On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives

Houas, Mohamed; Samei, Mohammad Esmael; Sundar Santra, Shyam; Alzabut, Jehad

doi:10.1186/s13660-024-03093-6

Journal of Inequalities and Applications

Table 5 Numerical results of \(\acute{\Delta} = ( \delta \varpi _{1}+\varpi _{2} ) \Gamma _{q} ( \eta +1 ) +\varpi _{2}\), \((\ast ) : \Gamma _{q}(\theta +\vartheta +\lambda +1)\Gamma _{q} ( \eta +1 )\), and UH stable \(\Sigma _{g^{\ast },h^{\ast }}\) of \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (24) with \(q\in \{ \frac{3}{10}, \frac{1}{2}, \frac{8}{9}\}\) in Example 5.1

From: On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives

n	\(q = \frac{3}{10}\)			\(q = \frac{1}{2}\)			\(q = \frac{8}{9}\)
n	Δ́	(∗)	\(\Sigma _{g^{\ast },h^{\ast}}\)	Δ́	(∗)	\(\Sigma _{g^{\ast },h^{\ast}}\)	Δ́	(∗)	\(\Sigma _{g^{\ast },h^{\ast }}\)
1	0.0129	2.2803	0.6042	0.0160	5.5396	0.3528	0.0674	764.7056	0.0149
2	0.0128	2.2125	0.6184	0.0158	4.9566	0.3856	0.0634	516.8007	0.0207
3	0.0128	2.1928	0.6226	0.0157	4.7011	0.4024	0.0608	387.8897	0.0263
4	0.0128	2.1870	0.6239	0.0156	4.5811	0.4108	0.0588	311.0775	0.0316
5	0.0128	2.1852	0.6243	0.0156	4.5230	0.4151	0.0574	261.0760	0.0366
6	0.0128	2.1847	0.6244	0.0156	4.4943	0.4172	0.0562	226.4491	0.0413
7	0.0128	2.1845		0.0156	4.4801	0.4183	0.0553	201.3523	0.0456
8	0.0128	2.1845	0.6245	0.0156	4.4731	0.4188	0.0545	182.5200	0.0495
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
10	0.0128	2.1845	0.6245	0.0156	4.4678	0.4192	0.0534	156.5566	0.0564
11	0.0128	2.1845	0.6245	0.0156	4.4669		0.0529	147.3789	0.0594
12	0.0128	2.1845	0.6245	0.0156	4.4664	0.4193	0.0526	139.9099	0.0620
13	0.0128	2.1845	0.6245	0.0156	4.4662	0.4193	0.0522	133.7573	0.0644
14	0.0128	2.1845	0.6245	0.0156	4.4661	0.4193	0.0520	128.6372	0.0666
15	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0517	124.3393	0.0685
16	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0515	120.7051	0.0703
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
42	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0501	97.7327	0.0841
43	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	97.6267	0.0842
44	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	97.5327	0.0842
45	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	97.4493	0.0843
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
65	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.8484	0.0847
66	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.8415	0.0847
67	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.8353
68	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.8298	0.0848
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
117	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.7860	0.0848
118	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.7859	0.0848
119	0.0128	2.1845	0.6245	0.0156	4.4660	0.4193	0.0500	96.7859	0.0848

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