Dynamic chaos generated by linear systems
It is shown that regimes with chaotic behavior are also inherent in linear systems. A dynamic chaos which can arise in essentially quantum systems (not quasi-classical) is of special interest. In particular, it is shown that the diffusion of quantum systems in the energy space can be considerably mo...
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irk-123456789-1082302016-11-01T03:02:41Z Dynamic chaos generated by linear systems Buts, V.A. Section F. Nonlinear Dynamics and Chaos It is shown that regimes with chaotic behavior are also inherent in linear systems. A dynamic chaos which can arise in essentially quantum systems (not quasi-classical) is of special interest. In particular, it is shown that the diffusion of quantum systems in the energy space can be considerably more effective than transitions at multiphoton processes. It is shown that taking into account the singular solutions allows to realize regimes with chaotic behavior even in systems with "one'' degree of freedom. Показано, что режимы с хаотическим поведением присущи также линейным динамическим системам. Особый интерес представляет динамический хаос, который может возникать в существенно квантовых системах (не квазиклассических). В частности, показано, что диффузия квантовых систем в пространстве энергии может быть значительно более эффективной, чем переходы при многофотонных процессах. Показано, что учет особых решений позволяет реализовать режимы с хаотическим поведением даже в системах с «одной» степенью свободы. Показано, що режими з хаотичним поводженням властиві також лінійним динамічним системам. Особливий інтерес представляє динамічний хаос, що може виникати в істотно квантових системах (не квазікласичних). Зокрема, показано, що дифузія квантових систем у просторі енергії може бути значно більш ефективною, ніж переходи при багатофотонних процесах. Показано, що облік особливих рішень дозволяє реалізувати режими з хаотичним поводженням навіть у системах з «одним» ступенем свободи. 2012 Article Dynamic chaos generated by linear systems / V.A. Buts // Вопросы атомной науки и техники. — 2012. — № 1. — С. 333-336. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 05.45.-a; 05.45.Mt http://dspace.nbuv.gov.ua/handle/123456789/108230 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Section F. Nonlinear Dynamics and Chaos Section F. Nonlinear Dynamics and Chaos |
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Section F. Nonlinear Dynamics and Chaos Section F. Nonlinear Dynamics and Chaos Buts, V.A. Dynamic chaos generated by linear systems Вопросы атомной науки и техники |
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It is shown that regimes with chaotic behavior are also inherent in linear systems. A dynamic chaos which can arise in essentially quantum systems (not quasi-classical) is of special interest. In particular, it is shown that the diffusion of quantum systems in the energy space can be considerably more effective than transitions at multiphoton processes. It is shown that taking into account the singular solutions allows to realize regimes with chaotic behavior even in systems with "one'' degree of freedom. |
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Buts, V.A. |
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Dynamic chaos generated by linear systems |
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Dynamic chaos generated by linear systems |
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Dynamic chaos generated by linear systems |
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Dynamic chaos generated by linear systems |
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Dynamic chaos generated by linear systems |
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dynamic chaos generated by linear systems |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Section F. Nonlinear Dynamics and Chaos |
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Dynamic chaos generated by linear systems / V.A. Buts // Вопросы атомной науки и техники. — 2012. — № 1. — С. 333-336. — Бібліогр.: 7 назв. — англ. |
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Вопросы атомной науки и техники |
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DYNAMIC CHAOS GENERATED BY LINEAR SYSTEMS
V.A. Buts∗
National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 27, 2011)
It is shown that regimes with chaotic behavior are also inherent in linear systems. A dynamic chaos which can arise
in essentially quantum systems (not quasi-classical) is of special interest. In particular, it is shown that the diffusion
of quantum systems in the energy space can be considerably more effective than transitions at multiphoton processes.
It is shown that taking into account the singular solutions allows to realize regimes with chaotic behavior even in
systems with “one” degree of freedom.
PACS: 05.45.-a; 05.45.Mt
1. INTRODUCTION
Practically till now it is considered that the regimes
with dynamic chaos (DC) arise only in nonlinear sys-
tems. For this reason nobody carries out the search
of regimes with DC at the study of linear systems.
However, it is well known that by the transformation
of the dependent variable the linear systems often
can be transformed into nonlinear systems. In such
systems regimes with DC are widely presented. Well-
known examples are transitions from the equations of
quantum mechanics to the equations of classical me-
chanics, and also transitions from the Maxwell equa-
tions to the equations of geometrical optics.
Thus, today there are known at least two exam-
ples at which in linear systems at certain values of
their parameters the regimes with DC are possible.
In works [1-3] it is shown that this situation is con-
siderably more widespread and that the regimes with
dynamic chaos are inherent to a large number of lin-
ear systems. In the present work the results of the
analysis of this feature for a number of classical and
quantum systems are given from unified positions. A
possible mechanism of occurrence of unpredictability
is discussed. For these purposes it is convenient to
use the concept of a measure. The introduction of
stochastic conjugated functions is also useful. These
functions are irregular. However, some of their com-
binations behave regularly. The examples of such
functions are given.
The significant role of singular solutions in the
origin of chaotic dynamics is shown.
2. DYNAMIC CHAOS AT INTERACTION
OF THREE LINEAR OSCILLATORS
For definiteness we shall consider the most simple and
at the same time one of the most important linear
physical system, in which the mode with DC is possi-
ble. This system is three connected linear oscillators:
q̈0 + q0 = −μ1q1 − μ2q2; q̈1 + q1 = −μ1q0;
q̈2 + (1 + δ) q2 = −μ2q0,
(1)
where q̇ ≡ dq
dτ , μi � 1, δ � 1.
The r.h.s terms (factors of connection) of the sys-
tem are small. That is why the solutions of the sys-
tem (1) are convenient to be searched as:
qi = Ai(τ) exp (iωit) . (2)
For finding complex amplitudes Ai(τ) it is possi-
ble to get the following system of equations:
2iȦ0 = −μ1A1 − μ2A2 exp(iδτ),
2iȦ1 = −μ1A0, 2iȦ2 = −μ2A0 exp(−iδτ). (3)
The system of equations (3) is linear. For further
analysis of dynamics of Ai(τ) we shall present them
as:
Ai(τ) = ai(τ) exp(iϕ(τ)), (4)
here ai, ϕi are real amplitudes and real phases. Sub-
stituting (4) in (3) for finding ai and ϕi we shall get
the following equations:
ȧ0 = − (μ1/2)a1 · sin Φ − (μ2/2) · a2 · sin Φ1;
ȧ1 = (μ1/2)a0 · sin Φ; ȧ2 = (μ2/2)a0 · sin Φ1;
Φ̇ =
(μ1
2
)(
a0
a1
− a1
a0
)
cosΦ −
(μ2
2
)(
a2
a0
)
cosΦ1,
Φ̇1 =
(μ2
2
)(
a0
a2
− a2
a0
)
cosΦ1 −
(μ1
2
)(
a1
a0
)
cosΦ+δ,
(5)
where Φ ≡ ϕ1 − ϕ0, Φ1 ≡ ϕ2 − ϕ0 + δτ .
The system (5) is already nonlinear. Basically,
the dynamics of such system can be chaotic. The
analysis of system (5) gives the following estimation
for the condition of occurrence of regimes with DC:
(μ1 + μ2) > δ. It is possible to find more detailed
information about the system (5), and also about the
results of numerical investigation in [1-5]. Let us note
∗E-mail address: vbuts@kipt.kharkov.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 333-336.
333
only that at fulfilling the conditions for onset of DC
the dynamics of amplitudes (ai) and phases (ϕi) had
been found as chaotic. The spectra are wide, correla-
tion functions fall down quickly, the main Lyapunov
indexes are positive.
3. WAVES SCATTERING IN THE
MEDIUM WITH WEAK PERIODIC
INHOMOGENEITY
Properties similar to the considered case of regular or
chaotic dynamics in the system of three linear cou-
pled oscillators take place in a large number of other
linear systems. Here we consider a model of an elec-
tromagnetic wave scattering on a non-uniform dielec-
tric medium filling the half-space (z > 0 for unam-
biguity). The wave comes from the upper (z < 0)
homogeneous half-space. The wave of electrical field
satisfies the well known wave equation:
ΔE + k2εE = 0,
where k = ω/c.
We consider the case that the permittivity is de-
scribed by the formula:
ε = 1 +
2∑
i=1
μi · cos (�κi · �r) .
It is assumed that the heterogeneity is small,
μi � 1. In this case the electromagnetic wave scat-
tering gives diffracted waves of the minus first order
of diffraction. The complete field can be represented
in the form:
E =
2∑
i=0
Ei =
2∑
i=0
Ai(z) · exp
(
i�ki�r
)
.
Here the first term corresponds to the incident
wave, the second and third ones correspond to the
minus first order of diffraction by the first and the
second heterogeneities accordingly. Let the relation
between the wave vectors satisfy the following expres-
sions:
�k1 = �k0 − �κ1, �k2 = �k0 − �κ2 + �δ, �δ = (0, 0, δ) .
In this model we consider the case that all the
wave amplitudes depend only on the coordinate z di-
rected into the lower half- space. Substituting equa-
tion for the field into the wave equation and applying
averaging to find the slow varying amplitudes of the
interactive waves we obtain the following system of
equations:
2iA′
0 = μ1A1 + μ2A2 exp(iδz),
2iA′
1 = μ1A0 · (k0z/k1z) ,
2iA′
2 = μ2A0 · (k0z/k2z) exp(−iδz),
(6)
where A′ ≡ dA/dz and the following dimension-
less parameters and independent variables are intro-
duced: μ1 ≡ μ1 · k/k0z, μ2 ≡ μ2 · k/k0z, δ ≡ δ/k,
z ≡ k · z.
The system (6) is similar to the system of equa-
tions (3) if the derivative with respect to time is sub-
stituted by the derivative with respect to the coor-
dinate z. Therefore, the dynamics of the systems
(3) and (6) has the same qualitative description.
They both have the areas of parameters in which
their behaviour is chaotic. The chaos criterion, for
example for the system (6), in the symmetric case
(μ1 ≈ μ2 = μ), is given by the inequality:
δ < (k2 · μ)/4
√
k1z · k2z .
4. CHAOS IN QUANTUM SYSTEMS
Anyone who uses words “quantum” and
“chaos” in the same sentence should be
hung by his thumbs on a tree in the park
behind the Niels Bohr Institute
Joseph Ford
This epigraph (from [6]) reflects the attitude of many
scientists to researches on quantum chaos. This atti-
tude, basically, is caused by the fact that the equa-
tions of quantum mechanics are linear equations. Be-
low we shall show that at the appropriate transforma-
tion of dependent variable the equations of quantum
mechanics become nonlinear, in which the regimes
with DC exist.
Let us consider a quantum system which is under
perturbation. Its Hamiltonian is Ĥ = Ĥ0 + Ĥ1(t).
We will use the conventional theory of perturbation.
In this case the solution can be represented in such a
form:
ψ(t) =
∑
n
An(t) · ϕn · exp(iωnt).
Unknown amplitudes An(t) must be found from the
following system of equations:
ih̄ · Ȧn =
∑
m
Un m(t) · Am, (7)
where
Un m =
∫
ϕ∗
m Ĥ1(t)ϕn exp[i t (En − Em)/h̄] dq.
Let us consider a more simple case when in our
system a harmonic perturbation Ĥ1(t) = Û ·exp(iΩt)
is present. In this case the matrix elements of inter-
action get such forms:
Un m = Vn m exp{i · t · [(En − Em)/h̄+ Ω]},
Vn m =
∫
ϕ∗
n · Û · ϕmdq.
(8)
Let us consider the dynamics of three-
level system (|0〉, |1〉, |2〉) (see the figure).
Scheme of energy levels
334
We will suppose that the frequency perturbation and
eigenvalue of energies of these levels satisfy such re-
lations:
m = 1, n = 0, h̄Ω = E1 − E0, m = 2, n = 0,
h̄(Ω + δ) = E2 − E0, |δ| � Ω.
(9)
Using these relations one can limit oneself only to
three equations in the system (7):
ih̄Ȧ0 = V01A1 + V02A2 exp(iδt),
ih̄Ȧ1 = V1 0A0, ih̄Ȧ2 = V2 0A0 exp(−iδt). (10)
The system (10) at Vi 0 = V0 i, ( i = 1, 2) is equivalent
to the system (3).
Let us make such replacement of the dependent
variable Ai(τ) = ai(τ) exp(iϕ(τ)). Then for a de-
finition of variables ai and ϕi we shall get a sys-
tem of nonlinear equations which are equivalent to
the system (5). A condition of the development of
DC in it will be the inequality (μ1 + μ2) > δ (here
μi = 2 ·V0 i/h̄ ·Ω). This inequality has a simple phys-
ical sense. It means that Rabi frequency (V0, 1/h̄)
of transitions between zero and first levels should be
more than the distance between the first and the sec-
ond levels.
Let us now assume that the conditions of realiza-
tion of DC are fulfilled. The investigated systems in
this case will casually wander on power levels. It is
interesting to give an estimation of time for transi-
tion of the system from one level to another and to
compare this value to the regular transitions. The
dimensionless time in the regular case is equal to the
ratio of frequency of external perturbation to Rabi
frequency: τr ∼ 1/μ = h̄Ω/U01.
We can get such estimation for probability to find
our system on first energetic level
〈
a2
1
〉 ∼ μ2 · 〈a2
0
〉 · τ
in stochastic regime. The squares of amplitudes
vary from 0 to 1. From this one can estimate the
average transition time between the levels in a sto-
chastic regime by such expression: τch ∼ (τr)2 ∼
(h̄ · Ω/U01)
2. Thus, the time of diffusion in energy
space at distance ΔE can be estimated by formula:
τD ∼ (ΔE/h̄Ω) (h̄Ω/U01)
2. It is interesting to com-
pare this time to the time of multiphoton transitions.
The time of multiphoton transitions at distance ΔE
can be estimated as: τMPh ∼ (h̄Ω/U01)
(ΔE/h̄Ω). If
the inequality (ΔE/h̄Ω) � 1 holds, the efficiency
of transitions at the stochastic motion will be much
higher than the one due to the multiphoton transi-
tions: (τD,CH/τMPh) � 1.
5. CHANGE OF PROBABILITY AT
TRANSFORMATIONS OF DEPENDENT
VARIABLES
Such question arises: which replacement of depen-
dent variables will lead to a system of nonlinear equa-
tions which have regimes with DC? Now there is no
exhausting answer to this question. However, there
are considerations that the measure and density of
probability can be useful in this case. Really, let’s
introduce the measure Δμ = p (�xi) · Δ�x. Here Δ�x
is the volume of the phase space, p (�xi) is the proba-
bility to find our system in the point �xi of the phase
space. Let’s make the replacement �zk = f(�xi). Thus
the new density of probability is connected to the old
density of probability by the following formula (see,
for example, [7]):
g(�z) =
∑
i
p (�xi)
Δ�xi
Δ�z
=
∑
i
p (�xi)
|J | , (11)
here J is the Jacobian of transformation from the old
variables to the new ones.
If initially we have a linear system, then for
its density of probability it is possible to choose
function: p(�xi) ∼ δ(�xi − �xi(t)). From the for-
mula (11) it follows that we can lose determi-
nacy if Jacobian tends to infinity. Let us con-
sider which Jacobian has our replacement (5). In
our case we have Ak = A′
k + iA′′
k = ak exp (iϕk);
A′
k = ak cosϕk, A
′′
k = ak sinϕk. The Jacobian of
this transformation is |J | = 1/ |ak|. Thus, if as a
result of dynamics of investigated system the am-
plitude tends to zero (ak → 0), the Jacobian will
infinitely grow. This area of the new phase space will
be a source of uncertainty.
6. CONCLUSIONS
It is unconditional that the most important result of
the carried out researches is the fact that the trans-
formation of the dependent variable in linear systems
can transform them in nonlinear systems which can
have regimes with DC. Let us note that such regimes
appeared in the systems which were not exposed to
coarsening.
Above we saw that the dynamics of new depen-
dent variable can be chaotic. At the same time
the initial dependent variable obeys the linear equa-
tions. Therefore their movement should be regular.
Really, the numerical investigation shows that de-
spite of the fact that functions ai and ϕi behave ir-
regularly being separate, their combination such as
A′
k = ak cosϕk, A
′′
k = ak sinϕk behaves quite regu-
larly. Such combination of stochastic functions can
be named as stochastic conjugated functions. But
it is turned out that sometimes the combination of
functions A′
k = ak cosϕk, A
′′
k = ak sinϕk behaves
irregularly. There is a question, to what is it con-
nected? The analysis shows that it occurs only in
the case when the amplitudes ai pass through zero.
We saw that the information about initial dynamics
in this case is lost. However, another factor can be
included in the game: the one of violation of unique-
ness of the solution. Really, at passage of functions
through zero, as it is easy to see, the conditions of
the theorem of uniqueness are broken.
Looking at all these investigations a question can
arise: why is it necessary to transform a simple linear
system into a complex nonlinear system? (We have
335
accustomed to the return transformations, when non-
linear equations are transformed to linear ones. A
classical example now is the method LA pair). The
answer to this question is such: first of all at these
transformations new sides of dynamics of the investi-
gated systems are revealed, and secondly, as soon as
we have defined the area of parameters of investigated
system, in which the regimes with DC are realized,
for the analysis of behavior of investigated system in
new variable the methods of statistical physics can
be used. In many cases it allows us to understand
essentially better the behavior of considered system.
In particular, above at the consideration of the dy-
namics of a quantum system we have seen that as
soon as the Rabi frequency became higher than the
frequency between two close levels, the DC develops.
Moreover, the efficiency of the diffusion process in the
energy space can be much greater than at multipho-
ton processes. Already this result justifies the use of
such transformations.
Above we have considered only one type of trans-
formations of dependent variable. Such transforma-
tion is most widely used in all physical researches.
However it is obvious that other transformations can
play a similar role too. As example of such new trans-
formation one can point on the one which realizes a
transformation of an ordinary second-order linear dif-
ferential equation into a Ricatti equation.
It is also necessary to pay attention to the fact
that now there is a common agreement that chaotic
dynamics is possible only in the systems with the
number of degrees of freedom equal or more than 1.5.
This statement is caused by the fact that there were
considered the systems for which the conditions of
the uniqueness theorem were fulfilled. The systems
having singular solutions were dropped out from the
investigation. The systems in which the phase space
had areas with violation of the uniqueness theorem
were dropped out too. If we take into account such
systems then we find out that the chaotic behavior
will be inherent even in systems with one degree of
freedom. Such example we saw above. More exam-
ples and more details of this situation are included in
the next work.
Acknowledgements
The author thanks Yu.L. Bolotin, V.F. Kravchenko,
A.A. Sanin, K.N. Stepanov, and V.V. Yanovsky for
useful remarks and discussions.
References
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