Journal of Modern Physics, 2013, 4, 1381-1392
http://dx.doi.org/10.4236/jmp.2013.410166 Published Online October 2013 (http://www.scirp.org/journal/jmp)

Identification of Structural Model for Chaotic Systems
Evgeny V. Nikulchev, Oleg V. Kozlov
Moscow Technological Institute, Moscow, Russia
Email: nikulchev@mail.ru
Received August 5, 2013; revised September 6, 2013; accepted September 28, 2013
Copyright © 2013 Evgeny V. Nikulchev, Oleg V. Kozlov. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.

ABSTRACT
This article is talking about the study constructive method of structural identification systems with chaotic dynamics. It
is shown that the reconstructed attractors are a source of information not only about the dynamics but also on the basis
of the attractors which can be identified and the mere sight of models. It is known that the knowledge of the symmetry
group allows you to specify the form of a minimal system. Forming a group transformation can be found in the reconstructed attractor. The affine system as the basic model is selected. Type of a nonlinear system is the subject of calculations. A theoretical analysis is performed and proof of the possibility of constructing models in the central invariant
manifold reduced. This developed algorithm for determining the observed symmetry in the attractor. The results of
identification used in real systems are an application.
Keywords: Chaotic Dynamics; Attractors; Symmetry; Identification

1. Introduction
The first work on the reconstruction of the strange attractor from the time series has been publishing the results on hydrodynamics [1]. The article shows that you
can get a satisfactory picture of the strange attractor of
the geometric dimensions of a small, if the variables x,
appearing in the equations of the dynamical system
dx dt  F  x  , use the m-dimensional vectors, derived
from the elements of time series of the same principle,
which in the problems of autoregression. That same year,
F. Takens reported on his theorem, which was published
a year later [2]. That it is the basis of all algorithms for
time series analysis methods of nonlinear dynamics. The
problem of determining the form of a dynamical system
from its one-dimensional realization belongs to a class of
incorrect problems. Unlike the problem of analyzing this
issue is ambiguous, since there are infinitely many dynamical systems of various kinds which can play the existing signal with a given degree of accuracy.
The method of the global reconstruction of a dynamic
system of equations for its one-dimensional realization
was proposed in [3,4]. The algorithm is as follows.
One-dimensional realization of the process is in a system,
which is considered a “black box” recovered phase portrait on the Takens theorem, topologically equivalent to
the attractor of the original system. A priori given equation is the method of least squares which is a set of unCopyright © 2013 SciRes.

known coefficients.
Now there are considerable publications, developing
and constantly improving the proposed the method [5-11].
For example, in R. Brown and others [12] to reconstruction dynamic equations on the experimental time series
with a broadband continuous spectrum use additional
information about the dynamic and statistical properties
of the original system contained in the implementation.
In obtaining equation takes into account the values of
Lyapunov exponents and the probability density, calculated from the original time series. However, the resulting evolution equations have a very cumbersome, inconvenient to use. In [13] used the hidden variables to write
a model equation. In [12] describes a method for synchronizing the model with the original data. In a number
of O.L. Anosov proposed reconstruction algorithm scalar
differential equations for systems with delay.
However, the feature of many studies is that the proposed methods are illustrated with examples of simple
low-dimensional model systems when we know in advance what should be the result of global reconstruction.
It does not show substantial benefits given by the modifications of the basic method [3]. Described in the publications of the algorithms are tested on a number of
known model systems with small dimension and a simple
form right sides. The efficiency of the method is demonstrated by the example of time series generated by the
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E. V. NIKULCHEV, O. V. KOZLOV

real “black boxes”.
The challenge is the need to work with noisy data in
the processing of experimental time series. On the one
hand, more desirable is the use of sequential differentiation to restore the phase trajectory, because it can get a
model that contains, in general, approximately n times
smaller than the coefficients of the various non-linear
than when using the method of delays. But differentiation
will inevitably lead to increased noise components of
high order. Without pre-filtering the time dependence of
the second derivative can be noise-like process. In addition, methods of attachment are obvious flaws in the
analysis significantly heterogeneous implementations,
i.e., signals in which areas with fast motion alternating
with areas of slow motions.
Arbitrary choice of nonlinearities, as a rule, does not
allow for a successful reconstruction of the dynamic
equations for real systems. In particular, in [14] indicated
the presence of three typical cases:
1) Reconstruction of locally describe the phase trajectory of the original system. In this case, the reconstructed
model is unstable in the sense that the solution of these
equations reproduces the signal under investigation only
in a short period of time.
2) There is poor local predictability of the phase trajectory, but there is visual similarity of the phase portraits. Reconstruction of solution is stable in the sense of
Poisson. In this case, the attractor of the reconstructed
model has metric characteristics similar to those of the
original attractor.
3) There is a good local predictability of the phase trajectory of any point in time values exceeding the characteristic correlation time. The phase portrait reconstructed
model is identical to the original, and the system is Poisson stable.

2. Group Properties for the Construction of
Equivalent Mapping
Consider the research questions of the global structure of
orbits of dynamical systems, which do not depend on the
choice of coordinate system, i.e., admit a symmetry
transformation. From a global point of view, the change
of coordinates is a diffeomorphism (in the case of a
smooth structure) or a homeomorphism (in the topological situation) phase spaces. Thus, we can introduce a
natural equivalence relation between the dynamical systems associated with different classes of coordinate
changes, and interpret the problem of describing the
structure of orbits as a classification problem of dynamical systems up to these equivalence relations.
Given the equivalence of the trajectories on the torus
[15], we can show that every periodic point of the map,
whose spectrum does not contain a unit, defines certain
1 -modules. Since these periodic orbits are separated
Copyright © 2013 SciRes.

from each other, their spectra can be outraged, regardless,
at least, for any finite set of points. The modules from a
variety of periodic orbits a independent.
On the other hand, at least in some cases, the local
spectrum is a complete invariant with respect to a smooth
conjugacy. Different approaches to the problem of local
smooth conjugacy shown in [16].
For the global structure of orbits there are ways to
build structures of independent modules associated with
periodic orbits. In the case of infinitely many periodic
orbits, like expanding mapping and hyperbolic automorphism of the torus, there are infinitely many invariants of
the local 1 -equivalence. For these two cases, which
represent the simplest examples of hyperbolic systems,
the spectra of periodic orbits form a complete system of
invariants for 1 - and  -equivalences in the vicinity
of these maps, respectively. Satisfactory description of
the set of possible values for the eigen values of periodic
points remains an open problem.
A variety of modules offer a significant, although not
complete information about the smooth equivalence in
the vicinity of the rotation. Every real analytic compression stores a unique way an affine structure. For the discrete map , two structures identified near the ends of
the segment are found in the middle. The transition functions between the two structures in any of the fundamental domain  a,  a   generate infinite moduli space 
[16]. In practice, this can be interpreted to mean that
there are changes of coordinates that will  affine mapping from 0, –1  0   in the [0, a] and   а  ,1 in
the  2  а  ,1 . Coordinates are uniquely determined up
to two factors, one at each end. Then the map
  а  ,1   2  а  ,1 can be normalized so that
 a :  0,1   0,1 :

a 

  a  t   a   a     a 
 2  a    a 

,

which can be extended to the whole real line by the formula

 a T  k    a T   k , k   .
Thus, the two diffeomorphism 1 and 2, for which

1  0   2  0  0
are equivalent if

 2 T    1 T  s   1  s 
for some s in [0,1]. Constructed above the map transition
depends on the factors that determine the linearization 
in the vicinity of the endpoints, as well as the choice of
the reference point а as well. It is obvious that the change
does not alter the linearization map transition if a
amended accordingly. Changing the reference point leads
to the substitution mapping the transition to the equivaJMP

E. V. NIKULCHEV, O. V. KOZLOV

lent.
The notion of smooth equivalence in accordance with
[16] can be easily moved in case of continuous time.
Flow equivalence is a contingency flows as a differentiable actions of a group  of real numbers. The structure of the orbits of the flow, unlike the case of discrete-time systems, has two distinct aspects: 1) The relative behavior of points on different orbits and 2) evolution of the initial condition along the orbit over time.
There is a natural way to change the thread, keeping the
first aspect of the structure of its orbits, and it does not
change its orbit.
For the two streams  t and  t will define the time
change by using the vector fields of infinitesimal generator of the symmetry group:



d
dt

t

and  
t 0

d
dt







Copyright © 2013 SciRes.

3. Geometric Method for the Reduction of
Systems
According to [15] on centrally-stable manifold, the system around an equilibrium point O takes the form
y  By  f1  x, y, z  ,
z  Cz  f 2  x, y, z  ,

(1)

x  Ax   0  x, y, z  ,

.

From the uniqueness of solutions of differential equations, it follows that the zeros of the vector fields are
fixed points of the corresponding flow. Thus, we conclude that (х) = 0 if and only if (х) = 0. In addition, if x
is not a fixed point, then the tangent vector to the curve
 t  x  and  t  x  , does not vanish and have the
same direction.
It is natural to try to describe all the changes of time
given flow modulo trivial substitutions. This problem is
essentially equivalent to the problem of describing the
space of all sufficiently smooth positive functions up to
adding features that are derivatives of other smooth functions in the direction of flow.
It makes sense when determining equivalence to consider the conservation of structural stability for flows.
One way of determining is based on the equivalence of
all the perturbations [16]. Not being fully degenerate, this
requirement is rarely performed, for example, in the presence of periodic orbits of periods are modules in this
sense. We assume that the local topological equivalence
should preserve the structural stability, thus transforming
the homeomorphism can be quite close to the identity for
small perturbations.
For all the above concepts in this section of the compact phase spaces is immaterial. In addition, a natural
way to modify these definitions for the cases where for
some points of the dynamical system is defined only for
a finite time interval. Such a generalization leads to the
concepts of local and semilocal structural stability.
For any contraction phase space can not have a smooth
structure, so that these concepts are not directly applicable. However, the contraction mapping of a small disc in
the Euclidean space, structural stability, as well as a hyperbolic linear map in the vicinity of the fixed point.
Symmetry rotation is not structurally stable. Since the
topological conjugacy preserves the periodic orbits, and



the transformation of rotation by an irrational angle can
not be associated with the transformation of rotation, for
which the corresponding number is rational. But since
the rational numbers, and the set of irrational numbers
are dense, then some arbitrarily small perturbations of the
turn to rational angle you can find the rotation by an irrational angle, and vice versa.

t

t 0

1383

where

 j  1, m  ;
 0,  j  m  1, k  ;
 0,  j  k  1, n  ;

x  m , y  k , z  nmk
spectr A  m 1 , , k  , Re  j
spectr C  k 1 , , n  , Re  j

r -functions f1, f2 and  0 together with its first derivatives vanish at the origin.
In this case, the right side of the system may depend
on the control u either continuously (in this case, smooth
manifolds, discussed below, depend continuously on u),
or smooth. In the latter case, u is included in the number
of “central” variables x and, thus, further investigated the
variety and the foliation will have the smoothness of u,
equal to the smoothness of x.
Theorem 1. In a small neighborhood of the equilibrium state there exists a (m + k)-dimensional center-stable invariant manifold. Central-stable manifold is
not uniquely defined, but for any two manifolds W1sC
and W2sC functions 1sC and 2sC the class r that
contains the point O and concerns at this point subspace
{z = 0}, define the same the same symmetries group of
the point O.
Proof. Without loss of generality, we consider the second-order system. Let the system defined on a manifold
gives rise to a one-parameter transformation group generated by the operator

A  1  x 



 2  x 
,
x1
x2

with infinitesimal operator
X  1  x 



 2  x 
.
x1
x2

We need to show that the operator
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E. V. NIKULCHEV, O. V. KOZLOV

1384



 2  x, 
A  1  x, 
.
x1
x2

Similarly,

system defined by W
an equivalent ( - group option).
We write the transformation of the group as a Li series
(the operator exponential):

 2 A
  A, X  , X  .
 2   0 

sC
2

x1  e X x1 ,
x1  e X x1,
x2  e X x2 .

We represent A in the previous coordinates. For this
we compute:

 


x  x  x 

 A e  X x1





 Ax
2
x1



x  x  x 



x  x  x 


x2


 A e  X x2
x1





x  x  x 


x2

,

it follows that

 A e
 A e

 X

 X


x 
x1

2

x  x  x 
x  x  x 

 1  x  ,
 2  x  .

Since 1,2 do not depend on , as determined by the
operator X, we have
d   X
Ae x1  0 ,
d





so
A  X
 e  X x  XA e  X x  0 .
e x1  AX
1
1


A similar formula holds for the second coordinate,
which together form a differential equation:
A 
 AX  XA   A , X 

with initial condition
A  x , 

 0

 A  x  .

Solution of the resulting Cauchy problem can be obtained by expanding the operator A  x ,  in a Taylor
series in powers of :
A
 2  2 A

 ,
A  x ,   A  x   
   0 2!  2   0

thus, we have:
A
  A, X  .
   0
Copyright © 2013 SciRes.


A  A    A, X    A, X  , X  , .
2!
2

By Theorem from [15] function 1sC and 2sC determine the same expansion in Taylor series, which suggests field commutations:

x2  e X x2 ,


A  Ax
1

Finally, a number will look like in the Hausdorff form:

 A, X   0 ,
i.e.
A  A.

Thus, the theorem is proved.
At the reference time t  –t of A, B and C go into respectively A, B and C. Thus, part of the spectrum of
characteristic exponents corresponding to the variable z,
is now left of the imaginary axis, and part of the spectrum, corresponding to the variable y—the right of it. To
the system obtained from system (1) by time reversal, we
can again apply the theorem about the Central stable
manifold and get the following theorem on center-unstable manifold.
Theorem 2. In a small neighborhood of the equilibrium state О there exists an (n – k)-dimensional invariant
manifold is a  r -smooth. Center unstable manifold includes all trajectories that remain in a small neighborhood of O for all negative values of time. For any two
W1usC and W2usC functions 1usC and 2usC , the class
 r containing the point O and tangent at this point subspace {y = 0} define the same symmetries group.
Intersection of center-stable and unstable manifolds of
a central  r -smooth m-dimensional invariant center
manifold, defined by the equation of the form (y, z) =
C(x). Function C with all derivatives, in particular, the
Taylor expansion of functions C at O is uniquely determined by the system.
Straightening of the central stable and center-unstable
manifolds, as well as rectification of the strong stable and
strong unstable invariant foliations on these manifolds
leads to the good fact.
Theorem 3 [15]. With a  r 1 -smooth transformation
of the system (1) can locally be reduced to
y   A  F1  x, y, z   y,
z   C  F2  x, y, z   z ,
x  Bx   0  x   1  x, y, z  y   2  x, y, z  z,

where  0 —is a  r -smooth function, which together
with its first derivative vanishes at x = 0; F1,2 are functions that vanish at the origin:
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E. V. NIKULCHEV, O. V. KOZLOV

1,2   r 1 ; 1  x, y, 0    2  x, 0, z   0 .

Here, the local center-unstable manifold given by the
equation {y = 0}, the local center-stable manifold—{z =
0}, and a local center manifold—{y = 0, z = 0}. Strong
stable foliation consists of surfaces {x = const, z = 0},
and the layers are strongly unstable foliation of the form
{x = const, y = 0}.
A similar theory is constructed for discrete systems.
For systems that admit the group of symmetries and
construct models of systems, reduced to a central invariant manifold results can be summarized as follows [17].
In the local region of the qualitative dynamic behavior
of systems typologically equivalent system, reduced to
the central manifold
x  Ax  t    0  x  ,

(2)

where  0  x  —is a r -smooth function

   x      x   0, x
0

0

0

0

0

 0 ,

whose structure is determined by the symmetry transformation.
For a discrete system, the local center manifold is determined by the system:
x  t  1  Ax  t   0  x  ,

(3)

where 0  x  —is a r -smooth function

  x      x   0, x
0

0

0

0

0

 0

defined on the basis of symmetry groups constructed on
the reconstructed attractor.
For the reconstruction of a nonlinear system in the
forms (2), (3) proposed the allocation of local regions of
phase trajectories 1  x  and 2  x  that are close to
periodic, and the construction of finite-transformations
transform one area to another. That is, the construction of
the symmetry group of phase trajectories, which is characterized by the transformation of graphs:
graph 1  x   graph 2  x  .

The resulting transformations determine the structure
of the desired evolution equations.

4. Algorithms and Applications
The result of reconstruction algorithms [1] is a set of
points belonging to the attractor of the system. The task
is an automatic search for symmetries of the local sections of the phase trajectories. We seek the symmetry of
translation, rotation, stretching and compression.
You must select the initial sequence collection sites so
that when reducing them to a single scale, position and
angle of rotation, they would be most similar to each
other, as well as numerical indicators of transformations
Copyright © 2013 SciRes.

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taking one piece to another without breaking of symmetry, to give a numerical estimate degree of symmetry
breaking. The Figure 1 shows a schematic illustration of
the problem.
The task of allocating a set of similar fragments in the
circuit is defined as follows:
Input: sequence of n points v1 , v2 , , vn , the function Dist V1 , V2  , which gives an assessment of similarity between any two fragments of the original sequence
Vi  v j , v j 1 , , v j  p .
Output: a set of disjoint fragments of the original sequence S  V1 , V2 , , Vk  , such that for any other set of
fragments S   V1 , V2 , , Vh  of the condition:

 k
1
1
   Vi  
S   S : max 
Vi,V j S   1  Dist V , V  
 i j   i 1 m


 h
1
1
   Vi 
 max 
Vi ,V j S  1  Dist V , V

m
 i j   i 1


Number of different fragments, which can be identified in the original sequence:
m

1
n  n  1
2

Number of arbitrary sets of fragments, determined by
the size of the set of all subsets of the set V1 , V2 , , Vm :
u    m   2m

Based on this challenge

 .

T  n    2n

2

Marking. In the simplest case for the selection of fragments of trajectories satisfying the decision to consider
the fragments located between all possible pairs of contour points. In most problems, this approach is not possible because of limited computer resources. So it makes
sense to consider as a start / end fragment only some
points of the contour, for example, extremes of functions
of each of the coordinates and evenly distributed along
the length of the markers on the slowly varying parts
(Figure 2). This approach provides a sufficient rational
partition of the circuit, although it suffers from some
redundancy.
Partitioning algorithm can be anything important to
exclude monotonous plots and identify the markers
characteristic sections of the circuit. Extracting fragments.
To isolate fragments of contour to study the simplest case,
you can use brute force pairs of markers. In this case, the
number of fragments needed to be addressed:
Fполн 

n2  n
,
2
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E. V. NIKULCHEV, O. V. KOZLOV

1386

Figure 1. Schematic illustration of the problem.

of points, the distance between the maximum in the current space projection.
Rotation of the fragment so that its axis were parallel
to the coordinate axes.
The construction restricting fragment of the n-dimensional parallelepiped and the shift of the fragment so that
the center of the parallelepiped coincides with the center
coordinates.
Scaling the fragment so that the length of its main axis
is equal to 1.
Fragment of an n-dimensional the contour F consisting
of m real points, represented as a matrix m  n :
Fb   X 1

X2  Xn  ,
T

where X i   xi ,1 , xi ,2 , , xi , m  .
For a compact description of the transformation fragment
as a single matrix, we give it to homogeneous coordinates using matrix transformations, obtain:
Fc   X 1 X 2  X n 1
In the normalization process uses three types of transformations: translation, scaling and rotation.
Translation transformation in a matrix form is:

Figure 2. Example of marking.

where n — number of markers on the contour.
It makes sense to limit the length of a fragment in the
original contour points as the top and bottom as well as
fragments, the length of the approaching to the original
contour, are not of interest to identify patterns of distribution symmetry, and too small fragments do not reflect
the specific behavior of the system. In addition, the introduction of restrictions on the length of the above increase the number of fragments for consideration is
linear, as opposed to quadratic, with the full brute force.
Selected fragments for the possibility of a comparison
between a need to interpolate in a way that they all described the same fixed number of points (Figure 3).
Normalization. After reducing to one the number of
points each of the fragments should be subjected to the
normalization procedure. The aim of the procedure is to
convert the fragment into a descriptor—the image that is
invariant with respect to the transport, rotation, and scaling the original fragment, and obtaining the numerical
parameters of this transformation (Figure 4).
The proposed normalization procedure is a sequence
of the following:
Isolation in a fragment of the so-called “axis”—pairs
Copyright © 2013 SciRes.

Figure 3. Interpolation of selected fragments.
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Figure 4. Three initial symmetrical 3-dimensional fragment (top) and their total descriptor (bottom).

M sh

 1
 0

 

 0
  gsh,1


0
1






0


1
  gsh,n

 gsh,2

0
0


0
0 
 ,

0
1 

where g rt —angle, a significant elements of the matrix
are located in cells (i, i), (i, j), (j, i) и (j, j).
(4)

1) Initialize the total transformation matrix of normalization:

where gsh,i  i  1, , n  —the shift along the axis X i .
Transformation of the scaling down and scaling up
 gsc
0

M sc   

0
 0

0  0
gsc  0
  
0  gsc
0  0

0
0 
 ,

0
1 

M norm

(5)

Copyright © 2013 SciRes.

(6)

1
0



0

0  0
1  0 
.
  

0  1

2) Setting the order of normalization k  1 .
3) F k  Fr .
4) Finding the fragment Ak —
 xi ,1
Ak  
 xi ,1

where gsc —a scaling factor.
Rotation transformation is:
0 0


1 0
0 
0
0 


cos  g rt  sin  g rt 

M rt  
,
 sin  g rt  cos  g rt 


0 0
 0


0 1 


0 0

Normalization Algorithm Is as Follows

xi ,2  xi , n 1
x j ,2  x j , n 1

the largest segment connecting two points of the loop
F k in space. With a simple enumeration pairs of points
fragment of F k sought a pair with the greatest distance:
max

n

  xd , j  xd ,i 

2

d k

5) Transfer A

k

so that it coincided with the first
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E. V. NIKULCHEV, O. V. KOZLOV

1388

point of origin. Conversion is performed by multiplying
with the matrix translation M sh,k form (4) with coefficients
gsh,i  Ak 1, i  :
Ashk  Ak  M sh, k .

6) Complement the overall normalization of the current transformation matrix:
M norm  M norm  M sh,k .

7) Rotation Ashk so that it coincided with the direction
of the axis X k .
a) Initialize rotation matrix:

M rt, k

1
0



0

10) Installation of the next order of normalization
k  k 1 .
11) If k  n , then go to step {4}.
12) Formation of the bounding parallelepiped fragment B is described by the matrix 2  n :
 b1,1 b1,2  b1, n 
B
,
b2,1 b2,2  b2, n 
b1,i  min  X i  , b2,i  max  X i  .

13) Transfer fragment so that the center of the bounding parallelepiped it coincided with the origin. Conversion is performed by multiplying with the translation
matrix M sh form (6) with coefficients
gsh,i    B 1, i   B  2, i   2 :

0  0
1  0 
  

0  1

Fcn  F k  M sc .

14) Scaling of the fragment in a way that restricts its
parallelepiped fit into the unit cube. This conversion
scaling matrix M sc form (5) with a coefficient

b) Setting the plane of rotation p  n .
Art,k p  Ashk

gsc  B  2, i   B 1, i  :
k
rt, p

c) Calculating the angle between the projection A
on the plane OX 1 X p and the axis X 1 of the formula:

k , p


 Ak  2, k  
arctan  k
 ,

 A  2, p  

 Ak  2, k   


arctan
 k
  ,


 A  2, p   2

 Ak  2, k  

arctan
 k
  ,

 A  2, p  


 k

 arctan  A  2, k    3 ,
 Ak  2, p   2





Ak  2, k   0, Ak  2, p   0;
A  2, k   0, A  2, p   0;
k

Ak  2, k   0, Ak  2, p   0;
Ak  2, k   0, Ak  2, p   0.

g rt  k , p :
Art,k p 1  Art,k p  M rt, p .

e) Complement the current rotation matrix transformation:
M rt, k  M rt, k  M rt, p .

f) Setting the next plane of rotation p  p  1 .
g) If p  1 , then go to step {d}.
8) Complement the overall normalization of the current transformation matrix:
M norm  M norm  M rt, k .

9) The transformation of the fragment by the current
matrix of normalization:
F

Copyright © 2013 SciRes.

 F  M norm
k

15) Complement the overall normalization of the current transformation matrix:
M norm  M norm  M cn  M sc .

k

d) Rotation Art,k p with a matrix form (6) by an angle

k 1

Fnorm  Fcn  M sc

End.
The result of the normalization procedure is the descriptor the fragment Fnorm , the matrix M norm is converted into a fragment of the descriptor and a set of indicators and all the intermediate transformation matrices:
n  1 matrices
M sh, k ,

n

2



n 2

matrices M rt, p and one matrix M sc .
In what is easily obtain the transformation matrix of
one fragment to another:
1
M transAB  М normA  M norm
B.

Evaluation of symmetry breaking. After receiving the
handles of the two supposedly symmetrical fragments
FA and FB can give a numerical estimate of divergence between them DAB , for it is proposed to use the
properties of Fourier series. Also, this indicator can be
used as an estimate of the degree of symmetry breakdown.
Successively translate the descriptor in both frequency
representation, using the discrete Fourier transform for
each point:
m

sd , k   xd , p e

2 i
 k 1 p 1
m

,1 d  n .

p 1

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E. V. NIKULCHEV, O. V. KOZLOV

For an n-dimensional the descriptor in this case applies
n-dimensional Fourier transforms independently for each
coordinate. Result of the transformation will range following structure:
Sd   sd ,0



sd ,1  sd , m  , 1  d  n .





Couples
 S1 Sn 1  ,  S2 Sn  2  , , S n 1 2 Sn 2
are complex conjugate numbers. The transition from the
spectrum to the original contour by using the inverse
discrete Fourier transform of the form:
2 i

xd , k 

 k 1 p 1
1 m
,1 d  n .
 sd , p e m
m p 1

For the purpose of comparing the contours of the Fourier transformation is convenient because it provides a
selection of principal components. The closer a pair of
conjugate elements in the middle of the spectrum, the
smaller, high frequency, especially circuit it describes.
Each of the pairs of conjugate elements of the spectrum
determines when restoring an ellipse, and all recovered
shape is a superposition of these ellipses. Each of the
next pair will be added to restore the image all the more
finer details of the original contour. The first, odd element of the spectrum indicates the position of the center
of the figure for comparison; the contours are not being
used.
To calculate the distance between the fragments is proposed to summarize the relevant elements of the divergence of the spectra of these contours, after Fourier
transformation. Higher-frequency pair (close to the center of the spectrum), the elements of the spectrum have
less impact on the indicator closeness, than the low frequency.
Corresponding criterion is:
q 
D A, B    

i 1 


n

n





j 1

j 1





 I 2   R 2   i 



R   Re  S
  Re  S  

I  Im  S B ,i , j   Im  S A,i , j  ,
B ,i , j

A, i , j

where q—the number of conjugate pairs of elements of
the spectrum, n—the dimension of phase space;  i —
discount factors  i  1, , q  , which determine the degree of importance of components of the spectrum of
frequencies—the relevant elements of the spectrum contours A and B. Coefficients  i are chosen empirically,
based on the conditions of the problem.
Once the fragments have been normalized circuit,
must choose from a variety of sequences of one the most
satisfying the requirements of the problem—containing
the most balanced between fragments. Number sequences that can be composed of non-overlapping fragments, even for small problems with the number of
Copyright © 2013 SciRes.

1389

markers about 100 very large, so it is proposed to solve
by using a genetic algorithm.

5. Examples
5.1. Detection of Symmetries for a Model
Example
Source data set consists of 300 points. These are the result of Ressler system simulation
x1    x2  x3  ,
x2  x1  0.2 x2 ,

x3  0.2  x3  x1  2.6  .

The observed parameter x3 reconstructed attractor
(Figure 5).
Attractor has been subjected to alignment markers.
The total number of markers: 144. After labeling of the
10238 possible paths have been selected 1375, the others
did not satisfy the restrictions in length: from 5 to 50
points of the original contour. Each segment was interpolated 60-th evenly spaced points along its length and
subjected to the normalization procedure. After that was
constructed adjacency matrix fragments (Figure 6). Been
carried out selection of solutions by using genetic algorithm (population size 200, the probability of mutation
rates   0.3 and   0.1 , limiting the number of
steps lengths: 5, an elite selection of a new generation,
the type of parental choice—outbreeding, unique individual in the population) and selected the winning solution.
The time of the preprocessing step: 3 min. 27 sec.
Time step of genetic selection decisions: 4 min. 36 sec.,
the population converged 5 times for 533 iterations, the
decision of the winner: the similarity of the fragments
0.3917, length 197.
The results are shown in Figure 7. Part of the trajectory for which were not symmetrical parts shown dotted.

5.2. Reconstruction of the Model to
Experimental Data
Original data set consists of 200 points. The data represent the traffic network. Smoothing was not applied. The
reconstructed attractor has been subjected to alignment
markers. The total number of markers: 192. After labeling of the 18153 possible paths have been selected 2633,
the others did not satisfy the restrictions in length: from
15 to 30 points of the original contour. Each segment was
interpolated 60-th evenly spaced points along its length
and subjected to the normalization procedure in Figure
8.
Was carried out the evolutionary selection of solutions
(population size 200, the probability of mutation rates
and limiting the number of steps lengths: 8, an elite seJMP

E. V. NIKULCHEV, O. V. KOZLOV

1390

Figure 5. Attractor.

Figure 7. Decision-winner in the time series (top) and in the
attractor (botom).

Figure 6. Attractor with markers.

lection of a new generation, the type of parental choice—
outbreeding, unique individual in the population) and
selected the winning solution. The time of the preprocessing stage: 12 min. 43 sec. Time step of genetic selection decisions: 6 min. 11 sec., The population converged
8 times for 652 iterations, the decision of the winner: the
similarity of fragments of 0.576, length 125. The results
are shown in Figure 9.
The presence of the symmetry of the system determined the structure of the form (3). Parameter identification system using the method of least squares, gives the
following result:
Copyright © 2013 SciRes.

Figure 8. Labeled attractor.
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(a)

Figure 10. Comparison of dynamics.

sion can be close enough to the identity for small perturbations.

6. Conclusion
The use of symmetric properties of evolution equations
can be successfully used for the reconstruction of the
system. Developing a constructive method for structural
identification systems with chaotic dynamics based on
geometric analysis of the experimentally obtained phase
portrait of the system.

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Figure 9. Decision-winner at baseline (a) and the attractor
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 0.9413 0.1805 0.1164 0.0295 
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A
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(b)

 0.0399 
 0.0463 
 exp t 0.0001 sin t 0.4
0  
 0.4848


 0.1851

 

C  104  2.1037

 0.0124

   ,

0.1202  0.0302 .

Figure 10 shows a comparison of the dynamics of the
original system and the reconstructed model.
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should preserve the structural stability, and the converCopyright © 2013 SciRes.

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