On the formulation of fractional Adams-Bashforth method with Atangana-Baleanu-Caputo derivative to model chaotic problems

• The proposed two-step fractional Adams-Bashforth scheme numerically approximates the Atangana-Baleanu operator.The derivative replaces the systems’ standard local time derivative.Source: Leading paragraph · p. 1
• Fixed point theory and approximation establish existence and uniqueness for a general two-component dynamical system.Source: Leading paragraph · p. 1
• Numerical results are presented for different fractional orders α.The modeled processes are chaotic and hyperchaotic.Source: Leading paragraph · p. 1

• Fractional calculus has applications across fields, while its use is described as poorly understood or lagging behind its developed theory.Source: §1 · p. 2
• The paper motivates applying the Atangana-Baleanu derivative in the Caputo sense to mathematical models of chaotic phenomena.Source: §1 · p. 2
• In the system formulation, α ∈ (0, 1] determines the differential-system order, and the coupled nonlinear functions account for local kinetics.Source: §1 · p. 2
• The paper outlines a review of fractional calculus, a fractional Adams-Bashforth method, and numerical experiments on three chaotic fractional differential equations.Source: §1 · p. 2

• The review sets its fractional-calculus properties in the context of continuous functions whose fractional operators are well defined.Source: §2 · p. 3
• The section introduces left- and right-sided Riemann-Liouville integrals and derivatives, alongside Caputo fractional derivatives.Source: §2 · p. 3
• For zero boundary data, the stated left- and right-sided Caputo and Riemann-Liouville derivatives coincide, under the passage’s integrability condition.Source: §2 · p. 3
• The Atangana-Baleanu derivative is introduced as having nonsingular and nonlocal properties, with its normalization and Mittag-Leffler function noted.Source: §2 · p. 4

• The derivation applies the fundamental theorem at successive times, subtracts the resulting expressions, and uses an approximation.Source: §3 · pp. 4–5
• The derived equation is identified as a two-step fractional Adams-Bashforth scheme for the Atangana-Baleanu derivative in the Caputo sense.Source: §3 · p. 5
• The scheme is rewritten using t_n = nh and t_(n+1) = (n + 1)h, then collected in powers of h.Source: §3 · p. 5
• The section outlines convergence and stability results previously obtained in, including a stated stability theorem under a Lipschitz assumption.Source: §3 · p. 6

4 Uniqueness and existence of solution via chaotic process

• The analysis starts from a general two-component system and uses the fundamental calculus theorem along its x and y components.The proof then constructs a compact set and uses the infinite norm in its estimates.Source: §4 · pp. 6–7
• The proof checks conditions for the fractional operator to be well-defined and establishes Lipschitz conditions for x(t) and y(t).The passage states the Lipschitz requirements separately for f with respect to x and g with respect to y.Source: §4 · pp. 7–8
• The procedure is stated to generalize from the two-component case to multicomponent fractional differential equations.The proof setup identifies initial-value intervals and a function over the compact region.Source: §4 · pp. 7–8

5 Numerical experiments

• The numerical experiments replace the classical time derivative with the Atangana-Baleanu fractional derivative in the sense of Caputo and apply the two-step scheme.The update uses function evaluations at the current and preceding time steps.Source: §5 · p. 8

5.1 Example 1

• The novel three-dimensional system shows a highly chaotic and oscillatory attractor for φ = 12, ϕ = 16, ψ = 5, σ = 96, and δ = 10.The reported simulations use α = 0.75 and 0.95, with h = 0.01 and final computational time t = 100.Source: §5.1 · p. 9

5.2 Example 2

• The three species in the second example oscillate chaotically in phase regardless of variation in α and computational time t.The simulations use α = 0.75 and 0.95, with the initial chaotic and spatiotemporal patterns shown at α = 0.68 and t = 5.Source: §5.2 · p. 9

5.3 Example 3

• The four-component simulation at α = 0.93 through t = 150 produces projections with the shape of four-wing attractors.The bottom-row result gives time-series solutions at t = 1 and t = 10.Source: §5.3 · p. 11

6 Conclusion

• The paper analyzes simulations of a newly proposed Adams–Bashforth scheme with the Atangana–Baleanu derivative in the Caputo sense, reporting strange attractors at some fractional-power values α.Figure 3 shows the three species of system (26) interact chaotically in phase as a function of time at α = 0.89.Source: §6 · p. 11, Figure 3 · p. 12
• The reported existence-and-uniqueness result for general two-component differential equations is slated for extension to multidimensional problems.Source: §6 · p. 12

Introduction

• The paper supplies reviewer-prompted corrections to equations (10)–(12) of the earlier article.Source: Introduction · pp. 12–13
• The general Cauchy problem has a unique solution when f is continuous, bounded, and Lipschitz in y; bounded continuous functions form a Banach space.Source: Introduction · pp. 15–17

Error for classical- vs fractional-derivatives in Caputo sense

• Fractional error depends on α and retains a memory kernel, whereas the α=1 classical error depends only on Δt.For the Atangana–Baleanu case, the paper says the error analysis matches the Caputo case up to an α/AB(α) coefficient.Source: Introduction · p. 20, Error for classical- vs… · pp. 22, 24

Example 1

• Example 1 establishes contraction-based existence and uniqueness and compares exact and approximate solutions across fractional indices.Source: Example 1 · p. 20, Figure 8 · p. 21

Example 2

• Example 2 also establishes existence and uniqueness after showing its mapping is a contraction.Source: Example 2 · p. 22

Error for classical- vs fractional-derivatives in Caputo sense

• The fractional scheme adds power-law kernel contributions at adjacent time nodes while approximating f with Lagrange polynomials.The error derivation assumes f(t, y(t)) is twice differentiable.Source: Error for classical- vs… · pp. 22–24