International Journal of Advances in Intelligent Informatics
Vol. 3, No 3, November 2017, pp. 117-124

ISSN: 2442-6571

117

Transformation of the generalized chaotic system into
canonical form
Roman Voliansky
Electric Engineering Department, Dniprovsky State Technical University,
2, Dniprobudivska Str, Kamyanske, 51918, Ukraine
voliansky@ua.fm

ARTICLE INFO

ABSTRACT

Article history:
Received September 30, 2017
Revised October 11, 2017
Accepted October 11, 2017

The paper deals with the development of a numerical algorithm for
transforming a generalized chaotic system into its canonical form.
Such transformation allows us to simplify control algorithm for
chaotic system. This algorithm is defined by using Lie derivatives
for output variable and a solution of nonlinear equations. The use of
the proposed algorithm is one of the ways for discovering new
chaotic attractors. These attractors can be obtained by transformation
of known chaotic systems into various state spaces. Transformed
attractors depend on both parameters of chaotic system and sample
time of its discrete model.

Keywords:
Chaos
Dynamical system
Differential equations
Generalized chaotic system
Nonlinear coordinate transformation

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All rights reserved.

I. Introduction

The theory of chaotic systems is a fast-growing branch of the dynamic system theory. This
branch has a wide application in various spheres of human activities, such as robotic [1],
communication [2], cryptography [3], meteorology [4], economy or business application [5], and so
on. Great interest to the chaotic systems was caused by their unique properties. Microcontroller one
can use these sequences in various ways. For example, they can be used for setting up secure data
transmission, planning path of mobile robot, investigating exchange rate fluctuations. This list can
be continued for pages.
Wide ranges of applications of chaotic systems have caused a great number of its researches. One
can find a lot of papers on researches on dynamics and implementations of integer-order [1]–[3] and
fractional-order [6] chaotic systems in continuous-time and discrete-time domains. These researches
proposed the novel chaotic systems [7] and investigated existing ones [1]–[3] [6].
One of the directions of the chaotic systems theory is control of chaotic systems. So many
publications on chaos control [7][8] and chaos systems synchronization [2][3][9] can be found in
scientific press today. The great interest to chaos control is caused by the possibility to test novel
control algorithms for nonlinear unstable dynamical objects. If these algorithms work correctly for
chaotic systems, they will work for various industrial objects with stable dynamics likewise.
The feedback linearization [10] is one of the effective control technique for nonlinear controller
construction, but the main drawback of this linearization is the use of the object’s complete state
vector. This fact makes the researcher to set up and to use tons of different sensors. It is obvious that
the control system becomes more complex and difficult to configure.
To avoid this drawback, we propose to transform a chaotic system’s dynamic into a canonical
form. It allows us to use only one sensor in the control system feedback. The transformation of the
chaotic system into the canonical form is known only for one class of chaotic systems [11][12] and
it is hard to use it for another one.
In this paper, we propose to perform transformation of an arbitrary chaotic system into a
canonical form by using generalized approach based on differential geometry methods and nonlinear
algebraic equations’ solution. We suggest using numerical methods while the mentioned
DOI: http://dx.doi.org/10.26555/ijain.v3i3.113

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transformation is being performed. It avoids us to use complex mathematical apparatus and gives
numerical algorithms, which can be used as numerical routines while control system is being
programmed on microcontroller.
Our paper is organized as follows: firstly, we get a transformation procedure for a general
dynamical object given in the continuous-time domain. We then adapt the mentioned procedure for
discrete-time domain. Finally, we show usage of proposed approach for transformation continuoustime and discrete-time dynamics of Lorenz system into canonical form.
II. Method
A. Continuous-time Transformation Algorithm for A Generalized Dynamical Object

Let us consider a generalized n-th order continuous-time dynamical object given in the following
way
x j  f j xi  , i , j  1, , n ,

(1)

where xi , x j are state variables of dynamical object, f j xi  are some nonlinear functions.
We assume that these functions are differentiable in all state variables x i for n times. This
assumption allows us to transform (1) into canonical form
y j  y j 1 ;

j  1, , n  1

y n  g n  yi  ,

(2)

where y i are new state variables, g n  y i  are nonlinear functions.
One can perform the above mentioned nonlinear coordinate transformation by using the
following algorithm:
1. One state variable x k is selected as output variable
y 1  x k ; k  1, , n ,

(3)

where k is the number of output variable.
2. This variable is differentiated for n times and Lie derivatives are defined [10]:
y i 1  Lif x k ; i  1, , n ,

(4)

where f is an (n x 1)-size matrix of functions f j xi 
f   f 1 xi 

f 2 xi  

f n xi T .

(5)

3. The interrelations between new y i and old x i state variables are defined as solution the
first n-1 equations of (4) for x i thus
xi  A y i  , i  1, , n  1 ,

(6)

where A y i  is some nonlinear operator.
4. The unknown function g n  y i  is defined from n-th equation of (4) by substituting into
the Lie derivative Lnf x k (6).
The given algorithm allows us to get transformed equations of a nonlinear object given by (1)
into a canonical form. The main drawback of the proposed method is the difficulty in analytically
determining the A y i  -operator. This operator in the elementary functions can be defined only for the
short range right-hand expressions in (1). The determination of the A y i  -operator is associated with
the usage of non-elementary functions in general case. The definition of these functions is a separate
nontrivial scientific problem with a weak practical usage due to the usage of complex mathematical
apparatus.

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119

We propose to simplify the determination of the A y i  -operator by transition into discrete-time
domain and using numerical methods.
B. Discrete-time Transformation Algorithm for A Generalized Dynamical Object

The known numerical methods are based on various approximations of the differentiation
operator. These approximations are built on the basis of future, current, and past values of state
variables.
We use a following general approximation of differentiation operator [13]:
x  dx / dt  d xi  q , xi  q  1, , xi , xi  1, , xi  w ,

(7)

where xi  is the value of state space variable x in i-th time interval, xi  q  is the value of state
variable x on q-th time interval in the future, and xi  w is the value of state x on w-th interval in
the past; or in z-form:





x  d z q x , z q 1 x , , x , z 1 x , , z  w x ,

(8)

where z 1 is the one step backward shift operator and z 1 is the one step forward shift operator.
An approximation for j-th order differential operator can be written down by using (8) in the
following way:
x  j   d j z q x , z q 1 x , , x , z 1 x , , z  w x , 2q  j ,2 w  j .
(9)





One can rewrite (4) by using (9) thus





d i 1 z q y1 , z q 1 y1 , , y1 , z 1 y1 , , z  w y1  Lif x k ; i  1, n .

(10)

Solution of (10) allows us to determine interrelations between the new coordinate y 1 and old
one x i . We propose to use for solution of these equations iterative numerical methods like NewtonRaphson method [14]. This method allows us to write down the following iterative expression for
state variables:
xi  z 1 xi 

where





Fi
, i  1, , n ,
Fi



(11)



Fi z 1 xi , y1  d i 1 z q y1 , z q 1 y1 , , y1 , z 1 y1 , , z  w y1  Lif x k ; i  1, , n .

(10)

Function g n  y i  can be defined by substituting (11) into Lie derivative Lnf x k . This function is
used while we are making the transformation of the differential equations (1) into algebraic ones:


d z y , z


y , , y , z y , , z y   g  y .

d j z q y1 , z q 1 y1 , , y1 , z 1 y1 , , z  w y1  y j ; j  1, , n  1,
n

q

1

q 1

1

1

1

1

w

1

n

(12)

i

Numerical solution of (12) allows us to define canonical state variables y j in general case.
III. Results and Discussion

Now we show two examples of using a proposed approach to transform the differential equations
in normal form into canonical one.
We consider a well-known Lorenz system, which is given by the following equations [15]:
x 1  x1  x 2 ;
x 2  x1  x 2  x1 x 3 ;
x 3  x1 x 2   x 3 ,

where  , , are some coefficients and x i are state variables.
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Equations (12) describe nonlinear objects with chaotic dynamic. Let us transform (12) into the
classical matrix form;
  f X  ,
X

(13)

where
X   x1

x2

x3 T ;

f X    x1  x 2

x1  x 2  x1 x3

(14)

x1 x 2   x 3  .

We consider transformations of (12) into the canonical form for x1 state variables.
A. Analytical Transformation of The Lorenz Equations for x1 Variable

After selecting x1 variable as output, we use y i as new state variables. The new state variables
y i are defined as Lie derivatives of the output variable x 1
y 1  x1 ;
y 2  Lf x1 ;

(15)

y 3  L2f x1 ,

where
Lf x1  x1  x 2 ;

(16)

L2f x1   x1  x 2   x1  x 2  x1 x3 .

Let us substitute (16) into (15)
y 1  x1 ;
y 2  x1  x 2 ;

(17)

y 3   x1  x 2   x1  x 2  x1 x3

or
y 2  y1  x 2 ;

y 3    y1    1x 2  y1 x3

.

(18)

We solve (18) for the variables x 2 and x 3
y2
 y1 ;

.
1  y2
y3

x3    1   1  

  y1 y1


x2 

Now let us find the 3-rd Lie derivative for variables x1





(19)





L3f x1  -x12 x 2 +   + 2 + 1x3 - 2 -  2 -  x1 +  - x3  +  +  2 +  + 1 x 2 .

(20)

We define an unknown function g 3  y1 , y 2 , y 3  by substituting (19) into (20):
y 2  + 1 + y 2 y 3
.
g 3  y1 , y 2 , y 3   -y13 - y12 y 2 +  - 1 y1 - y 2   1 - y 3     1 + 2
y1

(21)

Finally, we can write down the Lorenz system’s dynamic in canonical form:
y 1  y 2 ;
y 2  y 3 ;
y 3  -y13 - y12 y 2 +  - 1 y1 - y 2   1 - y 3     1 +

(22)





y 22  + 1 + y 2 y 3
y1

.

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121

It is simple to transform 3-rd order system of differential equations (22) into one 3-rd order
equation
y 2  + 1 + y 1 y1
y1  -y13 - y12 y 1 +  - 1y1 - y 1   1 - y1     1 + 1
.
y1

(23)

We call the equation as Lorenz equation in the canonical form and the corresponding dynamical
system as a continuous-time canonical Lorenz system.
Analyzing (22)-(23) allows us to formulate the following statement:
Statement 1: Equations of nonlinear system’s dynamic in canonical form are more complex than
in normal one. Thus, contrary to linear systems, whose mathematical model is simpler in canonical
state space, the transformation of a nonlinear system into another state space does not allow us to
simplify it.

y1 , x1 , x1  y1

Numerical solutions of (12) (curve 1) and (22) (curve 2) are shown on Fig. 1.

2

3
1

t , sec
Fig. 1. Results of numerical sollution of (12) and (22) for x 1 variable.

The complete coincidence of the shown curves is clearly understood. This coincidence is
approved by near zero values of error curve 3. Thus, we can claim the correct performing of
transformation of the Lorenz equation into the canonical form by using the proposed approach.
The usage of the proposed approach ensures a coincidence of normal and canonical state spaces
by only one variable. That is why other variables are differing. This difference cause different
attractors in different state spaces. For example, a Lorenz attractor in the canonical state space and
its projections are shown on Fig. 2. It is clearly understood the significant difference between the
shown and well-known classical Lorenz attractors.
B. Numerical Transformation of The Lorenz Equations for x1 Variable

We define the following functions as in (24).
F1  y 1  y1  x 2 ;
.
F2  y1     y1    1x 2  y1 x3

(24)

Let us transform (24) into discrete-time domain by using the simplest backward difference
approximation of the differential operator:
d 1  z 1

;
dt
T
d2
dt 2



1  2 z 1  z 2

,

T2

where T is the sample time,

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ISSN: 2442-6571

Fig. 2. Lorenz attractor in canonical state space.

as follows:
y1  z 1 y1
 y1  x 2 ;
T
y  2 z 1 y1  z 2 y1
F2  1
     y1    1x 2  y1 x3
T2

F1 

(26)

or
1
1

F1      y1  z 1 y1  x 2 ;
T
T

 1

2 1
1 2
F2  
     y1 
z y1 
z y1    1x 2  y1 x3 .
2
2
T
T2
T


(27)

At first, we define x 2 variable by using the following iterative algorithm based on NewtonRaphson method (28).
1

z y1
1

 x 2
    y1 
T
T

x 2  z 1 x 2  
.


(28)

This algorithm can be simplified as follows:
1

z y1
1

z 1 x 2      y1 
T
T


x2 
.
2

At last, we define x 3 variable by using similar procedure to (29) algorithm:

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2 1
1 2
 1

 2      y1  2 z y1  2 z y1    1x 2
z x3  T
T
T

x3 

.
2
2y1
1

(30)

Equations (29)-(30) allows us to write down the following iterative canonical equations for the
Lorenz system given in discrete-time domain:
y1  z 1 y1  Ty 2 ;
y 2  z 1 y 2  Ty 3 ;
y3  z

1

y 3 - Ty12 x 2 + T

where



 + 2 + 1x - 2 -  - y + T  +  +  + 1 - x x ,
2

3

(31)

2

1

3

2



x 2  z 1 x 2  1 / T    y1  z 1 y1 / T / 2;
x3 

z





x3 1 / T      y1  2 / T 2 z 1 y1  1 / T 2 z 2 y1    1x 2

.
2
2y1

1

2

(32)

Equations (31) and (32) are simpler than (22). These equations allow us to define both canonical
y i and normal x i variables by solving the appropriate algebraic equations by using the following

algorithm:
1. Current values of canonical variables y 1 and y 2 are defined by using the first and
second expressions of (31).
2. Current values of normal variables x 2 and x 3 are defined by using (32) in iterative way.
3. Current value of canonical variable y 3 is defined by using the third equation of (31).
4. The cycle is repeated for all simulation time.
Similar to (23), we call equations (31)-(32) discrete-time Lorenz equations in canonical form.
It is clearly understood the simplicity of the proposed approach contrary to the solution of
differential equations (22). Equations (31)-(32) depend on the sample time T as well as coefficients
of equations (12). So, we claim the following statement:
Statement 2. The dynamic of the discrete-time Lorenz system in the canonical form depends not
only on its parameters but also on the used numerical method.
This statement is proved by the numerical solution results of (31) and (32) for different sample
time (Fig. 3-4).

Fig.3 Results of the canonical Lorenz system
simulation with sample time T  10 3 sec

Fig.4 Results of the canonical Lorenz system
simulation with sample time T  10 4 sec

We claim following as the result of all given mathematical expressions:

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Statement 3: If a dynamic system has a chaotic attractor in one state space, it has chaotic
dynamic in other state spaces.
IV. Conclusion

The dynamic of a generalized chaotic system can be transformed into canonical form by defining
n-th Lie derivatives and solving n-1 nonlinear algebraic equations. This transformation can be
simplified by using numerical methods. One can develop numerical transformation algorithm as a
part of controller software by using the mentioned numerical methods. The use of the proposed
algorithm is one way of new chaotic attractors’ discovering. These attractors can be obtained by
transformation of known chaotic systems into various state spaces.
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Roman Voliansky (Transformation of the generalized chaotic system into canonical form)