Chaotic motion of dynamic systems with “one” degree of freedom
It is shown that the regimes with chaotic motion are also inherent in dynamic systems having only one degree of freedom. Such opportunity arises because such systems in the phase space have areas in which the conditions of the uniqueness theorem are broken, in particular, in the presence of singular...
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Buts, V.A. 2016-10-31T20:38:05Z 2016-10-31T20:38:05Z 2012 Chaotic motion of dynamic systems with “one” degree of freedom / V.A. Buts // Вопросы атомной науки и техники. — 2012. — № 1. — С. 328-332. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 05.45.-a https://nasplib.isofts.kiev.ua/handle/123456789/108231 It is shown that the regimes with chaotic motion are also inherent in dynamic systems having only one degree of freedom. Such opportunity arises because such systems in the phase space have areas in which the conditions of the uniqueness theorem are broken, in particular, in the presence of singular solutions. Examples of such systems are given. Показано, что режимы с хаотическим движением присущи также динамическим системам, имеющим всего одну степень свободы. Такая возможность возникает благодаря тому, что такие системы в своем фазовом пространстве имеют области, в которых нарушаются условия теоремы единственности, в частности, при наличии особых решений. Приведены примеры таких систем. Показано, що режими з хаотичним рухом властиві також динамічним системам, що мають усього один ступінь свободи. Така можливість виникає завдяки тому, що такі системи у своєму фазовому просторі мають області, у яких порушуються умови теореми єдності, зокрема, при наявності особливих рішень. Наведено приклади таких систем. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section F. Nonlinear Dynamics and Chaos Chaotic motion of dynamic systems with “one” degree of freedom Хаотическое движение систем с "одной'' степенью свободы Хаотичний рух систем з "одним'' ступенем свободи Article published earlier |
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Chaotic motion of dynamic systems with “one” degree of freedom |
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Chaotic motion of dynamic systems with “one” degree of freedom Buts, V.A. Section F. Nonlinear Dynamics and Chaos |
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Chaotic motion of dynamic systems with “one” degree of freedom |
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Chaotic motion of dynamic systems with “one” degree of freedom |
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Chaotic motion of dynamic systems with “one” degree of freedom |
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Chaotic motion of dynamic systems with “one” degree of freedom |
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chaotic motion of dynamic systems with “one” degree of freedom |
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Buts, V.A. |
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Buts, V.A. |
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Section F. Nonlinear Dynamics and Chaos |
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Section F. Nonlinear Dynamics and Chaos |
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Вопросы атомной науки и техники |
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Хаотическое движение систем с "одной'' степенью свободы Хаотичний рух систем з "одним'' ступенем свободи |
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It is shown that the regimes with chaotic motion are also inherent in dynamic systems having only one degree of freedom. Such opportunity arises because such systems in the phase space have areas in which the conditions of the uniqueness theorem are broken, in particular, in the presence of singular solutions. Examples of such systems are given.
Показано, что режимы с хаотическим движением присущи также динамическим системам, имеющим всего одну степень свободы. Такая возможность возникает благодаря тому, что такие системы в своем фазовом пространстве имеют области, в которых нарушаются условия теоремы единственности, в частности, при наличии особых решений. Приведены примеры таких систем.
Показано, що режими з хаотичним рухом властиві також динамічним системам, що мають усього один ступінь свободи. Така можливість виникає завдяки тому, що такі системи у своєму фазовому просторі мають області, у яких порушуються умови теореми єдності, зокрема, при наявності особливих рішень. Наведено приклади таких систем.
|
| issn |
1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/108231 |
| citation_txt |
Chaotic motion of dynamic systems with “one” degree of freedom / V.A. Buts // Вопросы атомной науки и техники. — 2012. — № 1. — С. 328-332. — Бібліогр.: 3 назв. — англ. |
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2025-11-25T01:11:04Z |
| last_indexed |
2025-11-25T01:11:04Z |
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1850503510714482688 |
| fulltext |
CHAOTIC MOTION OF DYNAMIC SYSTEMS WITH “ONE”
DEGREE OF FREEDOM
V.A. Buts∗
National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 27, 2011)
It is shown that the regimes with chaotic motion are also inherent in dynamic systems having only one degree of
freedom. Such opportunity arises because such systems in the phase space have areas in which the conditions of the
uniqueness theorem are broken, in particular, in the presence of singular solutions. Examples of such systems are
given.
PACS: 05.45.-a
1. INTRODUCTION
Now there is a common agreement that the regimes
with chaotic motion in dynamic systems can exist
only if they hold the number of degrees of freedom
more or equal than 1.5. The main reason for existence
of such opinion is the restriction that in the phase
plane (system with one degree of freedom) the phase
trajectories should not intersect. In other words,
there is a requirement that the theorem of unique-
ness be held (see, for example, [1]).
If we are limited to a class of ordinary differential
equation (ODE) whose parameters satisfy this theo-
rem, such statement, certainly, is correct. However,
it is known that except the general solutions (for
which the theorem of uniqueness is held), ODE have
singular solutions. In points of the singular solutions
the theorem of uniqueness is not held. Therefore, if
we shall take into account dynamic systems in which
there are singular solutions, the formulated restric-
tions on number of degrees of freedom (1.5) can be
removed. In this case the modes with chaotic motion
can be realized in the systems with one degree of
freedom as well. In the present work we give exam-
ples of such systems.
2. SYSTEMS WITH SINGULAR
SOLUTIONS
Below we shall show that in the systems with one de-
gree of freedom at the presence of singular solutions
the modes with chaos are possible. It is necessary
to note that, as soon as the system is situated in
the points of singular solutions, it, in general case,
“does not know” its further trajectory. The choice of
the further trajectory is determined by any external,
even arbitrarily small perturbations. As a character-
istic example we shall consider the dynamics of the
system which is described by the following equations:
ẋ0 = x1; ẋ1 =
(
x2
1
2x0
)
− 0.5 · x0. (1)
The phase portrait of the system (1) is presented in
Fig. 1. The integral curves in this case are circles (see
below). Also, the centers of the circles are located
on the axis x1, and the radiuses of these circles are
equal to distance of these centers from the zero point
(x0 = 0; x1 = 0). This point is common for all cir-
cles. Besides, this point is the singular solution of the
system (1) (see below). The system (1) was analyzed
numerically. The results are presented in Figs. 1-6.
In the second and third figures the characteristic
time dependencies are presented. Comparing Fig. 1
with Fig. 2, it is possible to make the conclusion that
the representing point, moving to one of circles after
passage of a zero point, finds itself on another circle.
Moreover, the transition from one circle to another
occurs according to a random law. Really, the spec-
trum of this dynamics is wide (see Fig. 3), and the
correlation function falls down quickly enough (see
Fig. 4).
Fig. 1. Phase portrait of the system (1)
Let’s note that the casual transitions from one cir-
cle to another at passage of the zero point depend on
the accuracy of computing. The change, e.g., of the
step of calculations changes the concrete character of
these transitions. However, as a whole, statistically,
the dynamics remains the same.
∗E-mail address: vbuts@kipt.kharkov.ua
328 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 328-332.
Fig. 2. Evolution in time of the variable x0. One
can see the transitions from one circle to another
The system (1) is not unique. Below we shall show
how the sets of such systems can be constructed. But
now let’s consider the dynamics of one more system,
probably, more interesting from the point of view of
chaotic dynamics:
dx0
dt
= x0 · x1 + γ · x1 ≡ F1,
dx1
dt
= x2
1 − x4
0 − γ · x0 ≡ F2.
(2)
A phase portrait of the system (2) is presented in
Fig. 5.
Fig. 3. Spectrum of x0
Fig. 4. Correlation function for variable x0
Addition of linear terms (γ · xi) to the system (2)
has allowed to change the character of a special zero-
point. From the complex quasisaddle this point has
turned into a “centre” point. If such updating is not
done, the representing points will remain on the way
to this special point during an unlimited time. To get
rid of this feature this additional term has been intro-
duced into the model. The characteristic dependence
of dynamics variable on time is presented in Fig. 6.
Fig. 5. Phase portrait of the system (2) at γ = 0
Fig. 6. Characteristic view of realization of va-
riable x0
From this figure it is already visible that the dy-
namics of transitions from one integrated trajectory
to another at the passage in a vicinity of the zero
point is complex. Moreover, these transitions occur
casually. The randomness of them is visible when
we are looking at the spectrum and the correlation
function (Figs. 7 and 8). In these two figures one can
see that the spectrum is wide enough and the cor-
relation function falls down while oscillating. Now
let us show how the set of systems with one degree
of freedom similar to the systems (1) and (2) can be
constructed. Let us take an integral curve which is
given by the equation ϕ (x0, x1) = 0.
Then the system of equations whose integral will
be this integral curve can be represented in the fol-
lowing form:
dx0
dt
= F1 (ϕ, x0, x1) − ∂ϕ
∂x1
M (x0, x1) ,
dx1
dt
= F2 (ϕ, x0, x1) +
∂ϕ
∂x0
M (x0, x1) ,
(3)
where Fs(ϕ, x0, x1) are arbitrary functions which
have such property: Fs(0, x0, x1) = 0; M (x0, x1)
is an arbitrary function.
329
Fig. 7. Spectrum of x0
Fig. 8. Correlation function for variable x0
Fig. 9. Dependence of amplitude on time
Using the system (3), it is possible to construct a
large variety of dynamic systems which will possess
the necessary properties. As an example we shall con-
sider a case that the integral curves are a family of
circles with the radius R:
ϕ = (x0 − R)2 + x2
1 − R2 = 0. (4)
The set of integral curves (4) is presented in
Fig. 1. By choosing the functions Fs and M it is pos-
sible to achieve an elimination of the parameter R
from the system (3). Really, we shall choose (for ex-
ample) such functions as: F1 = 0; F2 = ϕ ·f (x0, x1);
M = −x0 · f (x0, x1). Here f (x0, x1) is an arbitrary
function. Substituting these expressions in the sys-
tem (3), we shall get a set of systems of equations in
which the parameter R is already eliminated:
dx0
dt
= 2x0 · x1 · f (x0, x1) ,
dx0
dt
=
(
x2
1 − x2
0
) · f (x0, x1) .
(5)
Choosing function f (x0, x1) as f (x0, x1) = 1/2x0
we shall get the system of equations (1).
The dynamics of this system is chaotic. This is
visible from Figs. 2-4. It is necessary to notice the
following: the integral curves of the system (5) will
be a family of circles with a common point (0, 0).
Despite this fact, the character of this point and the
dynamics of the system will depend essentially on the
form of the function f (x0, x1).
Let’s show now that the zero point is a singular
solution of the system (1). Really, this point belongs
to the family of circles (4). The same circles are inte-
gral curves of the system (1). These integral curves
are convenient to be rewritten as: x2
1/x0 + x0 = R.
From these curves it follows that in the vicinity of
zero point the Lipchitz conditions for the system (1)
are broken. Really, the Lipchitz condition for the
system (1) can be written down as
∣∣∣∣ x̃
2
1
x̃0
− x2
1
x0
∣∣∣∣ ≤ L (|x̃0 − x0| + |x̃1 − x1|) , (6)
where L is a positive constant.
In the vicinity of zero the left part of the inequal-
ity (6) can be estimated as
∣∣∣R̃ − R
∣∣∣, where R̃ and R
are radiuses of two arbitrary circles. Generally, the
differences of these radiuses can be of any size. Thus,
in a zero point the Lipchitz conditions are not satis-
fied, i.e. the conditions of the theorem of uniqueness
for the system (1) are broken. Besides taking the
partial derivative of the function (4) with respect to
R and equating it to zero, we find that the point
(x0 = 0; x1 = 0) is a singular solution of the system
(1), and also envelope of the integral curves.
Let’s consider now the system of equations (2). If
γ = 0 then this system has the following family of
integral curves:
ϕ = x4
0 + x2
1 − Cx2
0 = 0. (7)
Here C is a certain constant.
The appearance of these integral curves is pre-
sented in Fig. 5. Using this formula the right part of
the second equation of the system (2) at γ = 0 can be
rewritten in the following form: F2 = −2 · x4
0 + Cx2
0.
Using this expression, it is easy to see that the left
part of the Lipchitz inequality in a vicinity of small
value of x0 will be proportional to x2
0
∣∣∣(C − C̃)
∣∣∣. Tak-
ing into account that the constants C and C̃ also can
be essentially different, in a vicinity of zero point (but
not namely in this point) the Lipchitz conditions will
not be satisfied. However such “cutting down” of the
system of equations (2) will behave quite regularly.
330
Moreover, very quickly all the initial points will find
themselves in the vicinity of the zero point and will
remain in its vicinity arbitrarily long time. To re-
move such feature of dynamics in the system (2), the
linear terms were added. They do not permit the
representing points to stop in the zero point.
Let’s point out another opportunity of construc-
tion of the systems of differential equations with one
degree of freedom whose dynamics can show chaotic
behavior. Let us have two families of integral curves
ϕ1 (x0, x1) = 0 and ϕ2 (x0, x1) = 0. Using the sys-
tem of equations (3), it is easy to find a system of
differential equations whose solutions are such inte-
gral curves:
dx0
dt
=
∂ϕ2
∂x1
F1(ϕ1, x0, x1, t)−
−∂ϕ1
∂x1
F2(ϕ2, x0, x1, t),
dx1
dt
= −∂ϕ2
∂x0
F1(ϕ1, x0, x1, t)+
+
∂ϕ1
∂x0
F2(ϕ2, x0, x1, t).
(8)
Here Fs(ϕs, x0, x1, t) are arbitrary functions pos-
sessing the property Fs(0, x0, x1, t) = 0. From the
system (8) it is possible to obtain the following equa-
tion for ϕs:
dϕs
dt
=
[
∂ϕ2
∂x1
∂ϕ1
∂x0
− ∂ϕ2
∂x0
∂ϕ1
∂x1
]
Fs ≡ Δ · Fs. (9)
From (9) it follows that if we want these integral
curves to be stable, the fulfillment of the following
inequalities is necessary:
[
Δ · ∂Fs
∂ϕs
]
ϕs=0
< 0. (10)
For our purposes (the occurrence of chaotic dy-
namics) is of interest the case that the integral curves
ϕ1 (x0, x1) = 0 and ϕ2 (x0, x1) = 0 have a common
point. However it is obvious, that if integral curves
are stable, and the inequality (10) is fulfilled, the
common points will be stable too. In this case, it
is obvious that the dynamics will be regular. There-
fore it is necessary to require that in common points
(in points of crossing of integral curves) the stability
must be broken. It can be achieved if one requires
[
Δ · ∂Fs
∂ϕs
]
ϕs=0
{
< 0
≥ 0.
(11)
The upper inequality in (11) should be fulfilled
in all points, where ϕ1 �= ϕ2. The lower inequality
should be fulfilled in the crossing points ϕ1 = ϕ2 = 0.
3. SYSTEM WITH BROKEN
CONDITIONS OF UNIQUENESS
Above we have considered the systems which have
singular solutions. In the points of singular solutions
the uniqueness theorem is broken. This fact also has
resulted in qualitative change of the dynamics of in-
vestigated system. Therefore it is clear that generally
there is no necessity in availability of singular solu-
tions. It is sufficient that in the phase space of the
investigated system there was an area in which the
uniqueness theorem is broken. Besides it is important
that the trajectories of the investigated system find
themselves often enough in this area of phase space.
As the elementary example, in which such scenario of
the development of chaotic dynamics can be realized,
we shall consider the following simple system:
ẋ = y, ẏ = −x. (12)
Let functions x(t), y(t) be complex. Then two
variants of substitutions are possible. The first one is
x = xR + i · xI , y = yR + i · yI . In this case the dy-
namics of real and imaginary parts are independent.
In this case new dynamics does not arise.
New dynamics arises at introduction of other de-
pendent variables, namely
x = x0 · exp (i · x2) y = x1 · exp (i · x3) . (13)
Here xk(t) k = {0, 1, 2, 3} are real functions.
Substituting (13) in (12) we shall get the following
system of ODE for finding xk(t):
ẋ0 = x1 cos (x3 − x2) , ẋ1 = −x0 cos (x3 − x2) ,
ẋ2 =
x1
x0
sin (x3 − x2) , ẋ3 =
x0
x1
sin (x3 − x2) .
(14)
These four equations can be rewritten as three
ones:
ẋ0 = x1 cosΦ, ẋ1 = −x0 cosΦ,
Φ̇ =
(
x0
x1
− x1
x0
)
· sin Φ,
(15)
where Φ = x3 − x2.
From the first two equations of the system (14) or
(15) it is visible that these equations have the follow-
ing integral:
x2
0 + x2
1 = const. (16)
Thus, the system of equations (15) has only one
degree of freedom. The system (14) was investigated
by numerical methods. The characteristics of their
dynamics are presented in Figs. 9-11. From these
figures it is visible that as soon as the value of am-
plitude appears in the vicinity of zero (x0 = 0), the
value of the phase (x2) can undergo a jump. Oc-
currence of these jumps is called by the fact that
in the point x0 = 0 the theorem of uniqueness for
the system (5) is not held. In this point arbitrarily
small fluctuations can define the value of a phase.
As a matter of fact, in the given elementary example
the choice is not so rich: the phase can either re-
main continuous or undergo a jump to a value of π.
The occurrence of these jumps, as it is visible from
Fig. 10, is a stochastic function. Similar dynamics
is observed for amplitude x1 and for a phase x3.
As a result of these random jumps the dynamics of
all system becomes chaotic. The degree of random-
ness can be characterized by correlation function (see
331
Fig. 11). For our case the correlation function falls
down quickly enough. It is possible to show that also
other statistical characteristics for the system (14)
(spectra, main Lyapunov index) are the same as for
the systems with chaotic dynamics.
Fig. 10. Evolution of phase in time
Fig. 11. Correlation function for variable x0
4. CONCLUSIONS
Thus, if we abandon the conditions of uniqueness of
the solutions, then in such systems the modes with
chaotic motion can appear. The simplest systems in
this case are the systems with one degree of freedom
which we have considered above. However, it is clear
that in the systems with a large number of degrees
of freedom such mechanism of occurrence of chaotic
regimes will appear too. It is necessary to say that
in many systems which, for example, are described
by the systems of equations (1) or (12) the random-
ness arises as a result of the presence of small fluctua-
tions at numerical calculation. And, the more precise
calculation is used, the smaller area in a vicinity of
zero will define the chaotic motion. Thus, chaotic
dynamics of these systems is practically caused by
the presence (even arbitrarily small) of casual forces.
Just for this reason the word “one” was taken in in-
verted commas in the title of this paper. However,
these cases, apparently, do not exhaust all opportu-
nities. If the dynamics of the system is such as the
area of non-uniqueness in the phase space is not lim-
ited to one point, the chaotic dynamics of the system
may not depend on numerical fluctuation. However,
these questions require an additional study. It is nec-
essary to notice that some aspects of the dynamics
considered above are similar to the dynamics of sin-
gular point mapping which were studied, for example,
in [2, 3].
References
1. A.J. Liechtenberg, M.A. Lieberman. Regular
and Stochastic Motion. New York: “Springer-
Verlag New York inc.”, 1983, 499 p.
2. S.V. Slipushenko, A.V. Tur, V.V. Yanovsky.
Mechanism of intermittency arising in singular
conservative systems // Izvestia vuzov “AND”.
2010, v. 18, N. 4, p. 91-110.
3. S.V. Slipushenko, A.V. Tur, V.V. Yanovsky.
Intermittency without chaotic phases // Func-
tional materials. 2006, v. 13, N. 4, p. 551-557.
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&&'
|