Abstract
We present a pseudospectral method application for solving the hyperchaotic complex systems. The proposed method, called the multistage spectral relaxation method (MSRM) is based on a technique of extending Gauss-Seidel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudospectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous hyperchaotic complex systems such as hyperchaotic complex Lorenz system and the complex permanent magnet synchronous motor. We compare this approach to the Runge-Kutta based ode45 solver to show that the MSRM gives accurate results.
1. Introduction
Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions and have complex and highly unpredictable profiles [1, 2]. Chaotic systems can be observed in a wide variety of applications. In 1982, the complex Lorenz equations were proposed by Fowler et al. [3], which extended nonlinear systems into complex space. After that, some research works in this field have been achieved [4–9]. With in-depth study of complex nonlinear systems, a variety of physical phenomena could be described by the chaotic or hyperchaotic complex systems, for instance, the detuned laser systems and the amplitudes of electromagnetic fields.
The nature of complex chaotic systems precludes the possibility of obtaining closed form analytical solutions of the underlying governing equations. Thus, approximate-analytical methods, which are implemented on a sequence of multiple intervals to increase their radius of convergence, are often used to solve IVPs modelling chaotic systems. Examples of multistage methods that have been developed recently to solve IVPs for chaotic and nonchaotic systems include the multistage homotopy analysis method [10], piecewise homotopy perturbation methods [11, 12], multistage variational iteration method [13], and multistage differential transformation method [14]. Other multistage methods which use numerical integration techniques have also been proposed such as the piecewise spectral homotopy analysis method [15–17] which uses a spectral collocation method to perform the integration process. Accurate solutions of highly chaotic and hyperchaotic systems require resolution over many small intervals. Thus, seeking analytical solutions over the numerous intervals may be impractical or computationally expensive if the solution is sought over very long intervals.
In this paper, we propose a piecewise or multistage spectral relaxation method (MSRM) for solving the hyperchaotic complex systems as an accurate and robust alternative to recent multistage methods. The proposed MSRM was developed using the Gauss-Seidel idea of decoupling systems of equations and using Chebyshev pseudospectral methods to solve the resulting decoupled system on a sequence of multiple intervals. The spectral relaxation method (SRM) was recently proposed in [18, 19].
The rest of the paper is organized as follows. In Section 2, we give a brief description of the proposed MSRM algorithm. In Section 3, we present the numerical implementation of the MSRM on two examples of hyperchaotic complex systems. Finally, the conclusion is given in Section 4.
2. Multistage Spectral Relaxation Method
In this section, we give a brief description of the numerical method of solution used to solve the nonlinear hyperchaotic complex. We employ the multistage spectral relaxation method (MSRM) proposed in [19]. The MSRM algorithm is based on a Gauss-Seidel type of relaxation that decouples and linearises the system and the use of spectral collocation method to solve the linearised equations in a sequential manner. For compactness, we express the system of nonlinear first order differential equations in the form subject to the initial conditions where are the unknown variables and are the corresponding initial conditions, are known constant input parameters and is the nonlinear component of the th equation and the dot denotes differentiation with respect to time .
The scheme computes the solution of (1) in a sequence of equal subintervals that makes the entire interval. We define the interval of integration as and divide it into a sequence of nonoverlapping subintervals , where and . We denote the solution of (1) in the first subinterval as and the solutions in the subsequent subintervals as . For obtaining the solution in the first interval , (2) is used as the initial condition. By using the continuity condition between neighbouring subintervals the obtained solution in the interval is used to obtain the initial condition for the next subinterval . This is applied over the successive subintervals; that is, the obtained solution for each subinterval is used to obtain the initial condition for the next subinterval . Thus, in each interval we must solve subject to where is the Kronecker delta. As mentioned earlier, the main idea behind the MSRM scheme is decoupling the system of nonlinear IVPs using the Gauss-Seidel idea of decoupling systems of algebraic equations. The proposed MSRM iteration scheme for the solution in the interval is given as subject to the initial conditions where is the estimate of the solution after iterations. A suitable initial guess to start the iteration scheme (5) is one that satisfies the initial condition (6). A convenient choice of initial guess that was found to work in the numerical experiments considered in this work is
The Chebyshev spectral method is used to solve (5) on each interval . First, the region is transformed to the interval on which the spectral method is defined by using the linear transformation, in each interval for . We then discretize the interval using the Chebyshev-Gauss-Lobatto collocation points [20]: which are the extrema of the th order Chebyshev polynomial:
The Chebyshev spectral collocation method is based on the idea of introducing a differentiation matrix which is used to approximate the derivatives of the unknown variables at the collocation points as the matrix vector product where and are the vector functions at the collocation points .
Applying the Chebyshev spectral collocation method in (5) gives with where is an identity matrix of order . Thus, starting from the initial approximation (7), the recurrence formula can be used to obtain the solution in the interval . The solution approximating in the entire interval is given by
3. Numerical Examples
In this section, we consider two examples which demonstrate the efficiency and accuracy of the proposed method. In particular, we use the MSRM algorithm as an appropriate tool for solving nonlinear IVPs; we apply the method to two complex nonlinear chaotic systems.
Example 1. The hyperchaotic complex Lorenz system can be described as
where , , , , , and are the conjugates of and . When the parameters are chosen as , , , and , the system (16) is hyperchaotic [21].
Replacing the complex variables in system (16) with real and imaginary number variables, one can get an equivalent system as follows:
For (17), the parameters and are defined as with all other and for .
Through numerical experimentation, it was determined that collocation points and 5 iterations of the MSRM scheme at each interval were sufficient to give accurate results in each interval. Tables 1 and 2 show a comparison of the solutions of the hyperchaotic complex Lorenz system computed by the MSRM and ode45. In Figures 1, 2, and 3, the MSRM graphical results are also compared with ode45 and good agreement is observed. The MRSM phase portraits in Figures 4 and 5 were also found to be exactly the same as those computed using ode45. This shows that the proposed MSRM is a valid tool for solving the hyperchaotic complex Lorenz system.
Example 2. State equations of a permanent magnet synchronous motor system in a field-oriented rotor can be described as follows [22, 23]: where , and are the state variables which represent currents and motor angular frequency, respectively; and are the direct-axis stator and quadrature-axis stator voltage components, respectively; is the polar moment of inertia; is the external load torque; is the viscous damping coefficient; is the stator winding resistance; and are the direct-axis stator inductors and quadrature-axis stator inductors, respectively; is the permanent magnet flux; and is the number of pole-pairs; the parameters , , , , , , are all positive.
When the air gap is even, and the motor has no load or power outage, the dimensionless equations of a permanent magnet synchronous motor system can be depicted as where are both positive parameters. If the current in the system (19) is plural and the variables in the system (20) are complex numbers, by changing cross coupled terms and to conjugate form, Wang and Zhang got a complex permanent magnet synchronous motor system as follows [24]: where , , , , and are the conjugates of and . Replacing the complex variables in system (21) with real and imaginary number variables, Wang and Zhang got an equivalent system as follows (see [24]): where are positive parameters determining the chaotic behaviors and bifurcations of system (22). When the parameters satisfy , , there is one positive Lyapunov exponent, two Lyapunov exponents of zero, and two negative Lyapunov exponents for system (22), which means system (22) is chaotic [24]. The values of parameters and initial values are , , and , , , , .
For (21), the parameters and are defined as with all other and for .
The results obtained were compared to those from the MATLAB inbuilt solver, ode45. The ode45 solver integrates a system of ordinary differential equations using explicit 4th and 5th Runge-Kutta formula. Tables 3 and 4 show a comparison of the solutions of the complex permanent magnet synchronous motor computed by the MSRM and ode45. In Figures 6, 7, and 8, the MSRM graphical results are also compared with ode45 and good agreement is observed. The MRSM phase portraits in Figures 9 and 10 were also found to be exactly the same as those computed using ode45. This shows that the proposed MSRM is a valid tool for solving the complex permanent magnet synchronous motor.
4. Conclusion
In this paper, we have applied a spectral method called the multistage spectral relaxation method (MSRM) for the solutions of hyperchaotic complex systems. The proposed MSRM was developed using the Gauss-Seidel idea of decoupling systems of equations and using Chebyshev pseudospectral methods to solve the resulting decoupled system on a sequence of multiple intervals. The proposed MSRM was used to solve the hyperchaotic complex Lorenz system and complex permanent magnet synchronous motor. The accuracy and validity of the proposed method was tested against Matlab Runge-Kutta based inbuilt solvers and against previously published results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.