be as external, and to be generated by own random dynamics of a system. The most important features of such systems dynamics are chosen. First of all, their behavior is characterized by an...
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| Date: | 2002 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Cite this: | Instabilities, induced by a noise / V.A. Buts // Вопросы атомной науки и техники. — 2002. — № 5. — С. 55-56. — Бібліогр.: 4 назв. — англ. |
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| author | Buts, V.A. |
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| citation_txt | Instabilities, induced by a noise / V.A. Buts // Вопросы атомной науки и техники. — 2002. — № 5. — С. 55-56. — Бібліогр.: 4 назв. — англ. |
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| description | The dynamics of systems, parameters of which are subject to casual forces are considered. The casual action can be as external, and to be generated by own random dynamics of a system. The most important features of such systems dynamics are chosen. First of all, their behavior is characterized by an alternation. In these systems (even linear) the properties, which are characteristic for a stochastic resonance, can be to exist. Instabilities, which are induced by a noise practically are not stabilized by nonlinearities.
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| first_indexed | 2025-11-29T14:13:35Z |
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INSTABILITIES, INDUCED BY A NOISE
V.A. Buts
NSC « Kharkov Institute of Physics and Technology », 61108, Kharkov, Ukraine
E-mail: abuts@kipt.kharkov.ua.
The dynamics of systems, parameters of which are subject to casual forces are considered. The casual action can be
as external, and to be generated by own random dynamics of a system. The most important features of such systems
dynamics are chosen. First of all, their behavior is characterized by an alternation. In these systems (even linear) the
properties, which are characteristic for a stochastic resonance, can be to exist. Instabilities, which are induced by a noise
practically are not stabilized by nonlinearities.
PACS: 52.35.-g
The real physical systems are under action of
fluctuations. It is useful to select a special class of
oscillatory systems, on which the effect of fluctuations
revealed in development of parametrical instability. It
happens when the fluctuations are multiplicate, i.e.
randomly change parameters of an investigated oscillatory
system. Frequently in such systems all moments will
became unstable. And the increment of each following
moment is more previous. In this case process of
instability has an alternated character. It is necessary to
mark, that, apparently, this case is unique case, in which
the higher moments acquire the concrete physical
contents. Really, in overwhelming majority of cases for
the description of a physical system it is enough to know a
behavior of first two moments. Below we shall formulate
the most important features of systems, instability of
which are induced by a noise.
1. Dynamics of a linear system, which has an alternated
character, is similar to nonlinear dynamics. As an
example on Figure 1 the dependence of a linear
oscillator amplitude is represented, which frequency is
subject to random disturbances. The equation of such
oscillator has a form
( ) ( )x t x+ ⋅ + ⋅ =ω ξ2 1 0 (1)
Where ξ ( )t - white noise.
103
103
xn 1,
700400 xn 0,
400 450 500 550 600 650 700
1000
500
0
500
1000
Fig.1 An example of an alternation in oscillations of a
unstable linear oscillator.
All moments of such oscillator (beginning with second)
are unstable. The increments of each following are more
previous. To the equation (1) the analysis of large number
of plasma systems is reduced. From Figure it is visible,
that dynamics of such linear oscillator has an alternated
character. Such feature of linear dynamics is necessary to
take into account because it is similar to dynamics
generated by nonlinear processes. Let’s give the brief
proof of the formulated statements. For determinancy we
shall be count, that the function ξ has the following
properties: ξ ξ ξ δ( ) , ( ) ( ) ( )t t t D t t= ⋅ = −0 21 1 . Here
angular brackets mean statistical averaging on a casual
ensemble ξ ( )t .
Using variational method (see, for example, [1]) from
the equation (1) it is possible to receive the following
system ( )n + 1 ordinary differential equations for
determination of moments of the order n for
displacements ( x t( ) ) and velocities ( y x= ):
( )d
dt
y x p y x p p
D y x n p y x
p n p p n p
p n p p n p
⋅ = − ⋅ ⋅ + − ⋅
⋅ ⋅ + − ⋅
− − − +
− − + + − −
ω
ω
2 1 1
4 2 2 1 1
1
2( ) ( )
The analysis of a system (2) shows, that beginning from
the second moment all moments are unstable. And the
increment of each following moment is more previous.
The behaviour of such system, as is known [2], is
characterized by an alternation. This statement can be
easy to proof for a unstable oscillator ( )ω 2 0< . For the
proof it is convenient a set of equations (2) to transform in
a vector form. For this purpose we shall enter function
Y y xp
p n p= ⋅ − . Using this function, is possible to
present set of equations (2) as:
dY
dt
A Yp
p i i= ⋅, p i n, , , , .....= 0 1 2 (3)
The matrix A p i, under condition of ω 2 0< is non-
negative, undecomposable matrix. From the Perron-
Frobenius theorem [3] follows, that the greatest positive
eigenvalues of such matrix will increase when anyone
from elements A p i, of this matrix will be magnified.
Magnification of intensity of fluctuations D , and also the
magnification order n of the moments results in
magnification of these elements.
2. Important is that fact that the casual modification of
parameters can happen as result of dynamic chaos
development. The important example of such systems is
the movement of a charged particle in an external
magnetic field and in a field of an external
electromagnetic wave. As is known [4], when the
amplitude of the wave is sufficiently large the movement
of particles becomes randomly. The magnitude of a
Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 55-56 55
particle energy randomly varies. The energy of a particle
is one from main parameters for description of particle
dynamics in an external constant magnetic field. The
casual modification of this parameter can result in
development of instability with features, which we have
described above. As an example we shall consider the
most simple configuration, when the external flat
electromagnetic wave is propagate in a direction that is a
perpendicular to the direction of an external homogeneous
magnetic field. And, the polarization of this wave is those,
that a magnetic component of this wave field coplanar to
an external constant magnetic field. Let, besides in an
initial time the particles did not have component of a
velocity directed along a constant magnetic field. In this
case equations describing dynamics of particles in such
fields are most simple and look like:
( )[ ]
/ ,
cos /
x p p
p x p
x z
x H y
= =
= ⋅ ⋅ − + ⋅
γ
ω α τ γ
0
1 (4)
( )[ ]
( )
cos /
cos
p x p
H x
y H x= − ⋅ ⋅ − + ⋅ +
+ ⋅ −
ω α τ γ
τ
1
where α ≡ H Hw / 0 - is the ratio strength of wave
magnetic field to the strength 0f external uniform
magnetic field. In (4) we have used such dimensionless
variables: x kx t p p mc→ → →, , / ,τ ω
H eH mcw= / ω -is dimensionless parameter of wave
force, ω ωH eH mc= ⋅0 / .
The set of equations (4) was analyzed numerically for
following parameters: ω H = 05, H=0.9, α = 1.8.
Characteristic dynamics of a particle impulse changing is
represented in fig.2. It is visible, that it has an alternation
character. This dynamics, besides is characterized by local
instability.
3. The instabilities, induced by a noise, have features of a
stochastic resonance. It is revealed, for example, so. Let
parameters of a system vary simultaneously under the
regular (periodically) law and on noise. Let, besides, the
periodic modification of parameters is such, that the
parametrical instability can develop. If the amplitude of a
periodic modification of parameters is insufficiently great
for reaching a threshold of instability, the introduction of
casual modulation of these parameters can result in
instability arising. Thus, the energy of external casual
force can promote development of instability in a
considered system. As an example we shall consider
dynamics of an oscillator, which is described by the
following equation
( )[ ] ( ) cosx x t A t x+ ⋅ + + + ⋅ ⋅ =ν ω ξ ω2 1 2 0 (5)
In this equation A - amplitude of regular parametrical
force; the function ξ ( )t characterizes noise action.
Using this equation it is easy to find such values of
parameters, which under operation of one regular
parametrical force does not result in instability; does not
result in development of instability and effect only of
noise force. However joint operation of these two forces
results in development of instability. And, dynamics of
this instability is characterized by an alternation. The
higher level of casual force, the more clear alternation.
As an example on fig.3 the solution of the equation (5) for
ν = = =0 01 0 003 011. , . , .A A N is represented.
38.151927
38.152933
Z 3<>
n
5 103.0 Tn
0 2000 4000 6000
50
0
50
Fig.2. Dependence of transversal impulse on time for
ν = 0.01, A = 0.003, AN = 0.11
.
0.06749
0.068009
xn 1,
N 1( ) dT.
1
0 xn 0,
0 500
0.1
0.05
0
0.05
0.1
Fig.3 Modification of an oscillator amplitude for
simultaneous operation of regular and noise forces
ν = 0.01, A = 0.003, AN = 0.11
4. The special role the multiplicate fluctuations play when
they act on unstable systems. In this case they can
radically change dynamics of a system. Practically always
in these cases the alternation will be realized. Under it
only on the certain time frame (or distance) regular
dynamics is saved. Outside of this interval dynamics - is
chaotic. The large intensity of a noise, the shorter this
interval. It is important, that instabilities, which are
induced by a noise are not stabilized by nonlinearities.
Really, in this case nonlinear frequency drift is
compensated by a broad spectrum of a noise signal.
REFERENCES
1. Klyathkin V.I. The statistic description of dynamics
system with fluctuating parameters. M. «Science»,
1975, 240 p.
2. Molchanov S.A., Ruzmaikin A.A., Sokolov D.D.//
Progress of Physical Siences, 1985, v.145, N4.
3. Gantmaher F.P. Theory of matrics - M. «Science»,
1988.
4. Balakirev V.A., Buts V.A., Tolstoluzhskii
A.P., Turkin Yu.A. Zh. Eksp. Teor. Fiz.,
1983, Vol. 84, pp. 1279-1289.
56
INSTABILITIES, INDUCED BY A NOISE
References
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| id | nasplib_isofts_kiev_ua-123456789-77878 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-29T14:13:35Z |
| publishDate | 2002 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.A. 2015-03-08T20:21:43Z 2015-03-08T20:21:43Z 2002 Instabilities, induced by a noise / V.A. Buts // Вопросы атомной науки и техники. — 2002. — № 5. — С. 55-56. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.35.-g https://nasplib.isofts.kiev.ua/handle/123456789/77878 The dynamics of systems, parameters of which are subject to casual forces are considered. The casual action can be as external, and to be generated by own random dynamics of a system. The most important features of such systems dynamics are chosen. First of all, their behavior is characterized by an alternation. In these systems (even linear) the properties, which are characteristic for a stochastic resonance, can be to exist. Instabilities, which are induced by a noise practically are not stabilized by nonlinearities. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Instabilities, induced by a noise Article published earlier |
| spellingShingle | Instabilities, induced by a noise Buts, V.A. Basic plasma physics |
| title | Instabilities, induced by a noise |
| title_full | Instabilities, induced by a noise |
| title_fullStr | Instabilities, induced by a noise |
| title_full_unstemmed | Instabilities, induced by a noise |
| title_short | Instabilities, induced by a noise |
| title_sort | instabilities, induced by a noise |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/77878 |
| work_keys_str_mv | AT butsva instabilitiesinducedbyanoise |