Rössler attractor-based numerical solution of the fractal-fractional operator: A fixed point approach

Authors

Keywords:

An extended $(\alpha, c)$-interpolative metric space, Rössler attractor., Fractal-Fractional operator

Abstract

In this article, we introduce an extended (α, c)-interpolative metric space and present an associated fixed- point theorem. With our main theorem, we present the existence of a solution for a fractal-fractional differential equation with a power-law kernel. In the end, we analyze the dynamics of the chaotic R¨ossler system.

References

[1] A. Granas, Fixed point theory. Springer Monographs in Mathematics/Springer-Verlag (2003).

[2] R. P. Agarwal, M. Meehan, Donal O’Regan, Fixed point theory and applications. Cambridge University Press (2001).

[3] W. Kirk , N. Shahzad, Fixed point theory in distance spaces, Springer Cham (2014).

[4] E. Karapinar , R. P. Agarwal, Fixed point theory in generalized metric spaces, Springer Cham (2022).

[5] M. D. Ortigueira, Fractional calculus for scientists and engineers, Springer Dordrecht (2011).

[6] A. Atangana , I. Koca, Fractional differential and integral operators with respect to a function, Springer Singapore (2025).

[7] J.L. Echenaus´ıa-Monroy, H.E. Gilardi-Vel´azquez, R. Jaimes-Re´ategui, V. Aboites, G. Huerta-Cuellar, A physical inter- pretation of fractional-order-derivatives in a jerk system: Electronic approach, Commun. Nonlinear Sci. Numer. Simul. 90(2020), Article ID. 105413.

[8] S.K. Panda, A. Atangana, T. Abdeljawad, Existence results and numerical study on novel coronavirus 2019-ncov/sars- cov-2 model using differential operators based on the generalized mittag-leffler kernel and fixed points, Fractals 30 (08) (2022).

[9] K. S. Nisar, M. Farman, M. Abdel-Aty, J. Cao, A review on epidemic models in sight of fractional calculus, Alexandria Eng. J. 75 (2023), 81-113.

[10] A. Khan, J.F. Gomez-Aguilar, T.S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos, Solitons Fractals 122 (2019) 119–128.

[11] I. Petr´aˇs, The fractional-order Lorenz-type systems: A review. Fract Calc Appl Anal 25(2022), 362–377.

[12] K. Sun, J. C. Sprott, Bifurcations of fractional-order diffusionless Lorenz system, arXiv 2009.

[13] He, S., Sun, K. & Banerjee, S. Dynamical properties and complexity in fractional-order diffusionless Lorenz system. Eur. Phys. J. Plus 131, (254) (2016).

[14] I.A.Bakhtin, The contraction mapping principle in almost metric spaces. Funct. Anal. 30(1989), 26–37.

[15] S.Czerwik, Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostra. 1(1993), 5–11.

[16] T.Kamran, Samreen, M.; UL Ain, Q. A generalization of b-metric space and some fixed point theorems, Mathematics, 5(19)(2017).

[17] M. Berzig, First results in suprametric spaces with applications, Mediterr. J. Math., 19 (2022).

[18] S. K. Panda, Ravi P. Agarwal, E. Karapinar, Extended suprametric spaces and Stone-type theorem, AIMS Math. 8(2023), 23183-23199.

[19] S. K. Panda, E. Karapınar, and A. Atangana, A numerical schemes and comparisons for fixed point results with applications to the solutions of Volterra integral equations in dislocated extended b-metricspace, Alex. Eng. J. 59(2020), 815–827.

[20] S. K. Panda, I. Khan, V. Velusamy, S. Niazai, Enhancing automic and optimal control systems through graphical structures. Sci Rep. 14 (2024), Article ID.3139.

[21] T.Abdeljawad, N.Mlaiki, H.Aydi, N.Souayah, Double controlled metric type spaces and some fixed point results. Mathe- matics, 6(320) (2018).

[22] N.Mlaiki, H. Aydi, N.Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle. Math- ematics, 6(194)(2018).

[23] E. Karapınar, Ravi P. Agarwal, Some fixed point results on interpolative metric spaces, Nonlinear Anal Real World Appl, 82(2025), Article ID. 104244.

[24] A.Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals, 102(2017), 396–406.

[25] M.Toufik, & A.Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models. Eur. Phys. J. Plus 132(2017), 1–16 .

[26] N.Khan, Z.Ahmad, J.Shah, et al. Dynamics of chaotic system based on circuit design with Ulam stability through fractal- fractional derivative with power law kernel. Sci Rep 13 (2023).

[27] S.K.Panda, T.Abdeljawad, & F. Jarad, (2023). Chaotic attractors and fixed point methods in piecewise fractional deriva- tives and multi-term fractional delay differential equations. Results in Physics, 46(2023), Article ID. 106313.

[28] Z. Ali, F. Rabiei, K. Shah, T. & Khodadadi, Qualitative analysis of fractal-fractional order COVID-19 mathematical model with case study of Wuhan. Alex. Eng. J. 60 (2021), 477–489.

[29] O. E.R¨ossler, An equation for continuous chaos, Phy. Let, 5(1976), 397–398.

[30] O. E. R¨ossler, An equation for hyperchaos, Phy. Let, 2(1979) 155–157.

Downloads

Published

2025-02-17

Issue

Section

Articles