Investigation of the Quaternion Dynamical System

The quaternion Mandelbrot set is one of the most important sets in mathematics. In this paper we first give some properties of the quaternion algebra. Then, we introduce the quternion dynamical system. We are concerned with analytical and numerical investigation of the quaternion dynamical system.


Introduction
The well-known concept of quaternions was introduced by Hamilton in 1866 [1].Quaternions are the generalization of complex numbers.A quaternion has four components, i.e., one real part and three imaginary parts: , r i j k q q q i q j q k = + + + where , , i j k obey the rules as below: Every quaternion q has its conjugate, r i j k q q q i q j q k * = − − − .From (1), we can find that the multiplication rule of quaternions is not commutative.
Quaternion algebra is an excellent tool for the treatment of three-and fourdimensional rotations.Therefore studying quaternion is an important task.Moreover it plays an important role in areas of computer graphics, biomechanics, molecular modeling and mesh deformation.
The topic of Mandelbrot dynamical system [2] was studied by many authors [3] [4] [5], but still interesting.In fact, we found that there is an strong relation between the well known Mandelbrot dynamical system and the new one, the so called quaternion dynamical system.
In the following sections, we give an overview of Mandelbrot dynamical system and study quaternion dynamical system.

Mandelbort Dynamical System
The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape.Images of the Mandelbrot set are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it.More precisely, the Mandelbrot set is the set of values of C in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial remains bounded.That is, a complex number C is part of the Mandelbrot set if, when starting with 0 0 z = and applying the iteration repeatedly, the absolute value of remains bounded however large n gets.More Precisely given this be- ginning value 0 0 z = , and a value for C , one can generate 1 z .Then, using this value of 1 z , one can generate 2 z , and so on.The Equation ( 2) in itself is very simple but the resulting graph is so complicated.A computer tool is the most powerful tool we have to work with this system due to its fast computational capabilities.This dynamical system was studied by many other authors, see e.g.[2] [6].

Quaternions
This section is concerned with some algebraic properties of the non-commutative division quaternion algebra.Let  and  denote the field of the real and complex numbers respectively.Let  be a four-dimensional real algebra generated by the identity element e and the symbols , i j and k .So { } Quaternions are added together component by component, and quaternion multiplication is given by the quaternion relations (Hamiltonian conditions), given in (1).
If x and y are any (real) scalars, while u , v are any two of , , , e i j k , then the product ( )( ) au bv is defined as ( )( ) ab uv , see [5].

Some Properties of Quaternion Algebra 
(A) For any 2. The complex part of x , x − = .Then the algebra  is a division algebra.

The product of two quaternions
) ( ) xy x x i x j x k y y i y j x k x y x y x y x y x y x y x y x y i x y x y x y x y j x y x y x y x y k Hence  is a non-commutative.(B) Every quaternion q can be uniquely expressed as q z z j = + , where 1 z and 2 z are complex numbers by the maping x .
 is a real division algebra: pq p q = for all , p q ∈  .

Quaternion Dynamical System
In this section, we study a new dynamical system, the so called quaternion dynamical system.Define the following quaternion dynamical system , is quaternion constant.
Proposition 1.The orbit of system (3) is either bounded by 1 C + or unbounded. Proof.
, then the trajectory generated by ( 3) is unbounded.Hence quaternion Mandelbrot set is bounded by Hence the trajectory is unbounded.

Since
( ) > .Using part one, hence ( ) This completes the proof of the proposition.

The Equilibrium Point and Its Stability
In this section we use the 2 × 2 Pauli matrices approach, in which a quaternion can be written in matrix form as: . , where, , , are the Pauli 2 ×2 matrices and the vector ( ) Following the analogy of the complex quadratic map [7], we introduce the quaternionic quadratic map , The corresponding system of (4) in terms .
The equilibrium point for Equation ( 4) is defined by the equation The solution of equilibrium is given by and it is stable if all the eigenvalues λ of the matrix J , where   the system converges to cycles and chaotic behavior.
Here we give the Lyapunov exponents for the quaternion dynamical system (3).

4 . 5 .
The conjugate of x , 1 2 3 o x x x x i x j x k * = = − − − .The norm of x , .And x is said to be a unit quaternion if its norm is 1.Hence we can see that each non zero qua-

Figure 2 .
Figure 2. The maximal Lyapunov exponent with respect to c0 and c1.
C) Two quaternions x and y are said to be similar if there exists a non- zero quaternion v such that1v xv y − = , this is written as x y ∼ .In other words x and y are similar if and only if there is a unit quaternion v such that 1 v xv y − = , and two similar quaternions have the same norm.It is easy to check that ∼ is an equivalence relation on the quaternions.We represent the equivalence class containing x by [ ]