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 6 Numerical results of \(\Gamma _{q} (\rho +1 )\), here \(\mathrm{p} (\tau ) = \tau ^{(\rho )}\), \(\chi _{\mathrm{p}}\) and UHR 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	\(\Gamma _{q} (\rho +1 )\)	\(\chi _{\mathrm{p}}\)	\(\Sigma _{g^{\ast },h^{\ast }}\)	\(\Gamma _{q} (\rho +1 )\)	\(\chi _{\mathrm{p}}\)	\(\Sigma _{g^{\ast },h^{\ast }}\)	\(\Gamma _{q} (\rho +1 )\)	\(\chi _{\mathrm{p}}\)	\(\Sigma _{g^{\ast },h^{\ast }}\)
1	1.4983	0.9002	0.9064	2.0868	0.7341	0.7371	8.1718	0.1219	0.1219
2	1.5056	0.9257	0.9321	2.1162	0.8135	0.8170	8.4905	0.1754	0.1755
3	1.5078	0.9333	0.9399	2.1301	0.8542	0.8580	8.7209	0.2290	0.2291
4	1.5084	0.9356	0.9422	2.1368	0.8749	0.8788	8.8974	0.2811	0.2813
5	1.5086	0.9363	0.9429	2.1401	0.8852	0.8892	9.0373	0.3309	0.3310
6	1.5087	0.9365	0.9431	2.1417	0.8905	0.8945	9.1510	0.3777	0.3779
7	1.5087	0.9366	0.9432	2.1425	0.8931	0.8971	9.2449	0.4213	0.4216
8	1.5087	0.9366	0.9432	2.1429	0.8944	0.8984	9.3235	0.4617	0.4619
9	1.5087	0.9366	0.9432	2.1431	0.8950	0.8991	9.3899	0.4987	0.4990
10	1.5087	0.9366	0.9432	2.1432	0.8953	0.8994	9.4466	0.5325	0.5329
11	1.5087	0.9366	0.9432	2.1433	0.8955	0.8996	9.4951	0.5634	0.5637
12	1.5087	0.9366	0.9432	2.1433	0.8956	0.8997	9.5370	0.5913	0.5917
13	1.5087	0.9366	0.9432	2.1433	0.8956	0.8997	9.5732	0.6166	0.6170
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
66	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8348	0.8330	0.8337
67	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8349	0.8331	0.8337
68	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8349	0.8331	0.8338
69	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8350	0.8332	0.8338
70	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8350	0.8332	0.8338
71	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8350	0.8332	0.8339
72	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8350	0.8333	0.8339
73	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8351	0.8333	0.8339
74	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8351	0.8333	0.8339
75	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8351	0.8333	0.8340
76	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8351	0.8333	0.8340
77	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8352	0.8334	0.8340
78	1.5087	0.9366	0.9432	2.1433	0.8957	0.8997	9.8352	0.8334	0.8340

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