A robust computational framework for analyzing fractional dynamical systems

Khosro Sayevand; Valeyollah Moradi

doi:10.3934/dcdss.2021022

Article Contents

2021, Volume 14, Issue 10: 3763-3783. Doi: 10.3934/dcdss.2021022

This issue Previous Article Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative Next Article Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

A robust computational framework for analyzing fractional dynamical systems

Khosro Sayevand^, and
Valeyollah Moradi

Faculty of Mathematical Sciences, Malayer University, Malayer, Iran

^* Corresponding author: Khosro Sayevand
^* Corresponding author: Khosro Sayevand

Received: January 31, 2020

Revised: March 31, 2020

Early access: March 2021

Published: October 2021

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Abstract

This study outlines a modiﬁed implicit ﬁnite difference method for approximating the local stable manifold near a hyperbolic equilibrium point for a nonlinear systems of fractional differential equations. The fractional derivative is described in the Caputo sense of order $ \alpha\; (0<\alpha \le1) $ which is approximated based on the modified trapezoidal quadrature rule of order $ O(\triangle t ^{2-\alpha}) $. The solution existence, uniqueness and stability of the proposed method is discussed. Three numerical examples are presented and comparisons are made to conﬁrm the reliability and effectiveness of the proposed method.

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Mathematics Subject Classification: Primary: 41A58; 39A10 Secondary: 34K28; 41A10.

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Figure 1. Comparison of $ x_{1} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha = 0.3 $

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Figure 2. Comparison of $ x_{2} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha=0.3 $

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Figure 3. Comparison of $ x_{1} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha=0.7 $

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Figure 4. Comparison of $ x_{2} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha=0.7 $

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Figure 5. Comparison of $ x_{1} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.2 $

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Figure 6. Comparison of $ x_{2} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.2 $

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Figure 7. Comparison of $ x_{3} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.2 $

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Figure 8. Comparison of $ x_{1} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.9 $

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Figure 9. Comparison of $ x_{2} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.9 $

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Figure 10. Comparison of $ x_{3} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha = 0.9 $

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Figure 11. Comparison of $ x_{1} $ for different values of $ t $ in Example 3 and corresponding to results obtained by Ref. [8] for $ \alpha = 0.7 $ and $ h = 0.1 $

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Figure 12. Comparison of $ x_{2} $ for different values of $ t $ in Example 3 and corresponding to results obtained by Ref. [8] for $ \alpha=0.7 $ and $ h=0.1 $

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Figure 13. Comparison of $ x_{3} $ for different values of $ t $ in Example 3 and corresponding to results obtained by Ref. [8] for $ \alpha = 0.7 $ and $ h = 0.1 $

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Figure 14. Projection $ \sigma_3 $ versus $ \sigma_1 $ for different values of $ \alpha $ in Example 3

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Figure 15. Projection $ \sigma_2 $ versus $ \sigma_1 $ for different values of $ \alpha $ in Example 3

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Figure 16. Projection local stable manifold for different values of $ \alpha $ in Example 3

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Table 1. Numerical results of Example 1 for different values of $ t $ and $ \alpha = 0.3 $ and $ h = 0.1 $

			$ \alpha=0.3, a_{1}=0.015 $
	$ x_{1} $			$ x_{2} $
$ t $	Ref. [39]	$ OM $	$ AE $	Ref. [39]	$ OM $	$ AE $
$ 0.3 $	0.00824	0.00835	0.00010	-0.00002	-0.00003	0.00000
$ 0.6 $	0.00744	0.00753	0.00009	-0.00002	-0.00005	0.00003
$ 0.9 $	0.00697	0.00705	0.00008	-0.00002	-0.00008	0.00006
$ 1.2 $	0.00663	0.00676	0.00012	-0.00002	-0.00012	0.00009
$ 1.5 $	0.00638	0.00655	0.00017	-0.00001	-0.00017	0.00015
$ 1.8 $	0.00617	0.00637	0.00020	-0.00001	-0.0.00024	0.00022
$ 2 $	0.00605	0.00627	0.00021	-0.00001	-0.00030	-0.00028

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Table 2. Numerical results of Example 1 for different values of $ t $ and $ \alpha=0.7 $ and $ h=0.1 $

			$ \alpha=0.7 , a_{1}=0.05 $
	$ x_{1} $			$ x_{2} $
$ t $	Ref. [39]	$ OM $	$ AE $	Ref. [39]	$ OM $	$ AE $
$ 0.3 $	0.03226	0.03276	0.00049	-0.00037	-0.00029	0.00007
$ 0.6 $	0.02539	0.02596	0.00056	-0.00025	-0.00014	0.00011
$ 0.9 $	0.02109	0.02170	0.00060	-0.00019	-0.00007	0.00011
$ 1.2 $	0.01808	0.01884	0.00076	-0.00015	-0.00004	0.00011
$ 1.5 $	0.01584	0.01683	0.00098	-0.00013	-0.00003	0.00010
$ 1.8 $	0.01411	0.01523	0.00111	-0.00011	-0.00002	0.00008
$ 2 $	0.01315	0.01432	0.00116	-0.00010	-0.00002	0.00007

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Table 3. Numerical results of Example 2 for different values of $ t $ and $ \alpha=0.2 $ and $ h=0.1 $

				$ \alpha=0.2 , a_{1}=0.02,a_{2}=0.03 $
	$ x_{1} $			$ x_{2} $			$ x_{3} $
$ t $	Ref. [39]	$ OM $	$ AE $	Ref. [39]	$ OM $	$ AE $	Ref. [39]	$ OM $	$ AE $
$ 0.3 $	0.01064	0.01075	0.0011	0.01092	0.01101	0.0009	0.00000	0.00000	0.00000
$ 0.6 $	0.00994	0.01003	0.00009	0.00997	0.00998	0.00001	0.00000	0.00000	0.00000
$ 0.9 $	0.00952	0.00960	0.00007	0.00943	0.00941	0.00002	0.00000	0.00000	0.00000
$ 1.2 $	0.00923	0.00935	0.00012	0.00907	0.00907	0.00000	0.00000	0.00000	0.00000
$ 1.5 $	0.00901	0.00917	0.00016	0.00879	0.00880	0.00001	0.00000	0.00000	0.00000
$ 1.8 $	0.00882	0.00901	0.00018	0.00856	0.00859	0.00002	0.00000	0.00001	0.00001
$ 2 $	0.00872	0.00891	0.00019	0.00843	0.00846	0.00002	0.00000	-0.00001	0.00001

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Table 4. Numerical results of Example 2 for different values of $ t $ and $ \alpha = 0.9 $ and $ h = 0.1 $

				$ \alpha=0.9 , a_{1}=0.02,a_{2}=0.01 $
	$ x_{1} $			$ x_{2} $			$ x_{3} $
$ t $	Ref. [39]	$ OM $	$ AE $	Ref. [39]	$ OM $	$ AE $	Ref. [39]	$ OM $	$ AE $
$ 0.3 $	0.01416	0.01434	0.00018	0.00515	0.00536	0.00020	0.00000	0.00000	0.00000
$ 0.6 $	0.01062	0.01090	0.00028	0.00305	0.00327	0.00021	0.00000	0.00000	0.00000
$ 0.9 $	0.00817	0.00849	0.00031	0.00194	0.00211	0.00016	0.00000	0.00000	0.00000
$ 1.2 $	0.00640	0.00675	0.00034	0.00132	0.00144	0.00012	0.00000	0.00000	0.00000
$ 1.5 $	0.00511	0.00547	0.00036	0.00095	0.00104	0.00008	0.00000	0.00000	0.00000
$ 1.8 $	0.00413	0.00450	0.00037	0.00072	0.00078	0.00006	0.00000	0.00000	0.00000
$ 2 $	0.00362	0.00398	0.00036	0.00061	0.00066	0.00004	0.00000	0.00000	0.00000

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Table 5. Numerical results of Example 3 for different values of $ t $ and $ \alpha = 0.7 $ and $ h = 0.1 $

				$ \alpha=0.7, \sigma_{1}=0.001 $
	$ x_{1} $			$ x_{2} $			$ x_{3} $
$ t $	Ref. [8]	$ OM $	$ AE $	Ref. [8]	$ OM $	$ AE $	Ref. [8]	$ OM $	$ AE $
$ 0.3 $	0.00064	0.00065	0.00000	0.00000	0.00000	0.00000	-0.00020	0.00000	0.00000
$ 0.6 $	0.00051	0.00052	0.00001	0.00000	0.00000	0.00000	-0.00020	0.00000	0.00000
$ 0.9 $	0.00042	0.00043	0.00001	0.00000	0.00000	0.00000	-0.00020	0.00000	0.00000
$ 1.2 $	0.00036	0.00037	0.00001	0.00001	0.00001	0.00000	-0.00020	0.00000	-0.00001
$ 1.5 $	0.00031	0.00033	0.00002	0.00002	0.00003	0.00001	-0.00021	0.00000	-0.00003
$ 1.8 $	0.00028	0.00030	0.00002	0.00005	0.00009	0.00003	-0.00022	0.00002	-0.00007
$ 2 $	0.00026	0.00028	0.00002	0.00009	0.00016	0.00006	-0.00024	-0.00003	-0.00012

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