| Citation: |
Mati ur Rahman, Sonia Akram. HYBRID ANALYTICAL AND DYNAMICAL ANALYSIS OF A FRACTIONAL GENERALIZED DUFFING MODEL WITH ATANGANA'S CONFORMABLE DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1668-1694. doi: 10.11948/20250233
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HYBRID ANALYTICAL AND DYNAMICAL ANALYSIS OF A FRACTIONAL GENERALIZED DUFFING MODEL WITH ATANGANA'S CONFORMABLE DERIVATIVE
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Abstract
In this research, we analyze the fractional generalized reaction Duffing model (FGRDM) within the framework of Atangana’s conformable derivative, providing a novel approach to modeling memory and hereditary characteristics in nonlinear dynamical systems. To determine exact analytical solutions, we utlize a hybrid methodology combining the Riccati sub-equation method and the Riccati–Bernoulli sub-ODE method. This integrated approach successfully yields a wide variety of soliton solutions, such as dark, bright, M-shape, combo, periodic, singular, and mixed hyperbolic wave structures. Beyond the analytical construction of solutions, we present a detailed qualitative analysis of the governed model, both in its perturbed and unperturbed forms, through bifurcation and chaos investigations. To explore the nonlinear behavior and chaotic dynamics, we employ a suite of diagnostic tools, such as Poincaré sections, return maps, Lyapunov exponents, time series analysis, power spectra, strange attractors, recurrence plots, and fractal dimension estimation. These tools help uncover the rich and sensitive dependency of the system on initial conditions and parameter variations, suggesting transitions between periodic, quasi-periodic, and chaotic regimes. The importance of this work lies in its comprehensive analytical and numerical analysis of the FGRDM using fractional calculus. It offers insights into the interplay between memory effects, nonlinearity, and chaos, thereby enhancing the comprehending of complex dynamical systems. Our research have potential applications in nonlinear science, particularly in fields where fractional-order models are used to explain physical phenomena with inherent damping and memory characteristics.
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References
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Mati ur Rahman, Sonia Akram. HYBRID ANALYTICAL AND DYNAMICAL ANALYSIS OF A FRACTIONAL GENERALIZED DUFFING MODEL WITH ATANGANA'S CONFORMABLE DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1668-1694. doi: 10.11948/20250233
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Figure 1.
Graphical abstract.
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Figure 2.
Phase portraits corresponding to distinct cases of parameter values representing the nature of EPs.
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Figure 3.
Physical dynamics of chaos are illustrated using 2D and 3D, time series and poincare phase portraits for system (4.4) obtained by varying the values of
$ \Upsilon_1, \; \Upsilon_2 $ -
Figure 4.
Physical dynamics of chaos are illustrated using 2D and 3D, time series and poincare phase portraits for system (4.4) obtained by varying the values of
$ \Upsilon_1, \; \Upsilon_2 $ -
Figure 5.
Physical dynamics of chaos are illustrated using 2D and 3D, time series and poincare phase portraits for system (4.4) obtained by varying the values of
$ \Upsilon_1, \; \Upsilon_2 $ -
Figure 6.
Physical dynamics of chaos are shown using 2D phase portraits for system (4.4) obtained by varying the values of
$ \Upsilon_1, \; \Upsilon_2 $ -
Figure 7.
Physical dynamics of return map analysis of the system described by Eq. (4.1) achieved with the values of parameters as
$ \gamma = -7.73, \; \mu = -2.4, \; \kappa = 4.19, \; \beta=-3.9, \; \delta = 3.9, \; \Upsilon_1= 7.91, \; \Upsilon_2= 9.9 $ -
Figure 8.
Physical dynamics of strange attractor analysis of the system described by Eq. (4.1) achieved with the values of parameters as
$ \gamma = -7.73, \; \mu = -2.4, \; \kappa = 4.19, \; \beta=-3.9, \; \delta = 3.9, \; \Upsilon_1= 7.91, \; \Upsilon_2= 9.9 $ -
Figure 9.
Physical dynamics of various dynamical characteristics of the system described by Eq. (4.1) achieved with the values of parameters as
$ \gamma = -7.73, \; \mu = -2.4, \; \kappa = 4.19, \; \beta=-3.9, \; \delta = 3.9, \; \Upsilon_1= 2.91, \; \Upsilon_2= 7.21 $ -
Figure 10.
The system described by Eq. (4.1) was subjected to a sensitivity analysis by choosing two different initial conditions:
$ (0.5, \, 0, \, 0) $ $ (0, \, 0.7, \, 0) $ -
Figure 11.
The system described by Eq. (4.4) was subjected to a sensitivity analysis by choosing three different initial conditions:
$ (2.35, \, 0, \, 0) $ $ (0, \, 1.58, \, 0) $ $ (5.1, \, 0.1, \, 0) $ -
Figure 12.
Physical depiction of dark type solution (3.5) under certain parameter
$ \sigma =-0.5, \; \beta =0.9, \; \kappa =0.7, \; \mu =0.8, \; \delta =0.23 $ -
Figure 13.
Physical depiction of
$ M- $ $ \sigma =-1.3, \; \beta =0.6, \; \kappa =0.3, \; \mu =0.2, \; \delta =0.7 $ -
Figure 14.
Physical depiction of periodic solution (3.11) under certain parameter
$ \sigma =2.8, \; \delta =0.7, \; \kappa =0.3, \; Q=0.2, \; R=0.4, \; \mu =0.2 $ -
Figure 15.
Physical depiction of hyperbolic type solution (3.17) under certain parameter
$ \sigma =1.8, \; \delta =0.3, \; \kappa =0.8, \; \mu =0.32, \; R=0.2, \; Q=0.9 $ -
Figure 16.
Physical depiction of combo type solution (3.22) under certain parameter
$ \mathfrak{F} =-0.8, \; a=0.9, \; b=0.3, \; \kappa =0.62, \; \mu =0.7, \; c_1=0.3 $ -
Figure 17.
Physical depiction of periodic solution (3.24) under certain parameter
$ \mathfrak{F} =3.4, \; a=0.8, \; b=0.7, \; \kappa =0.62, \; \mu =0.7, \; c_1=-0.93, \; \beta =0.4 $ -
Figure 18.
Physical depiction of mixed trigonometric solution (3.30) under certain parameter
$ \mathfrak{F} =4.4, \; a=0.5, \; b=0.3, \; \gamma =0.2, \; \mu =0.43, \; \delta =0.3, \; \beta =0.4 $