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 GaussSeidel 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 RungeKutta based ode45 solver to show that the MSRM gives accurate results.
Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions and have complex and highly unpredictable profiles [
The nature of complex chaotic systems precludes the possibility of obtaining closed form analytical solutions of the underlying governing equations. Thus, approximateanalytical 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 [
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 GaussSeidel 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 [
The rest of the paper is organized as follows. In Section
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 [
The scheme computes the solution of (
The Chebyshev spectral method is used to solve (
The Chebyshev spectral collocation method is based on the idea of introducing a differentiation matrix
Applying the Chebyshev spectral collocation method in (
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.
The hyperchaotic complex Lorenz system can be described as
Replacing the complex variables in system (
For (
Through numerical experimentation, it was determined that
Numerical comparison between MSRM and






MSRM 

MSRM 

MSRM 


2  −2.91138  −2.91138  21.73155  21.73155  −3.24491  −3.24491 
4  −3.63001  −3.63001  6.52144  6.52144  −6.30884  −6.30884 
6  2.80571  2.80571  −2.77638  −2.77638  −2.37099  −2.37099 
8  0.01134  0.01134  2.09585  2.09585  −0.14880  −0.14880 
10  −0.80219  −0.80219  16.48559  16.48560  −0.06690  −0.06690 
Numerical comparison between MSRM and






MSRM 

MSRM 

MSRM 


2  23.96851  23.96851  44.32071  44.32071  26.54682  26.54682 
4  11.30830  11.30830  14.68007  14.68007  3.25221  3.25221 
6  4.65208  4.65208  39.34559  39.34559  12.99055  12.99055 
8  −4.99685  −4.99685  33.79560  33.79560  8.02232  8.02232 
10  1.98179  1.98179  50.59739  50.59740  24.48234  24.48234 
Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system.
Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system.
Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system.
Phase portraits of the hyperchaotic complex Lorenz system.
Phase portraits of the hyperchaotic complex Lorenz system.
State equations of a permanent magnet synchronous motor system in a fieldoriented rotor can be described as follows [
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
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 RungeKutta formula. Tables
Numerical comparison between MSRM and






MSRM 

MSRM 

MSRM 


3  −3.85711  −3.85711  −5.66683  −5.66683  −5.20445  −5.20445 
10  −0.33729  −0.33729  −0.49554  −0.49554  −0.49104  −0.49104 
17  0.12630  0.12631  0.18555  0.18557  0.15550  0.15551 
24  0.05091  0.05105  0.07480  0.07501  0.19500  0.19518 
31  −2.55034  −2.54878  −3.74694  −3.74465  −0.79819  −0.79326 
38  −3.93154  −3.73551  −5.77619  −5.48818  −5.33693  −5.20595 
Numerical comparison between MSRM and





MSRM 

MSRM 


3  −7.64635  −7.64635  15.05932  15.05932 
10  −0.72144  −0.72143  10.73663  10.73663 
17  0.22846  0.22848  14.25582  14.25583 
24  0.28649  0.28675  19.33844  19.33921 
31  −1.17270  −1.16545  25.34856  25.35739 
38  −7.84098  −7.64855  14.98250  14.03140 
Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor.
Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor.
Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor.
Phase portraits of the complex permanent magnet synchronous motor.
Phase portraits of the complex permanent magnet synchronous motor.
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 GaussSeidel 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 RungeKutta based inbuilt solvers and against previously published results.
The authors declare that there is no conflict of interests regarding the publication of this paper.