A new multistage numerical method based on blending a Gauss-Siedel relaxation method and Chebyshev pseudospectral method, for solving complex dynamical systems exhibiting hyperchaotic behavior, is presented. The proposed method, called the multistage spectral relaxation method (MSRM), is applied for the numerical solution of three hyperchaotic systems, namely, the Chua, Chen, and Rabinovich-Fabrikant systems. To demonstrate the performance of the method, results are presented in tables and diagrams and compared to results obtained using a Runge-Kutta-(4,5)-based MATLAB solver,
Systems exhibiting chaotic behaviour have received increasing attention from various researchers in recent years due to the challenge associated with computing their solutions and their applications in a number of areas in science such as in electrical circuits, lasers, fluid dynamics, mechanical devices, population growth, and many other areas. Chaotic behavior was first observed by Lorenz [
Chaotic systems are complex dynamical systems which are characterised by rapidly changing solutions and high sensitivity to small perturbations of the initial data and hence computing their solutions has proven to be challenging. A chaotic system consists of only one positive Lyapunov exponent. Chaotic systems with at least two positive Lyapunov exponents are said to be hyperchaotic. Hyperchaotic systems generally have more complex dynamical behaviours than ordinary chaotic systems. The concept of hyperchaos was first introduced by Rössler [
For rapidly oscillating dynamical systems like the chaotic and hyperchaotic systems, standard analytical iterative methods are not guaranteed to give solutions valid globally in time. Recent research efforts have been able to overcome this lack of global convergence by modifying the standard analytical iterative schemes. This is achieved by implementing the schemes on sequences of subintervals whose union make up the domain of the underlying problem. These modified methods are known as multi-stage (or piece-wise) methods. Examples which have recently been applied to chaotic systems include the multi-stage Adomian decomposition method [
In this work we present a new multi-stage iterative scheme which is based on blending a Gauss-Seidel type relaxation method with spectral collocation integration. The new method is called the multi-stage spectral relaxation method (MSRM). The MSRM is based on simple decoupling and rearrangement of the governing IVPs and numerically integrating the resulting equations in multiple intervals. We examine the applicability of the proposed MSRM in hyperchaotic chaotic systems of IVPs which include the Chua, Chen, and Rabinovich-Fabrikant systems. The computed numerical simulations are presented and it is noted that the method is accurate, efficient, and very easy to implement because its algorithm is easy to derive as it does not require any linearization of the original equations. The numerical results are also compared with some Matlab built-in Runge-Kutta solvers and good agreement is observed.
In this section, we give a brief description of how the multi-stage spectral relaxation method (MSRM) algorithm is developed for the solution of common chaotic systems governed by nonlinear systems of first order IVPs. Consider a chaotic system defined by
Following [
The MSRM algorithm uses the ideas of the Gauss-Siedel method to decouple systems of equations then apply the Chebyshev spectral collocation method to discretize and solve the resulting decoupled subsystems. We illustrate the development of the MSRM for a four equation IVP system (with
In the interval
A suitable initial guess to start the iteration scheme (
The system of (
We use the Chebyshev spectral method 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 apply the proposed MSRM to systems of IVPs with chaotic behavior to illustrate its effectiveness. In particular, we consider the hyperchaotic Chua, Chen, Rabinovich-Fabrikant systems. The results obtained are compared to results obtained by the built-in Matlab solvers,
The Chua system was originally created as an electic circuit by Chua in 1983 [
Based on the Chua oscillator, Rech and Albuquerque [
When
In this example, the parameters used in the MSRM iteration (
Qi et al. [
The system exhibits hyperchaotic behavior since it contains more than one Lyapunov exponent. Qi et al. [
Following Qi et al. [
In this example, the parameters used in the MSRM iteration (
The Rabinovich-Fabrikant system models the dynamical behavior arising from the modulation instability in a non-equilibrium dissipative medium [
In this example, the parameters used in the MSRM iteration (
Two cases are considered for this system:
We now present and discuss numerical results from the hyperchaotic systems introduced in the last section. The MSRM results are compared with those obtained from MATLAB solver
Comparison of results and computation times for the Chua system is depicted in Table
Comparison of the numerical solution of the hyperchaotic Chua system obtained by the MSRM and
|
MSRM |
|
MSRM |
|
---|---|---|---|---|
|
|
|||
2 | −0.170293 | −0.170293 | 1.551695 | 1.551694 |
4 | 3.990219 | 3.990219 | 2.525985 | 2.525985 |
6 | 5.159513 | 5.159513 | 3.058536 | 3.058536 |
8 | 4.648366 | 4.648366 | 1.766664 | 1.766664 |
10 | 2.583704 | 2.583704 | −0.980654 | −0.980654 |
| ||||
|
|
|||
| ||||
2 | 11.014651 | 11.014651 | −1.493298 | −1.493298 |
4 | 7.011997 | 7.011997 | −3.597274 | −3.597274 |
6 | −6.752382 | −6.752381 | −7.371016 | −7.371016 |
8 | −22.334344 | −22.334343 | −16.055827 | −16.055827 |
10 | −27.076591 | −27.076591 | −22.167142 | −22.167142 |
| ||||
CPU time | 0.387212 | 0.675014 |
Comparison between the MSRM (solid line) and
Phase plots for the hyperchaotic Chua system.
Three-dimesional phase plots for the hyperchaotic Chua system.
For the Chen system, computed MSRM results are compared with the piece-wise successive linearisation method (PSLM) [
Comparison of the numerical solution of the hyperchaotic Chen system obtained by the MSRM and
|
MSRM | Ref [ |
MSRM | Ref [ |
---|---|---|---|---|
|
|
|||
5 | −2.530836 | −2.530836 | −0.829573 | −0.829573 |
10 | −2.095053 | −2.095053 | −2.392373 | −2.392372 |
15 | −3.887545 | −3.887545 | −3.202475 | −3.202475 |
20 | −3.842229 | −3.842229 | −1.323089 | −1.323089 |
25 | −1.768250 | −1.768250 | −0.704882 | −0.704882 |
| ||||
|
|
|||
| ||||
5 | 10.015585 | 10.015585 | 1.616863 | 1.616863 |
10 | 9.146382 | 9.146382 | 0.846507 | 0.846507 |
15 | 9.332875 | 9.332875 | 2.232157 | 2.232157 |
20 | 10.065855 | 10.065855 | 2.561486 | 2.561486 |
25 | 9.841910 | 9.841910 | 1.065879 | 1.065879 |
Comparison of the numerical solution of the hyperchaotic Chen system obtained by the MSRM and
|
MSRM | Ref [ |
MSRM | Ref [ |
---|---|---|---|---|
|
|
|||
5 | −0.136746 | −0.136746 | −0.447315 | −0.447315 |
10 | −0.648525 | −0.648525 | −0.931221 | −0.931221 |
15 | −1.467115 | −1.467115 | −1.969698 | −1.969698 |
20 | −3.308084 | −3.308084 | −3.472248 | −3.472248 |
25 | −4.571305 | −4.571305 | −1.447157 | −1.447157 |
| ||||
|
|
|||
| ||||
5 | 8.739333 | 8.739333 | 0.411085 | 0.411085 |
10 | 8.366550 | 8.366550 | 0.219825 | 0.219825 |
15 | 8.000950 | 8.000950 | 0.451603 | 0.451603 |
20 | 7.962339 | 7.962339 | 1.747536 | 1.747536 |
25 | 9.377642 | 9.377642 | 3.588083 | 3.588083 |
Comparison of the numerical solution of the hyperchaotic Chen system obtained by the MSRM and
|
MSRM | Ref [ |
MSRM | Ref [ |
---|---|---|---|---|
|
|
|||
1 | 4.217177 | 4.217177 | 2.446151 | 2.446151 |
2 | −1.176404 | −1.176404 | −1.210601 | −1.210601 |
3 | −2.659744 | −2.659744 | −3.020974 | −3.020974 |
4 | 2.282134 | 2.282134 | 2.644847 | 2.644847 |
5 | −2.068296 | −2.068296 | −1.797169 | −1.797169 |
| ||||
|
|
|||
| ||||
1 | 7.927908 | 7.927908 | 3.766390 | 3.766390 |
2 | 7.107290 | 7.107290 | 1.268230 | 1.268230 |
3 | 6.258640 | 6.258640 | 1.532034 | 1.532034 |
4 | 6.293389 | 6.293389 | 1.304145 | 1.304145 |
5 | 7.022363 | 7.022363 | 1.860774 | 1.860774 |
Phase plots for the cyclic case of the hyperchaotic Chen system with
Phase plots for the periodic case of the hyperchaotic Chen system with
Phase plots for the chaotic case of the hyperchaotic Chen system with
For the Rabinovich-Fabrikant, two cases were considered depending on the value of
Comparison of the numerical solution of Rabinovich-Fabrikant equations obtained by the MSRM and
|
|
|
|
|||
---|---|---|---|---|---|---|
MSRM |
|
MSRM |
|
MSRM |
|
|
10 | −0.193770 | −0.193770 | 5.046955 | 5.046956 | 0.964453 | 0.964454 |
20 | 0.030179 | 0.030178 | −1.020136 | −1.020136 | 0.000192 | 0.000192 |
30 | 0.878727 | 0.878727 | −1.845389 | −1.845389 | 0.399523 | 0.399523 |
40 | −1.029824 | −1.029824 | 1.771524 | 1.771524 | 0.000093 | 0.000093 |
50 | 1.037551 | 1.037551 | −0.878634 | −0.878634 | 0.000091 | 0.000091 |
60 | 1.034005 | 1.034005 | 0.277127 | 0.277127 | 0.002862 | 0.002862 |
70 | 0.798540 | 0.798540 | −0.876128 | −0.876127 | 0.457949 | 0.457949 |
80 | 1.026191 | 1.026191 | −1.979857 | −1.979857 | 0.000876 | 0.000876 |
| ||||||
CPU time | 0.286794 | 0.501966 |
Comparison of the numerical solution of Rabinovich-Fabrikant equations obtained by the MSRM and
|
|
|
|
|||
---|---|---|---|---|---|---|
MSRM |
|
MSRM |
|
MSRM |
|
|
5 | 0.209579 | 0.209579 | −0.115166 | −0.115166 | 0.006755 | 0.006755 |
10 | −0.186073 | −0.186073 | −0.348447 | −0.348447 | 0.000415 | 0.000415 |
15 | −0.470037 | −0.470037 | 0.450639 | 0.450639 | 0.000030 | 0.000030 |
20 | 0.763142 | 0.763142 | −0.755078 | −0.755078 | 0.000002 | 0.000002 |
25 | 1.037421 | 1.037421 | −1.430741 | −1.430741 | 0.001753 | 0.001753 |
30 | 1.809575 | 1.809575 | −0.478596 | −0.478597 | 0.003993 | 0.003993 |
35 | 1.385747 | 1.385746 | −1.420675 | −1.420676 | 0.005921 | 0.005921 |
40 | −0.878223 | −0.878223 | 1.883510 | 1.883510 | 0.014795 | 0.014795 |
| ||||||
CPU time | 0.275718 | 0.469540 |
Comparison between the MSRM (solid line) and
Phase plots for the Rabinovich-Fabrikant equations for
Comparison between the MSRM (solid line) and
Phase plots for the Rabinovich-Fabrikant equations for
In this work we have successfully computed solutions of three hyperchaotic systems namely Chua, Chen, and Rabinovich-Fabrikant systems, using a new method which is based on blending Gauss-Siedel relaxation method and the Chebyshev pseudo-spectral method. The method, called the multi-stage spectral relaxation method (MSRM) is a multi-stage method which is adapted to solve complex dynamical systems like the hyperchaotic systems. The results presented in table and graphical form are comparable to results obtained using the Runge-Kutta-(4,5)-based MATLAB built-in solver,