Singular solutions and dynamic chaos
It is shown that singular solutions of ordinary differential equations may cause new dynamic chaos conditions and new dynamic chaos modes. In particular, these solutions may lead to the dynamic chaos modes in a completely integrable system. An example of a physical system with dynamics significantly...
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irk-123456789-1122272017-01-19T03:02:41Z Singular solutions and dynamic chaos Buts, V.А. Нелинейные процессы в плазменных средах It is shown that singular solutions of ordinary differential equations may cause new dynamic chaos conditions and new dynamic chaos modes. In particular, these solutions may lead to the dynamic chaos modes in a completely integrable system. An example of a physical system with dynamics significantly affected by the presence of singular solutions is analyzed. This example is the movement of particles in the central fields. Показано, що врахування особливих розв’язків систем звичайних диференціальних рівнянь може призводити до нових умов появи режимів з динамічним хаосом. Зокрема, показана можливість виникнення режимів з динамічним хаосом у системах, що повністю інтегруються. Наведено приклад фізичної системи, динаміка якої може істотно залежати від наявності особливих рішень. Такою системою є система, яка описує рух тіл у центральному полі. Показано, что учет особых решений систем обыкновенных дифференциальных уравнений может приводить к новым условиям появления режимов с динамическим хаосом. В частности, показана возможность возникновения режимов с динамическим хаосом в полностью интегрируемых системах. Приведен пример физической системы, динамика которой может существенно зависеть от наличия особых решений. Такой системой является система, которая описывает движение тел в центральном поле. 2015 Article Singular solutions and dynamic chaos / V.А. Buts // Вопросы атомной науки и техники. — 2015. — № 4. — С. 232-236. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 05.45.-a http://dspace.nbuv.gov.ua/handle/123456789/112227 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Buts, V.А. Singular solutions and dynamic chaos Вопросы атомной науки и техники |
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It is shown that singular solutions of ordinary differential equations may cause new dynamic chaos conditions and new dynamic chaos modes. In particular, these solutions may lead to the dynamic chaos modes in a completely integrable system. An example of a physical system with dynamics significantly affected by the presence of singular solutions is analyzed. This example is the movement of particles in the central fields. |
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| author |
Buts, V.А. |
| author_facet |
Buts, V.А. |
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Buts, V.А. |
| title |
Singular solutions and dynamic chaos |
| title_short |
Singular solutions and dynamic chaos |
| title_full |
Singular solutions and dynamic chaos |
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Singular solutions and dynamic chaos |
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Singular solutions and dynamic chaos |
| title_sort |
singular solutions and dynamic chaos |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах |
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| citation_txt |
Singular solutions and dynamic chaos / V.А. Buts // Вопросы атомной науки и техники. — 2015. — № 4. — С. 232-236. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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AT butsva singularsolutionsanddynamicchaos |
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2025-07-08T03:34:01Z |
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 232
SINGULAR SOLUTIONS AND DYNAMIC CHAOS
V.А. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
It is shown that singular solutions of ordinary differential equations may cause new dynamic chaos conditions
and new dynamic chaos modes. In particular, these solutions may lead to the dynamic chaos modes in a completely
integrable system. An example of a physical system with dynamics significantly affected by the presence of singular
solutions is analyzed. This example is the movement of particles in the central fields.
PACS: 05.45.-a
INTRODUCTION
Singular solutions of differential equations are well
known in mathematics. It is sufficient to point to the
fact that the term “singular solutions” has been intro-
duced by Joseph Louis Lagrange (1736-1813). In this
paper, by the term “singular solutions” we assume the
solutions of the system of ordinary differential equa-
tions (ODE) which have points of phase trajectories
where the conditions of uniqueness are violated (non-
Lipshitz condition). However, the analysis of mathemat-
ical models that describe physical processes, it seems,
always imposes either explicit or implicit condition of
uniqueness (author have not found cases which contra-
dicts this). This condition excludes the singular solu-
tions. As the example, we can provide the definition of
the basic properties of the phase space, in which regular
and stochastic dynamics of physical systems is investi-
gated [1]: "Trajectories in phase space do not intersect
at any particular moment of time…". This restriction
immediately eliminates all singular solutions. Such
attitude to the singular solutions appears, apparently,
due to the fact that they do not correspond to our con-
cept of specific solutions of physical problems. We also
note that, according to the literature, quite detailed
analyses of singular solutions were given by V.A.
Steklov [2]. However, this book is very difficult to find.
In [3] was a point out that the singular solutions can
generate chaotic dynamics in systems with known ana-
lytical solutions for all the phase trajectories (for fully
integrated systems). Thus, the presence of the analytical
solutions for the phase trajectories does not define the
regular dynamics of a system under investigation. It
may be assumed that the models considered in [3] to-
gether with the numerous models with singular solu-
tions in mathematics are interesting only from the math-
ematical point of view. Below, we show that such solu-
tions may be also important for solving physical prob-
lems. In the second section of this publication some
basic mathematical models with the dynamics deter-
mined by singular solutions are analyzed. It is shown
that their dynamics is chaotic. It is important that such
systems have only one degree of freedom and are fully
integrable. Thus, the one of the main paradigms of the
dynamic chaos theory, namely that the chaos can appear
only in a system with more than one degree of freedom,
breaks down. The third section contains the physical
example, which shows the influence of singular solu-
tions on the dynamics of the system. Such example is a
dynamics of the particles (bodies) in the central field. It
is shown that the singular solution of this problem sig-
nificantly affects the dynamics in this field at some
values of parameters. The fourth and final chapter
summarizes the main results and highlights the features
of chaotic modes that emerged as the result of account-
ing for singular solutions.
1. SPECIAL SOLUTIONS AND CHAOTIC
DYNAMICS, GENERATED BY THEM
Here is the simplest example of a mathematical
model that has singular solution:
dy y
dx
= . (1)
It is nonlinear first order ordinary differential equa-
tion. Its general solution has the following simple form:
( )21( , , ) 0
4
F x y C y x C= − + = . (2)
Fig. 1. Quadratic parabola of the solution of (1)
We obtain the set of particular solutions of this equa-
tion by changing the value of constant C . Each of these
particular solutions is the quadratic parabola (Fig. 1).
However, all these particular solutions do not contain
the solution 0y = . However, as it is easy to see from
the equation, this expression also satisfies the equation
(1), i.e. it is the solution of the equation. This is the
singular solution of equation (1). This feature of singu-
lar solutions that they cannot be obtained from the gen-
eral solution by changing the arbitrary constants was
one of the first definitions of these solutions. Later it
became clear that these solutions have more important
feature. Namely, the uniqueness theorem is violates at
the points of these solutions. It is easy to see, from
equation (1), that in the points of singular solutions
( 0y = ) Lipshitz conditions are not fulfilled. In addition,
the figure shows that all partial solutions pass through
the points of singular solutions ( 0y = ). There is a gen-
eral question: How to find singular solutions of differen-
tial equations? For ordinary differential equations of the
first order to do this is quite easily. There are several
ways to find such solutions. We consider only one of
mailto:vbuts@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №4(98) 233
them. By its nature, singular solutions are the trajectory
or the surface (of different dimensions), which envelope
the general solution of differential equations. Therefore,
finding such solutions can be associated with finding
envelopes of the overall solution. Let us demonstrate
this statement on the system of equation (1). To find the
envelope, as it is known, we need to differentiate the
general solution (2) by an arbitrary constant C and
equate the derivative to zero:
( )1 0
2
F x C
C
∂
= + =
∂
. (3)
From this expression, we must determine the con-
stant C and substitute it in the expression for the general
solution (2). As a result, we find a singular solution.
This solution in this simple example, of course, coin-
cides with the above solution. Unfortunately, such a
simple procedure for finding singular solutions is char-
acteristic only of the first order ODE. To illustrate the
difficulty of finding specific solutions for the ODE of
higher order, consider one of the possible algorithms for
their finding. To do this, write the ODE system in the
canonical form:
( ), xk
k
dx f t
dt
=
. (4)
Suppose that we have found general solution to this
system of equations. Let this solution has the form:
( , )k kx t Cϕ=
. (5)
It is seen that even at this first step in most real sit-
uations, we encounter difficulties. Now, by analogy
with the general principle of finding envelopes, we need
to find a vector of arbitrary constants C
. The easiest
way to do it is to use such way. We assume that these
constants are functions of time. Then the complete time
derivative of the general solution of (5) will have the
form:
1
n
kk i
i
k
i
dC
C dt
dx x
dt t
ϕ
=
∂
∂
∂
= +
∂ ∑ .
From this expression we can see that if the second
term on the right vanishes:
1
0,
n
k i
i i
dC
C dt
ϕ
=
∂
=
∂∑ { }1,2,.......k n= , (6)
then the general solution (5) is left as the solution of the
original equation (4) in spite of the dependence of the
constants on the time ( )C C t=
. The system of equa-
tions (6) will be the main for the determination of the
vector C
. First of all, note that the system (6) has a
nontrivial solution for derivatives /idC dt only when its
determinant will become zero:
1
1
......
det ..... .... ..... 0
.....
k k
n
n n
n
C C
C C
ϕ ϕ
ϕ ϕ
∂ ∂
∂ ∂
=
∂ ∂
∂ ∂
. (7)
This will be the second relation for finding the vec-
tor of constants C
. Suppose that we by using (7), found
one of the constants expressed through other constants:
1 2 1( , , ,..... )n nC F t C C C −= . (8)
Then their derivation will have such form:
1
1
n
n
n i
i i
dC F FC C
dt t C
−
=
∂ ∂′ ′= = +
∂ ∂∑ . (9)
Substitute expressions (8) и (9) into first (n-1) equa-
tions (6):
1 1
1 0
1
1
0,
n n
k i k k i
n
i ii n i
n
k i
in i
dC dC
C
C dt C C dt
dCF F
C t C dt
ϕ ϕ ϕ
ϕ
− −
= =
−
=
∂ ∂ ∂′+ = +
∂ ∂ ∂
∂ ∂ ∂
+ + = ∂ ∂ ∂
∑ ∑
∑
{ }1,2,....... 1k n= − . (10)
The system (10) is linear ODE for finding deriva-
tives /idC dt . We find after solving (10):
1 2 1( , , ... )i
i n
dC F t A A A
dt −= { }1,2,....... 1 ,i n= − (11)
where A
is vector of new independent constants.
The formulas (8) and (11) allow potentially find all
the components of the vector C
. Substituting them in
the expression for the general solution of (5), we will
find set of singular solutions.
As we can see, the implementation of the considered
algorithm for finding singular solutions is generally a
separate challenge. Only in rare cases it can be imple-
mented analytically.
Let us present a simple example of a nonlinear ODE,
on which can be illustrated the above algorithm to ob-
tain singular solutions, as well as all the difficulties that
can arise while. It is a model example [4]. So, let there
be given:
2 0,xy y z y′ ′ ′+ + − = (12)
0.z y y z z′ ′ ′+ − =
In accordance with the above algorithm, we need to
have a general solution of the system (12). In this case,
these solutions are easy to find:
2
1 1 2y c x c c= + + , (13)
2 1 2 .z c x c c= +
Let us find determinant (7) for these solutions:
( )
( ) ( ) ( )1
1 1 2
2 1
2 1
2 0
x c
x c x c c
c x c
+
= + + − =
+
. (14)
From (14) we will find 2c :
( )( )2 1 12 .c x c x c= + + (15)
Let us the expression for 2c substitute in system of
equations (6). For our example this system has the form:
( )1 1 22 0x c c c′ ′+ + = ,
2 1 1 2( ) 0c c x c c′ ′+ + = . (16)
As a result of the substitution we get 1 0.5c ′ = − −
from the first equation of the system (16). Using this
relation and the expression (15), we find one of the sets
of singular solutions:
2 20.25 3y x Ax A= − ⋅ + + ,
2 2 30.5 2 2z x A x A A= ⋅ + ⋅ ⋅ + . (17)
Expressions (17) are a one-parameter set of singular
solutions in which is a new parameter − arbitrary con-
ISSN 1562-6016. ВАНТ. 2015. №4(98) 234
stant A . Another set of special solutions can be obtained
using the second equation (16). A complete solution of
the model system (12) we shall not write out. Now we
can see, that use of the above algorithm for finding
singular solutions for real systems is a complex prob-
lem. We will not use it here. The main result that can be
get out of this algorithm is that an increase in the num-
ber of degrees of freedom of the system being studied
follow to the growing set of singular solutions. There-
fore, we can expect that for complex systems with many
degrees of freedom, they will play a more significant
role than for simple systems. In this paper we will be
interested, first of all, those particularly of singular
solutions that are in uncertainty of those phase trajecto-
ries that come to the points of these solutions. In most
cases, it is easy to solve problems. It is enough to ana-
lyze the right-hand sides of equations (4) for the imple-
mentation Lipshitz conditions for them. Even easier:
should look at the derivatives of right-hand sides from
the dependent variables in, for example, systems (4). If
these derivatives in some points of the phase space will
be infinitely large, then these areas will be those areas in
contact with which there appears uncertainty. When
taking into account the singular solutions it can be easi-
ly implemented modes of dynamic chaos. For this it is
necessary that the phase trajectories of the dynamical
system periodically get to the points of singular solu-
tions. In extreme cases, as we shall see in the following
section, these trajectories should be approached enough
close to points of singular solutions. As a typical exam-
ple, let us consider the dynamics of the system, which is
described by the following equations:
0 1x x= ;
2
1
1 0
0
0.5
2
xx x
x
= − ⋅
. (18)
The dynamics of such a system has been studied in
[3]. Here the most important characteristics of this
system are represented. First of all, it is easy to show
that the function ( )2 2 2
0 1 0x R x Rϕ = − + − = is an inte-
gral of the system (18). Moreover parameter R (the
radius of the circles) can take arbitrary values. The
system (18) is a model of an oscillator with nonlinear
friction. The phase portrait of the system (18) is shown
in Fig. 2. Integral curves in this case are the circles. The
center of these circles are located on the axis 1 0x = .
The radii of circles are equal to the distance of center to
the zero point ( 0 10; 0x x= = ). This point is common
point to all circles. In addition, this point is a singular
solution of (18). Looking at the integral equation (18), it
is difficult to imagine that the dynamics of the system
(18) may be irregular. However, numerical calculations
show that it is irregular. Indeed, Fig. 3 shows the de-
pendence variable 0x on the time. It is seen that the
phase trajectory after passing the point of singular solu-
tion ( 0 1 0x x= = ) can jump from one circle to another
circle. And these jumps occur randomly. Practically any
change accuracy of calculation change the time dynam-
ics of this system. In addition, spectral analysis of the
system (18) shows that her spectrum is broad and the
correlation function decays quite quickly (Fig. 4).
Fig. 2. Phase portrait of the system (18)
Fig. 3. Time dependence of the variable 0x .
One can see transitions image point from one circle
to another
Fig. 4. Autocorrelation function of the variable 0x
The system (18) is not unique. In [3] has been sta-
died the dynamics of other system:
0
0 1 1 1
dx x x x F
dt
γ= ⋅ + ⋅ ≡ ; 2 41
1 0 0 2
dx x x x F
dt
γ= − − ⋅ ≡ . (19)
It has been shown that in the vicinity of zero this
system has an area in which the uniqueness theorem is
break down. In addition, the phase trajectories are peri-
odically fall into this region. The dynamics of this sys-
tem is irregular. In [3] also shows how you can con-
struct set of systems that have the desired properties.
2. KEPLER PROBLEM
The above examples do not contain a clear physical
meaning. In this section we show that taking into ac-
count singular solutions can be substantial and for well-
known physical models. The first physical example of a
system with one degree of freedom is the problem of the
motion of the particle in a central field. This problem
due to the existence of the integral, which expresses the
law of conservation of angular momentum, is reduced to
the task having only one degree of freedom. Moreover,
take into consideration integral of the energy, this prob-
lem is fully integrated. Below we show, despite its
integrability in its dynamics can be observed modes that
resemble modes with dynamic chaos. Such dynamics
occur when the phase trajectories pass near the points of
singular solutions. The dynamics of particles in the
central field can be described by the following system
of equations:
r v=
ISSN 1562-6016. ВАНТ. 2015. №4(98) 235
( )2
2 3 2
cos tMv
m r mr rγ
β ωα ⋅ ⋅
= − − , (20)
2/ ,M mrϕ =
here 2M mr constϕ≡ = angular moment of the impulse.
For definiteness in (20) we introduced Coulomb (for
the motion of charged particles) or gravity (Kepler
problem) potential. In addition, we took into account a
small external perturbation 1β << . In particular, if
4,γ = 0ω = it can mean that the shape of the attracting
center differs from a strictly spherical shape. If
4, 0γ ω= ≠ it may be an ion that is in an external
periodic electric field.
It should be noted, that the third equation in (20)
does not affect the dynamics of the first two equations.
The dynamics of the angular variable is uniquely deter-
mined by the radial dynamics. The system (20) has one
degree of freedom in this case.
If we take into account the energy integral, the prob-
lem of the dynamics in the central field is fully integrat-
ed. Such solutions can be found in many of the courses
of mechanics (see, e.g., [5]).
Our task is to show that, despite the complete inte-
grability, the dynamics of such system can be very
complicated and, in some sense, chaotic. This case is
similar to that which was discussed in the previous
section. The difference lies mainly in the fact that it is a
well-studied physical problem. The second difference is
that the phase trajectories of the system are not strictly
go through a singular solution.
These trajectories are going close to the area of vio-
lations of the uniqueness theorem, not getting into it
directly. The value of proximity which necessary for
realization of chaotic regimes is determined by parame-
ters of the system, the accuracy of numerical calcula-
tions and the value of small external perturbations. Let
us illustrate these features. First, consider the dynamics
of the system (20) in the absence of disturbance 0β = .
Select the next parameter: (0) 10r = ; (0) 0.1r = ;
(0) 0ϕ = ; 0.3a = ; 2;b = There / ; /a M m b mα≡ ≡ .
Typical results of numerical calculations of a Kep-
lerian dynamics of the system (20) are shown in Fig. 5.
Fig. 5. The typical time dependence of the distance
of the particle from the center of attraction
To further illustrate the regular dynamics of a parti-
cle in Fig. 6 shows the result of the influence of asym-
metric form the center of attraction. It is known that the
deviation of the attractive potential of a spherically
symmetric shape the perihelion rotates at an angle dif-
ferent from 2π . In addition, the same feature perihelion
motion observed if we take into account the effects of
relativism. There arises, so-called rosette trajectory of
Somerfield (Fig. 6). From these figures it is clear that
the dynamics is regular. And it can be shown that the
smaller the value of the angular momentum (option), the
closer the trajectory of light particles will approach the
attracting center. The coordinates of the center of attrac-
tion is a singular solution.
Fig. 6. Rosette trajectory, similar to the Sommerfed
trajectory. The system parameters (20):
(0) 10r = ; (0) 0.1r = ; (0) 0ϕ = ;
0.1β = ; 0, 4ω γ= = ; 1.5; 2a b= =
At this point, the uniqueness theorem is violated.
Fig. 6 illustrates the fact that the integral curves in the
neighborhood of this point are "thicken". Therefore,
even a small perturbation of the trajectories in this area
may transfer moving body from one trajectory to anoth-
er trajectory. In this case previously closely spaced
particles in other regions of the phase space can be
located far away. The forces that relocate the moving
body from one trajectory to another can be very small
and can generally be random. But even in the case when
they are regular but periodic, and the period of these
forces does not coincide with a period of body motion
around the center of attraction, the dynamics of the body
may be random. Indeed, let us introduce a small regular
periodic perturbation ( 0.0001β = , 0γ = ). All other
parameters of the system (20) remain unchanged. The
dynamics of the moving body in this case is shown in
Fig. 7. From these figures it is seen that the moving
body jumps randomly from one trajectory to another
trajectory. The spectrums of this movement are wide,
and the correlation function decays rapidly. This result
we have obtained with high accuracy of calculations. If
the numerical calculations carried out at a lesser degree
of precision then a similar picture (the emergence of the
irregular dynamics) appears in the absence of external
perturbation.
Fig. 7. The typical time dependence of the distance of
the particle from the center of attraction
ISSN 1562-6016. ВАНТ. 2015. №4(98) 236
CONCLUSIONS
Here are the most important results obtained in this
work:
1. The most significant result is the result that singu-
lar solutions must take into account in the analysis of
many physical processes.
2. As shown in Section 2, set of singular solutions
increases with the number of degrees of freedom of the
system under study. This means that the role of singular
solutions is particularly high in the dynamics of com-
plex systems.
3. Draw attention that in general, the search for spe-
cial solutions is challenging. However, for practical
purposes, for the estimates, it is sufficient to determine
the region of phase space, in which is a violation of the
uniqueness theorem. In this case, if the phase trajectory
of the studied system in its dynamics often visits this
area, the dynamics of these systems will be chaotic.
4. Note that in such tasks as body motion in a cen-
tral field, the appearance of singular solutions is a natu-
ral and objective characterize the dynamics of the sys-
tem under study.
5. It should be noted that even in the case when the
phase trajectories do not fall into area with singular
solutions, but are enough close to it, consideration of
these singular solutions can also be very significant. As
we have saw in the example the motion of bodies in a
central field, in this case, even a very small, but the
unaccounted external forces can qualitatively change the
dynamics of the system.
6. Thus, the account of singular solutions signifi-
cantly expands the range of physical systems in which
can be realized regimes with chaos.
7. It should also pay attention to the fact that the
physical nature of the onset of chaos, which is generated
by singular solutions, is different from the nature of
occurrence of the usual dynamic chaos. The most signif-
icant difference is the appearance of chaotic dynamics
in fully integrated systems. Unpredictable divergence
phase trajectories occur only in the vicinity of singular
solutions.
8. In the present work, we focuses our attention on
the role of singular solutions in the emergence of re-
gimes with dynamic chaos. Singular solutions can play
a significant role in many other cases. One thing is clear
that they must be taken into account in the study of the
dynamics of physical systems.
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Article received 05.06.2015
ОСОБЫЕ РЕШЕНИЯ И ДИНАМИЧЕСКИЙ ХАОС
В.А. Буц
Показано, что учет особых решений систем обыкновенных дифференциальных уравнений может приво-
дить к новым условиям появления режимов с динамическим хаосом. В частности, показана возможность
возникновения режимов с динамическим хаосом в полностью интегрируемых системах. Приведен пример
физической системы, динамика которой может существенно зависеть от наличия особых решений. Такой
системой является система, которая описывает движение тел в центральном поле.
ОСОБЛИВІ РОЗВ’ЯЗКИ ТА ДИНАМІЧНИЙ ХАОС
В.О. Буц
Показано, що врахування особливих розв’язків систем звичайних диференціальних рівнянь може приз-
водити до нових умов появи режимів з динамічним хаосом. Зокрема, показана можливість виникнення ре-
жимів з динамічним хаосом у системах, що повністю інтегруються. Наведено приклад фізичної системи,
динаміка якої може істотно залежати від наявності особливих рішень. Такою системою є система, яка опи-
сує рух тіл у центральному полі.
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