On uniform stability and numerical simulations of complex valued neural networks involving generalized Caputo fractional order

The dynamics and existence results of generalized Caputo fractional derivatives have been studied by several authors. Uniform stability and equilibrium in fractional-order neural networks with generalized Caputo derivatives in real-valued settings, however, have not been extensively studied. In contrast to earlier studies, we first investigate the uniform stability and equilibrium results for complex-valued neural networks within the framework of a generalized Caputo fractional derivative. We investigate the intermittent behavior of complex-valued neural networks in generalized Caputo fractional-order contexts. Numerical results are supplied to demonstrate the viability and accuracy of the presented results. At the end of the article, a few open questions are posed.

• The authors of 5 have expanded the glucose molecule's graph representation and taken into consideration contemporary modeling of the fractional derivatives on each graph edge.By taking multiple constraints on the existing operators, the authors have provided two different insights in the context of integral boundary value requirements.• In light of fractional derivatives, the authors of 6 examined a wide range of physical structures, including thermal transmission, controllers with PID tuning, and network fabrication via stochastic algorithms.
A particular class of computations stimulated by data platforms referred to as artificial neural networks (simply, we called, ANNs) aims to imitate the functions and operations of the human brain.With the advancement of artificial intelligence (AI), in particular deep learning, ANN-based machine intelligence algorithms have substantial usage in various fields that permeate our everyday lives.Digital devices can now make realisticsounding speech and pitches for music for practical purposes by using tools like automated facial recognition.They can also divide the speech of several speakers into separate waves of sound for each individual speaker using this tool, as shown in 12 .Complex numbers are frequently used in a wide range of practical uses, including recognition of speech, processing of pictures, automation, and communication systems.This demonstrates the possibilities for utilizing numbers which are complex-valued to personify parameters like weights in various www.nature.com/scientificreports/Then d is called a complex-valued metric on M , and (M, d) is called a complex-valued metric space.
Theorem 1.1 40 Let (M, d) be a complex-valued metric space and let mappings A , B : M → M gratify where x, y ∈ M and ν, ̺ are nonnegative reals having the condition ν + ̺ < 1 .Then A and B have a unique common fixed point.
If we take ̺ = 0 in Theorem.1.1,we get contractive mapping theorem in the setting of complex-valued metric space.
Theorem 1.2 Let (M, d) be a complex-valued metric space and let mapping W : M → M gratify where x, y ∈ M and 0 < ν < 1 .Then W has a unique fixed point.
We adopt the subsequent assumptions in order to obtain the main results.Assumption A: www.nature.com/scientificreports/Assumption B : Let z = x + iy where i denotes the imagenary unit, that is i = √ −1 .Moreover, ϕ j (u) and ϕ j (u(ρ − τ )) by separating it into its real as well as imagined portions, it could potentially be demonstrated as where ϕ R j (., .) is maps from R 2 to R ; ϕ I j (., .)maps from R 2 to R ; ϕ R j (., .)maps from R 2 to R and ϕ I j (., .)maps from R 2 to R .To make formulas simpler, x(ρ − τ ) and y(ρ − τ ) despite being stated as x τ and y τ appropriately.
• In addition, the partial derivatives of ϕ j (., .)regarded to x, y : ∂ϕ R j /∂x, ∂ϕ R j /∂y, ∂ϕ I j /∂x and ∂ϕ I j /∂y are exist and they are continuous.
• ∂ϕ R j /∂x, ∂ϕ R j /∂y, ∂ϕ I j /∂x and ∂ϕ I j /∂y are bounded, i.e., where θ RR j , θ RI j , θ IR j and θ II j are non-negative constant numbers.• In addition, the partial derivatives(with time delay) ∂ϕ R j /∂x, ∂ϕ R j /∂y, ∂ϕ I j /∂x and ∂ϕ I j /∂y , they're deemed as bounded.In other words, additionally, there are some numbers that are consistently non-negative.ω RR j , ω RI j , ω IR j and ω II j so that We can infer the following from the mean value hypothesis regarding multi-variable features: where x, x ′ , y, y ′ ∈ R w .Assumption C : ϕ j (.) gratify the Lipschitz assertions, that is, for any l, k ∈ C, there is non-negative constant θ j such a way that

Uniform stability result of generalized Caputo fractional-order complex-valued neural networks
In the current section, we construct a necessary condition for the generalized Caputo fractional-order complexvalued neural networks with time delays to be uniformly stable.The existence and uniqueness results are then obtained through the contraction mapping theorem in complex-valued metric space.

Theorem 2.1 If Assumption
A and B hold, then (1.1) is uniformly stable.
The integral equation below is equal to the previous equation (2.5), Which implies, (2.1) www.nature.com/scientificreports/www.nature.com/scientificreports/The integral equation underneath is equal to Eq. (2.9) :   www.nature.com/scientificreports/where,   where It is evident that the solutions converge to an equilibrium point that is consistent with the obtained theoretical results.Table 2, shows the equilibrium points at various values of γ and µ for z k (t), k = 1, 2.Moreover, we com- pared the values for the obtained equilibrium points for our results at γ = 0.98 and µ = 1 with those obtained      18 .It's worth mentioning that the numerical method is based on a decomposition formula for the generalized Caputo derivative, for more details, see 22,23 .

Remark 3.1
• In recent past, many authors have demonstrated the various stability results of fractional order neural net- works in Banach spaces making use of the contractive map.(check 18,19 , for more info) -As compared to the above results, we have used the complex-valued metric space to demonstrate the uniform stability of the problem.• In order to investigate the stability of the equilibrium point and demonstrate its existence and uniqueness for generalized Caputo fractional-order, the authors of 20,21 have taken into consideration real-valued neural networks.
-As compared to the above results, we first utilized complex-valued neural networks in the setting of generalized Caputo fractional-order.

Open Questions
or disprove the following: • Synchronization and uniform stability of neural networks having complex valued fractional order; • Finite time stability of neural networks having Atangana-Baleanu fractional derivative with complex order 43 ; • Extend our results to complex orders.

Conclusion
The uniform stability of CVNNs under the setting of generalized Caputo fractional-order with time delays in complex-valued metric spaces is investigated.The results of CVNNs are also developed in the context of fractional-order with time delay in complex-valued metric spaces, which produces fixed-point and unique equilibrium-point results.Additionally, numerical examples are given to support and illustrate the theoretical results.Our findings are significant because they open up novel possibilities for studying neural systems and chaotic theory.The findings of this work provide additional research directions for: k−1) |� k (t)|.Vol:.(1234567890)Scientific Reports | (2024) 14:4073 | https://doi.org/10.1038/s41598-024-53670-4

Figure 1 .
Figure 1.The trajectories of the real and imaginary parts of z k , k = 1, 2 for Example 3.1 with the initial condition ϑ 1 and various values of γ and µ..

Table 1 .
The equilibrium points for system (3.1) at various values of γ and µ..

γ µ Equilibrium points as real and imaginary parts for z 1 and z 2 Example 3 . 3
Figures 3, 4 and 5 represent the time trajectory for the components of z k , k = 1, 2, 3 at γ = 0.95, µ = 0.9 , γ = 0.8, µ = 0.75 and γ = 0.65, µ = 0.5, respectively.The numerical simulations of the real against imaginary parts are shown in Figs.6 and 7.It is clear from these results that the system (3.2) displays chaotic behavior.Example 3.3Let the fractional-order CVNNs is described as follows:

Figure 2 .
Figure 2. The trajectories of the real and imaginary parts of z k , k = 1, 2 for Example 3.1 with the initial conditions ϑ 1 and ϑ 2 at γ = µ = 0.95..

Table 2 .
The equilibrium points for system (3.3) at various values of γ and µ..