Generating a New Hyperchaotic Pan System Via State Feedback Control

This paper proposes a new four-dimensional continuous autonomous hyperchaotic system based on the 3D Pan system by introducing a nonlinear state feedback controller. Dynamical behaviors of the new system are analyzed, both theoretically and numerically, including equilibrium points, Lyapunov exponents spectrum stability and bifurcation , finally, an illustrative example is given.

controller, and we investigated some basic properties and behaviors for this system numerically and analytically, including symmetry and invariance, dissipative and existence of attracter, equilibrium points, Lyapunov exponents spectrum, stability and bifurcation .

2-System Description
Pan system or L u  -like system [5] is found by L. Pan, D. Xu and W. Zhou in 2010 which is similar to the Lorenz system , but it is not topological equivalent with the Lorenz system [4]. And this system provides another interesting framework for advanced control techniques since it is more complex than the Lorenz system and Chua system. Moreover, it is more difficult to control the Pan system than the other than those already known chaotic systems due to the rapid change of the velocity in the z-direction [4].
The mathematical model of Pan system is a system of nonlinear ordinary differential equations which has the form: . The Lyapunov exponents spectrum and Phase portraits of the system (1) are shown in Fig.1 and Fig.2 respectively [5].
The Lyapunov dimension of system (1) is given as follows [5]:

3-Helping Results:
The first two remarks give us information about the roots of a polynomial and can be applied to the characteristic polynomial of a matrix.

Remark 1 (Coefficient Test) [1]:
Suppose that: where the coefficients are real. If any coefficient of ) ( f is either zero or negative, then, at least, one root has a nonnegative real part.

Remark 2 (Routh-Hurwitz Test) [1]:
All the roots of the indicated polynomial have negative real parts precisely when the given conditions are met. is negative (positive). Then, at least, one eigenvalue of 1 A has a negative (positive) real part. If the trace is zero, then either all eigenvalues have zero real parts, or there is a pair of eigenvalues whose real parts have opposite signs. 1) The coefficient test does not apply to the polynomial with a positive coefficient.
2) The Routh -Hurwitz test applies to polynomials of degree no more than four.
3) The trace test applies directly to matrix, so there is no need to find the characteristic polynomial. Finally, the Routh-Hurwitz test is the best from the coefficient test and trace test.

Remark 4 (Generating Hyperchaos) [2, 5, 6, 7, 8]:
To generate hyperchaos from the dissipatively autonomously polynomial systems by using a state feedback controller, the state equation must satisfy the following two basic conditions: The minimal dimension of the phase space of an autonomous system is at least four. The number of terms in the coupled equations giving rise to instability is at least two, of which, at least, one has a nonlinear function .

Remark 5 (Lyapunov Exponents) [2, 6]:
Assume that the Lyapunov exponents for a simple four system has an equilibrium point. In the context of ordinary differential equations ODEs, the word "Bifurcation" has come to mean any marked change in the structure of the orbits of a system (usually nonlinear) as a parameter passes through a critical value [1].

Remark 6 (Hopf Bifurcation) [9]:
Any system has a Hopf bifurcation if the following conditions are satisfied: 1-The Jacobian matrix has two purely imaginary roots and no other roots with zero real parts.

Remark 7 (Generating Hyperchaotic Pan system) [5]:
Hyperchaotic Pan system is generated by introducing an additional state u to the first equation of system (1). Then, we get the following four-dimensional hyperchaotic system: are constant parameters. This system is called hyperchaotic Pan  . The Lyapunov exponents spectrum and Phase portraits of the system (7) are shown in Fig.3 and Fig.4 respectively [5].

4-Main results:
Based on Pan system and remark 4,we can construct a new four dimension hyperchaotic system by introducing a state feedback controller, as follows: Add a nonlinear controller u to the second equation of system (1), let dy xz u + =  , then we obtain a new hyperchaotic system are constant parameters. For simplification, system (8) is called a new hyperchaotic Pan system in this paper.
In the following, we briefly describe some dynamical behaviors of the new hyperchaotic system (8).

4.1-Symmetry:
System (8) has a natural symmetry under the coordinates which persists for all values of the system parameters. This means that system (8) is symmetric about the z-axis.

4.2-Dissipative and existence of attracter:
Therefore, to make system(8) be dissipative, it is required that 0 b a  + .

4.3-Equilibrium Points:
In order to obtain the equilibrium points of system (8), , and we can obtain the following expressions (10) From the above equations, we obtain three equilibrium points ) the Lyapunov exponents spectrum and Phase portraits of the system (8) are shown in Fig. 5 and Fig. 6 respectively.
So, we can obtain the Lyapunov dimension of the new hyperchaotic system (8), it is described as  has the following cases: (8)  , the proof is completed. Due to the system is invariant under the transformation, so one only needs to consider the stability of any one of the both. The stability of the system (8) at equilibrium point 1 p is analyzed in this paper. Theorem 2: The solution of system (8) at the equilibrium point 1 p is always unstable. Proof: Now, to find Jacobian matrix at 1 p we need the following transformation under the linear transformation ) , , , The equilibrium point 1 p of the system (8)   . In this case, the solutions of equation (12)  , In the following, we will prove that the system (8)  Proof: , the real part and imaginary part of the ) ( 0 c c  respectively are: then either all roots have zero real parts, or there is a pair of roots whose real parts have opposite signs for all equilibrium points , then, at least, one root has a nonnegative real part; consequently where Corollary 3: System (8) has at least one root with nonnegative real part at equilibrium point 1 p if one of the following cases is satisfied:

…(21)
Proof: By coefficient test (remark 1), when one coefficient of equation (17) , then, at least, one root has a nonnegative real part. consequently where

-Illustrative Example:
Example: Investigate the stability and Hopf bifurcation at the equilibrium point

6-Conclusion:
This paper presents a new four dimensional hyperchaotic system, which is called a new hyperchaotic Pan system. This new hyperchaotic system is different from the hyperchaotic Lorenz system, hyperchaotic Lü system, another hyperchaotic Lü system as well as hyperchaotic system which was proposed by Pan in Ref [5]. Since, the new hyperchaotic system has more complex dynamical behaviors than the normal chaotic systems, it is believed that the system will have broad applications in various chaosbased information systems.