Transformation of the generalized chaotic system into canonical form

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.


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 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.

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 where are state variables of dynamical object,   i j x f are some nonlinear functions.
We assume that these functions are differentiable in all state variables i x for n times.This assumption allows us to transform (1) into canonical form  , where i y  are new state variables,   i n y g are nonlinear functions.
One can perform the above mentioned nonlinear coordinate transformation by using the following algorithm: where k is the number of output variable.
2. This variable is differentiated for n times and Lie derivatives are defined [10]: where f is an (n x 1)-size matrix of functions   3. The interrelations between new i y and old i x state variables are defined as solution the first n-1 equations of (4) for i x thus where   i y A is some nonlinear operator.

 
i n y g is defined from n-th equation of ( 4) by substituting into the Lie derivative 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   i y A -operator.This operator in the elementary functions can be defined only for the short range right-hand expressions in (1).The determination of the   i y A -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.
We propose to simplify the determination of the   i y A -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]: where   i x is the value of state space variable x in i-th time interval,   is the value of state variable x on q-th time interval in the future, and   is the value of state x on w-th interval in the past; or in z-form: where 1  z  is the one step backward shift operator and 1 z 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: One can rewrite (4) by using ( 9) thus Solution of (10) allows us to determine interrelations between the new coordinate 1 y and old one i x .We propose to use for solution of these equations iterative numerical methods like Newton-Raphson method [14].This method allows us to write down the following iterative expression for state variables: where

 
i n y g can be defined by substituting (11) into Lie derivative k n x L f .This function is used while we are making the transformation of the differential equations (1) into algebraic ones: Numerical solution of (12) allows us to define canonical state variables j y 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]: where are some coefficients and i x are state variables.
Equations ( 12) describe nonlinear objects with chaotic dynamic.Let us transform (12) into the classical matrix form; where We consider transformations of (12) into the canonical form for 1 x state variables.

A. Analytical Transformation of The Lorenz Equations for x 1 Variable
After selecting 1 x variable as output, we use i y as new state variables.The new state variables i y are defined as Lie derivatives of the output variable We solve (18) for the variables 2 x and 3 x Now let us find the 3-rd Lie derivative for variables We define an unknown function   Finally, we can write down the Lorenz system's dynamic in canonical form: It is simple to transform 3-rd order system of differential equations ( 22) into one 3-rd order equation 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.
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 x 1 Variable
We define the following functions as in (24).
Let us transform (24) into discrete-time domain by using the simplest backward difference approximation of the differential operator: where T is the sample time, as follows: At first, we define 2 x variable by using the following iterative algorithm based on Newton-Raphson method (28).
This algorithm can be simplified as follows: At last, we define 3 x variable by using similar procedure to (29) algorithm: Equations ( 29)-(30) allows us to write down the following iterative canonical equations for the Lorenz system given in discrete-time domain: Equations ( 31) and (32) are simpler than (22).These equations allow us to define both canonical i y and normal i x variables by solving the appropriate algebraic equations by using the following algorithm: 1. Current values of canonical variables 1 y and 2 y are defined by using the first and second expressions of (31).

Current values of normal variables 2
x and 3 x are defined by using (32) in iterative way.
3. Current value of canonical variable 3 y is defined by using the third equation of (31).
4. The cycle is repeated for all simulation time.
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).
We claim following as the result of all given mathematical expressions:

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.

Fig. 4 Fig. 3 Statement 3 :
Fig.4 Results of the canonical Lorenz system simulation with sample time sec 10 T 4  