Parametric stochastic instability of connected oscillator
Oscillators are used for investigation different physical processes, for example, oscillation in plasma, modeling of continuous medium ets. The large interest is fluctuation influence on oscillation processes. It is well known that multiplicative fluctuation causes the instability in thus systems. W...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2003
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| Цитувати: | Parametric stochastic instability of connected oscillator / V.A. Buts, I.K. Kovalchuk // Вопросы атомной науки и техники. — 2003. — № 4. — С. 104-108. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-1109832017-01-08T03:03:18Z Parametric stochastic instability of connected oscillator Buts, V.A. Kovalchuk, I.K. Нелинейные процессы Oscillators are used for investigation different physical processes, for example, oscillation in plasma, modeling of continuous medium ets. The large interest is fluctuation influence on oscillation processes. It is well known that multiplicative fluctuation causes the instability in thus systems. We investigated the systems of two and three connected oscillators. We have shown that it is possible considerably to increase the increment of this instability. The conditions when this is possible are obtained. The numerical simulation of processes in the system of two and three connected oscillators was carried out. Oscillation increment was increasing when correspond conditions were performed. 2003 Article Parametric stochastic instability of connected oscillator / V.A. Buts, I.K. Kovalchuk // Вопросы атомной науки и техники. — 2003. — № 4. — С. 104-108. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/110983 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы Нелинейные процессы |
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Нелинейные процессы Нелинейные процессы Buts, V.A. Kovalchuk, I.K. Parametric stochastic instability of connected oscillator Вопросы атомной науки и техники |
| description |
Oscillators are used for investigation different physical processes, for example, oscillation in plasma, modeling of continuous medium ets. The large interest is fluctuation influence on oscillation processes. It is well known that multiplicative fluctuation causes the instability in thus systems. We investigated the systems of two and three connected oscillators. We have shown that it is possible considerably to increase the increment of this instability. The conditions when this is possible are obtained. The numerical simulation of processes in the system of two and three connected oscillators was carried out. Oscillation increment was increasing when correspond conditions were performed. |
| format |
Article |
| author |
Buts, V.A. Kovalchuk, I.K. |
| author_facet |
Buts, V.A. Kovalchuk, I.K. |
| author_sort |
Buts, V.A. |
| title |
Parametric stochastic instability of connected oscillator |
| title_short |
Parametric stochastic instability of connected oscillator |
| title_full |
Parametric stochastic instability of connected oscillator |
| title_fullStr |
Parametric stochastic instability of connected oscillator |
| title_full_unstemmed |
Parametric stochastic instability of connected oscillator |
| title_sort |
parametric stochastic instability of connected oscillator |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2003 |
| topic_facet |
Нелинейные процессы |
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http://dspace.nbuv.gov.ua/handle/123456789/110983 |
| citation_txt |
Parametric stochastic instability of connected oscillator / V.A. Buts, I.K. Kovalchuk // Вопросы атомной науки и техники. — 2003. — № 4. — С. 104-108. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT butsva parametricstochasticinstabilityofconnectedoscillator AT kovalchukik parametricstochasticinstabilityofconnectedoscillator |
| first_indexed |
2025-07-08T01:28:04Z |
| last_indexed |
2025-07-08T01:28:04Z |
| _version_ |
1837040229936005120 |
| fulltext |
PARAMETRIC STOCHASTIC INSTABILITY
OF CONNECTED OSCILLATOR
V.A. Buts, I.K. Kovalchuk
NSC “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
Oscillators are used for investigation different physical processes, for example, oscillation in plasma, modeling
of continuous medium ets. The large interest is fluctuation influence on oscillation processes. It is well known that
multiplicative fluctuation causes the instability in thus systems. We investigated the systems of two and three
connected oscillators. We have shown that it is possible considerably to increase the increment of this instability.
The conditions when this is possible are obtained. The numerical simulation of processes in the system of two and
three connected oscillators was carried out. Oscillation increment was increasing when correspond conditions were
performed.
PACS: 52.35.Mw
1. INTRODUCTION
Oscillator is model, which widely is used for
studying of different oscillation processes. For example,
many of oscillation in plasma physics may be
considered as connected oscillators. This may perform
by separating temporal and spacing parts in solution of
according physical problem. Space part is described by
differential equation in partial derivatives. This allows
to define space structure of electromagnetic fields. On
other hand temporal processes dynamic is described by
means temporal ordinary differential equations, which
may be considered as describing the system of
connected oscillators. Such consideration is possible in
complicated electrodynamical structures filled by
plasma. For example, cavity filled by plasma. Such
system may be considered as two connected oscillators.
One of them is electromagnetic, i.e. this is natural
oscillation of cavity, and other is oscillation of plasma.
This system was considered. It was supposed that
plasma density was subjected to fluctuation action.
The dynamic as linear as nonlinear oscillators has
enough studied. The different physical systems are
subjected action of fluctuations. In oscillation ones they
may be as additive as multiplicative. One of the
important results is that multiplicative fluctuations may
cause instability. It increment usually is proportional to
fluctuation level. In the case of multiplicative
fluctuations the frequencies one or some oscillators are
random functions. This causes instability named as
parametric instability one. Ref. [1,2,3] are devoted
different physical processes that may be considered as
connected oscillators.
But it may be expected increment increasing of such
instability in some cases. We investigated some of these
ones. We showed that fluctuations instability increment
might be proportional to root of some power of
fluctuation level. It may be realized at some values of
parameters, which describe concrete physical system.
Values of these parameters are defined.
First of all we considered two connected oscillators
with multiplicative fluctuations. On other hand, we
studied three connected oscillators, where each of them
is connected with neighboring. Such system may be
employed for modeling of continuos matter. We also
made some conclusion when increment increase may be
realized. Besides, it will be shown that in the system of
three connected oscillators increment of stochastic
parametric instability may be proportional to root of
sixth power of fluctuation level. In the case of two
oscillators increment may be proportional to cubic root.
Essential increasing of increment was confirmed by
numerical simulation.
2. THE SYSTEM OF TWO CONNECTED
LINEAR OSCILLATORS
Let us consider two connected linear oscillators,
frequency one of them is subjected of random
perturbation action. The differential equations
describing this system are following:
( )
2
1 1 2 2
2
2 2 1
,
,
ξ ξ µ ξ α µ ξ
ξ ξ µ ξ
+ = − Ω − ∆
+ Ω + ∆ = −
&&
&& (1)
where 1 2,ξ ξ – generalized coordinates, describing these
oscillators, µ – connection coefficient ( 0 1µ≤ ≤ ), Ω –
dimensionless frequency of the second oscillator (the
dimensionless time is normalized on the first oscillator
period, 2∆ < < Ω – random addition to second oscillator
frequency, α – coefficient accounting additive
influence of second oscillator on first one ( 0 1α≤ ≤ ).
Random addition is Gaussian and delta-correlated,
)()()( ττδττ ′−=〉′∆∆〈 N (2)
where N – noise level.
The differential equation set for first moments which
are average values of 1 1 2 2, , ,ξ ξ ξ ξ& & , coincides with set
(1) when ∆ =0, therefore they are not interest for us. We
investigated second moments, which are average values
of all products above mentioned 1 1 2 2, , ,ξ ξ ξ ξ& & . The
amount of second moments is ten. The differential
equations set for them was obtained. The technique of
variational derivatives and Furutsu-Novikov formula
was used for obtaining of second moments equations
set. This technique is described in [2]. Averaging (1) for
obtaining equation set for first moments we have
addendum >∆< 2ξ . The set of equations for second
moments contains addendum >∆< kiξξ . Thus new
unknowns appear. For reducing this expressions
following Furutsu-Novikov formula is employed:
< > =∆
∆
( ) ( )
( )
( )
τ ξ τ
δ ξ τ
δ τ2
21
2
N ,
where
δ ξ τ
δ τ
2 ( )
( )∆
is variational derivative of ξ τ2 ( ) with
respect ∆ ( )τ at time τ . This formula is correct for
random Gaussian delta-correlated processes. To obtain
variational derivative equations (1) are integrated
formally.
For solution defining of these equations for second
moment in exponential form as exp( )λ τ , the
characteristic equation was obtained. It is algebraic
equation of tenth power. It may be presented such as
consisting of two parts:
10
(2)
1
( ) ( )
i
NPλ λ λ
=
− =∏ ,
where (2)λ – roots of left part, N – fluctuation level.
Left part is polynomial. Right part is polynomial too
proportional to fluctuation level. The right part of
characteristic equation in such form is conditioned by
fluctuations influence. The roots of left part polynomial
simply are connected with natural frequencies of initial
linear connected oscillators system without fluctuations,
which describing by set (1) when ∆ =0. These are all
different sums and differences of natural frequencies
multiplied on imaginary one. The characteristic
equation for (1) when N=0 is following:
( ) ( ) 011 22224 =−Ω+Ω++ µλλ ,
and its solutions are:
( )
( )
2(0) 2 2 2 2
1,2
2(0) 2 2 2 2
3,4
1 1 4 ,
2
1 1 4 .
2
i
i
λ µ
λ µ
= ± + Ω − − Ω + Ω
= ± + Ω + − Ω + Ω
They coincide with natural frequencies without
imaginary one i. Accordingly for (2)λ we obtain
following expressions:
( )
( )
( )
( )
Ω+Ω−+Ω++
+Ω+Ω−−Ω+±=
=
Ω+Ω−+Ω+±=
Ω+Ω−−Ω+±=
22222
22222)2(
8,7
)2(
6,5
22222)2(
4,3
22222)2(
2,1
411
411
2
0
4112
4112
µ
µλ
λ
µλ
µλ
i
i
i
( )
( )
Ω+Ω−+Ω+−
−
Ω+Ω−−Ω+±=
22222
22222)2(
10,9
411
411
2
µ
µλ i
When noise is low, the roots of characteristic equation
may be presented in following form: (2)λ λ δ= + . The
addend δ is conditioned by fluctuation influence. λ is
complex value. If real part of λ (i.e. real part δ ) is
positive, there is instability on the second moments in
the system consisting of two connected linear
oscillators. This instability is conditioned by
fluctuations, and it increment equals real part λ . In
general case this increment is proportional to noise
level. But it is possible that δ will be proportional to
root of some power of fluctuations level. It may be
when there are multiple roots among ones of left part
polynomial. In this case the addend δ is proportional to
root of power, which coincide with the left part equal
roots number. In general case the equal roots of left part
of characteristic equation may exist at some parameters
of initial set (1). To get parameter value when multiple
roots exist it is necessary to equate two or more roots
)2(λ . The obtained equalities are equations for
parameters defining when equal roots exist. For
example, two times multiple roots exist when following
condition for µ is right in the case 1=α :
2
22
2
100
)9)(19(
Ω
Ω−−Ω=µ . (3)
Numerical simulation results for this case will be
considered in section 5. But it is possible that one or
more roots of the right part polynomial )(λP are same
as )2(λ . In this case there is not expected increasing of
increment. The values of these parameters were defined.
When left part of characteristic equation has multiple
roots, increment of mentioned instability becomes more
large comparatively the ordinary case. We defined
increments of second moment instability, which were
proportional as cubic root as square root of fluctuation
level. The parameters values, when this is possible,
were defined.
The above considered results may be using for
concrete physical system. It is cylindrical cavity filled
by plasma with fluctuating density. Such system may be
considered as two connected linear oscillators. One of
them corresponds to electromagnetic oscillations of
cavity other corresponds to plasma oscillations. The last
is objected to plasma density fluctuations. There is
instability on second moments in such system. But there
is not expected increase of increment in cavity with
fluctuating plasma.
3. SOME CONCLUSION ABOUT
ARBITRARY OSCILLATORS NUMBER
Let us consider system of arbitrary number linear
connected oscillators described by following differential
equations set:
&& , ,...,
,
x x x i ni i i ij j
j j i
+ = =
= ≠
∑ω α2
1
1 2 (4)
where ix – generalized coordinate describing i-th
oscillator, iω – it's frequency, ija – connection
coefficients, n – oscillators number. We suppose that
there is frequency fluctuation in one of oscillators. The
above mentioned technique of variational derivatives
may be used for moments equations set obtaining. It
may be shown that first moments equation set coincides
with set (4) without fluctuations. The second moments
are interesting. The first moments number is 1 2M n= .
It may be shown that second moments number equals
2 (2 1)M n n= + . The differential equations set may be
obtained. It will be linear homogeneous set. Solutions
may be presented in exponential form as for two
oscillators. There is characteristic equation for second
moments. It is algebraic equations of 2M power. It may
be presented in form of two parts, as that performed for
two oscillators. The left part of this equation may have
more larger number of multiple roots than in two
oscillators case. The maximum multiplicity is defined
by as oscillators number as parameters number
describing this physical system. Maximum multiplicity
may equal k M M p= min( , )2 , where M p –
number of parameters of considered system.
In general case the system containing large number
of connected linear oscillators and described by large
number of parameters, and subjected action of
multiplicative fluctuations may have large multiple
roots number of characteristic equation left part for
second moments. Solutions of characteristic equation in
the case of low fluctuation level may be presented in
form (2)λ λ δ= + , as it was mentioned above. In the
case of left part roots with large multiplicity the addend
δ will be proportional to root of multiplicity power of
fluctuations level. The addend δ may be larger, if
multiplicity of corresponding root is large, in the case of
many oscillators comparatively case of two oscillators.
It was mentioned that the roots λ are complex values.
Therefore δ is complex too. If real part δ is positive
there is instability on second moments in system of
many oscillators with multiplicative fluctuations. If
multiplicity of mentioned root is large the instability
increment, which equals real part of δ , will be large
too. In the case of many oscillators it may be more
larger comparatively case of two connected linear
oscillators.
4. THREE CONNECTED OSCILLATORS
UNDER ACTION OF MULTIPLICATIVE
FLUCTUATIONS
In this section we shell be shown that in system of
three connected oscillators with multiplicative
fluctuations the essential increase of stochastic
parametric instability is possible. Such system is
describing by following:
2
1 1 1 12 2
2
2 2 2 21 1 23 3
2
3 3 3 32 2
,
,
,
x x x
x x x x
x x x
ω α
ω α α
ω α
+ =
+ = +
+ =
&&
&&
&&
(5)
where symbols are as in (4). The characteristic numbers
of this system are when frequencies equal:
1,2 3,4 5,,6, 1 , 1 ,i i A i Aλ λ λ= ± = ± + = ± −
where 12 21 23 32A a a a a= + .
We suppose that frequency of first oscillators 1ω is
under action of fluctuations, which are delta-correlative
with level 1N . The number of second moments is 21.
The differential equations set for them was obtained.
The technique of variational derivatives and Furutsu-
Novikov formula as in section 2 was used for obtaining
of second moments equations set too. As in section 2,
the characteristic equation was obtained. It was
presented in form:
21
(2)
1
1
( ) ( )
i
N Pλ λ λ
=
− =∏ . (6)
(2)λ are all possible sums of iλ . The roots were
analyzed. The goal of this analysis was to obtain the
existence multiple roots among (2)λ . There are roots
with different multiplicity. The parameters values when
this is possible were obtained. (2)λ =0 with maximal
multiplicity that equals 9 and (2) 2iλ = ± with
multiplicity 6. It possible when parameter A=0. But
such (2)λ may be roots of polynomial ( )P λ . To exhibit
this the right part of (6) was transformed. This right part
is determinant. The determinant order was reduced to
order of three. It was exhibited that polynomial ( )P λ
has multiple roots coinciding with (2)λ . Just it is 0. As
mentioned above in section 2 the addend δ is
proportional to root of multiplicity power of noise level
for multiple (2)λ . But this condition is violated when
polynomial ( )P λ has same roots. In this case the root
index in the expression for δ is less on number that
equals multiplicity of corresponding root of ( )P λ . In
our case such root is (2)λ =0. λ =0 is fourth
degenerated root of polynomial ( )P λ . Therefore
5
1Nδ ≈ for (2)λ =0 and 6
1Nδ ≈ for (2) 2iλ = ± .
There are complex δ with positive real part. There is
instability in this case. Fluctuations are faint. Therefore
we have fluctuation increment increasing when multiple
roots (2)λ exist. Thus as it see from this section there is
increasing possibility of stochastic parametric instability
increment for three connected oscillators comparatively
of two ones case. This confirms conclusions made in
section 3. Results of this section may be used when
fluctuation influence on continuos matter is
investigated. Continuos matter may be presented as
chain of connected oscillators.
5. NUMERICAL SIMULATION
In previous sections we analytically investigated
some examples when increment of parametric stochastic
instability may increase. These results relate to second
moments. In [2,3] the stochastic parametric instability
for one oscillator was considered in detail. The
characteristics of individual realizations were
investigated. It was shown that oscillation amplitude in
individual realization increases with some increment.
We numerically investigated some cases when
essential increasing of increment was expected. We
considered two and three connected oscillators for sets
(1) and (5). The parameter values corresponded to
conditions when instability increment increasing was
expected. The corresponding plots are demonstrating
our conclusions in section 3.
The random function was taken Gaussian with zero
mean value. Our consideration is correct when random
function is delta correlated. For numerical simulation
this means that correlation radius should be more less
then processes period. The correlation radius is of order
the mean period of random function. In our cases this
period was of order 0.16 whereas processes period was
0.2≈ .
For two oscillators was considered case for two
multiple roots. The corresponding condition given by
expression A=0. As it was mentioned in section 4 in the
case of three oscillators the degree of intrinsic was six.
The maximum increment corresponds to the optimal
condition (3) for two oscillators and 0=A for three
ones. Corresponding plots are shown on the fig. 1 for
two oscillators and fig. 3 for three oscillators. In these
plots there is a typical shape of generalized coordinates
dynamic for two and three oscillators.
Fig. 1. Dynamic of two connected oscillators in the case
of optimal conditions
As it’s seen amplitudes of each variables increase
with time in averaging. The connection coefficients for
three oscillators were following 05.0322112 === aaa
and 05.023 −=a .
For others parameter values increments were less. It
demonstrated in the fig. 2 for two oscillators and fig. 4
for three oscillators where 05.0322112 === aaa and
045.023 −=a .
Fig. 2. Dynamic of two oscillators (variable 2ξ ) when
parameter µ does not correspond to optimal
Fig.3. Dynamic of tree connected oscillators in the case
of optimal conditions
Fig. 4. Dynamic of three oscillators (variable 2ξ ) when
parameter A does not correspond to optimal
In equations (1) and (5) the nonlinear member was
added. In this case the oscillation amplitude is limited.
6. SUMMARY
It is well known that muliplikative fluctuations may
cause instability that named parametric stochastic one..
It was shown on the example of one oscillator, which
frequency is subjected to random action, in [2].
This instability exists on second moments for delta-
correlation fluctuation. In this repot possibility of
stochastic parametric instability increment increasing
was considered. It has been noted that such increasing
may take place in a system of large oscillators number,
which described by large parameters number. It was
considered on the examples of two and three connected
oscillators with multiplicative fluctuations. It is
necessary to note that in the case of three oscillators
instability increments may be more larger. The
necessary conditions of increment increasing were
investigated. The parameters when increment increasing
takes place were defined. The corresponding numerical
simulation was performed. It confirms the analytical
conclusion
REFERENCES
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cheskikh sistemakh s peremennimi parametrami. M.:
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Russian).
2. .I. Klyatskin. Statisticheskoye opisaniye dinamiches-
kikh system s fluktuiruyushchimi parametrami. M.:
"Nauka". Gl. red. Phys.-mat. lit. 1975. 240 s. (In
Russian).
3. V.I. Klyatskin. Stokhasticheskiye differentsial’niye
uravneniya i volni v sluchayno neodnorodnikh sredakh.
M.: "Nauka". 336 s. (In Russian).
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