Bifurcations in a fractional-order neural network with multiple leakage delays

Abstract

This paper expatiates the stability and bifurcation for a fractional-order neural network (FONN) with double leakage delays. Firstly, the characteristic equation of the developed FONN is circumspectly researched by employing inequable delays as bifurcation parameters. Simultaneously the bifurcation criteria are correspondingly extrapolated. Then, unequal delays-spurred-bifurcation diagrams are primarily delineated to confirm the precision and correctness for the values of bifurcation points. Furthermore, it lavishly illustrates from the evidence that the stability performance of the proposed FONN can be demolished with the presence of leakage delays in accordance with comparative studies. Eventually, two numerical examples are exploited to underpin the feasibility of the developed theory. The results derived in this paper have perfected the retrievable outcomes on bifurcations of FONNs embodying unique leakage delay, which can nicely serve a benchmark deliberation and provide a comparatively credible guidance for the influence of multiple leakage delays on bifurcations of FONNs.

Introduction

Neural networks (NNs) have been incessantly garnered keen concerns as a result of their major applications in a wide range of domains (Cao et al., 2019a, Cao et al., 2019b, Cao et al., 2020, Fan, Huang, Li, et al., 2019, Fan, Huang, Shen, and Cao, 2019, Liu et al., 2020, Wang et al., 2020). Owing to numerous benefits involving memory effects for fractional calculus, modeling dynamic systems by exploiting fractional calculus has been achieved considerable concerns (Fan et al., 2018, Čermák and Kisela, 2019, Yao et al., 2020). Delay-dependent stability switches of linear fractional-order systems with delay were investigated (Čermák & Kisela, 2019), and the explicit stability dependence on a changing time delay was depicted. Lately, fractional calculus has been incorporated into NNs and fractional-order neural networks (FONNs) have been constructed. This intervention can augment the capability of information processing among neurons. Delightedly, some worldly applications of FONNs have been further detected thanks to unremitting persistence of researchers, such as network approximation (Vyawahare et al., 2018), state estimation (Nagamani et al., 2020) and system identification (Aslipour & Yazdizadeh, 2019), robotic manipulators (Sharma et al., 2019), formation control (Liu et al., 2019), disease treatment (El-Sayed et al., 2020), and so on. There exist two highlighted advantages of the fractional-order elements for neurons. On one hand, fractional calculus possesses quite a good representation of the memory and hereditary properties compared with conventional one (Ali et al., 2019). On the other hand, parameters of fractional-order can enrich the system performance by augmenting one degree of freedom in Chen et al. (2019). Quite evidently, it is an incalculable amelioration by merging memory peculiarity into NNs. Some remarkable results have been derived of FONNs (Jia et al., 2020, Lundstrom et al., 2008, Zheng et al., 2018).

To acquire more dynamic characteristics of nonlinear systems, bifurcation methods are constantly viewed as nice candidates (Li et al., 2018, Wang et al., 2018, Wang et al., 2017). Employing bifurcation strategies, some prominent system performance can be captured, such as the generation of bifurcation can be retarded by adjusting a bifurcation parameter, and unstable periodic orbits can be stabilized. Bifurcation theory often is utilized to acquire the learning rules in NNs. Baird discovered that bifurcations play an essential role in procuring the learning rules (Baird, 1990). Various local periodic solutions can arise from the different equilibrium points of BAM NNs by adopting Hopf bifurcation techniques (Xiao et al., 2013). This implies that storage pattern or memory pattern is extremely important for obtaining learning rules in NNs. On account of higher accuracy, reliability and practicability in describing the dynamical properties of fractional order systems compared with conventional ones, numerous scholars have very actively involved in studying bifurcations of fractional-order systems, and some gratifying results have been derived (Alidousti, 2020, Huang, Li, and Cao, 2019, Huang, Nie, et al., 2019, Huang, Zhao, et al., 2019). In Huang, Nie, et al. (2019), the delay-induced bifurcation results were derived in a delayed fractional-order quaternion-valued NN. The generation of bifurcation for a fractional order predator–prey model was ingeniously controlled by varying extended feedback delay or fractional order (Huang, Li, & Cao, 2019).

There exist numerous realistic systems, such as population dynamics (Gopalsamy, 1992), NNs (Gopalsamy, 2007), genetic regulatory networks (Manivannan et al., 2020), etc. There perpetually encounters a representative time delay, which is excessively dissimilar to the conventional delays, called leakage delay (or forgetting delay), and it broadly exists in the negative feedback terms of the system which are identified as forgetting or leakage terms and it was formerly overlooked in modeling and researching a lot of models. Generally, the leakage delay oftentimes has an inclination to destabilize the NNs (Gopalsamy, 2007, Huang et al., 2017, Li et al., 2014). Broadly speaking, leakage delay cannot be ignored in the analysis of nonlinear system dynamics for procuring a more accurate description of the dynamic responses of the real systems. In Chen et al. (2017), it manifested that leakage delays are extremely difficult to tackle, which results in demolishing speedily the system performance. The high susceptibility of stability performance of NNs with the presence of leakage delay was further discovered (Li et al., 2010). Some extraordinary results considering the role of leakage delay for complex systems have been derived (Cao et al., 2019a, Popa, 2020, Suntonsinsoungvon and Udpin, 2020). Therefore, when considering the dynamics of NNs, if leakage delay is discarded, then the obtained results will be utterly imprecise. This kind of remedy also is deleterious for the design and applications of NNs. Henceforth, the study of integer-order NNs with leakage delays has emanated the great fascination (Balasundaram et al., 2016, Li and Cao, 2016, Popa, 2018). With the speedy advancement of the application research for fractional calculus, some marvelous outcomes of FONNs with leakage delays have been derived. In Wang et al. (2017), the results on finite-time stability of a fractional-order complex-valued memristor-based NNs with both leakage and time-varying delays were deduced. In Zhang et al. (2017), the uniform stability for fractional-order complex-valued NNs with both leakage and discrete delays was considered, and some delay-dependent criteria on the global uniform stability were derived of the investigated NNs.

Researchers normally deal with tangible problems by presuming the identical leakage delay in NNs for the convenience of theoretical analysis. Despite of ameliorating the stability performance of the developed systems to some degree, it becomes clear that this treatment gives rise to unrealistic and imprecise outcomes. This is extremely unfavorable to the design and engineering applications with regard to NNs. As a matter of fact, the sizes of leakage delays commonly are nonidentical with respect to different state variables in NNs as a result of the conspicuous difference of the lags on independent tranquillization for diverse neurons in NNs. Therefore, it is rational and imperative to consider the effects of different leakage delays in unequal state variables on the dynamical behaviors for NNs. The robust stability problem of uncertain fuzzy BAM NNs with different delays in the leakage term was researched, and the globally asymptotic stability results were procured (Li & Rakkiyappan, 2013). In Sakthivel et al. (2015), the authors investigated the problem of state estimation for BAM NNs with different leakage delays, and realized that leakage delays can subvert the stability performance for systems.

To gain an insight into the effects of leakage delay on bifurcation of NNs, some scholars have introduced leakage delay into FONNs to explore its impact on the bifurcations, and some exciting results have been reported (Huang and Cao, 2018, Lin et al., 2019, Yuan et al., 2019). In Huang and Cao (2018), it detected that the generation of bifurcations for a high-order BAM FONNs can be advanced by leakage delay. In Lin et al. (2019), the authors discovered that a lesser leakage delay may lead to the generation of bifurcations if selecting an appropriate parameter of fractional order. Noticeably, the authors merely considered the influence of leakage delay on bifurcations for FONNs based on the assumption that leakage delay is consistent with communication delay in Huang and Cao (2018) and Yuan et al. (2019). Actually, leakage delay and communication delay are usually not identical to NNs. In this connection, the effects of leakage delay on bifurcations of FONNs were further detected by viewing the unequals of leakage delay and communication delay in Lin et al. (2019). It is much to be regretted that the derived results are still imperfect due to merely taking into account the same leakage delay in state variables (Lin et al., 2019).

Stimulated by the above facts, the primary objective of this paper is to explore the effects of multiple leakage delays concerning bifurcation for FONNs. The merits of the thesis are generalized as follows: (1) Multiple leakage delays induced-bifurcation conditions are theoretically derived by introducing two different leakage delays in comparison with Huang, Zhao, et al. (2019). Moreover, discriminating from Huang, Zhao, et al. (2019), the bifurcation diagrams sharply check the precision of the developed bifurcation results. (2) It detects that the stability performance can be immensely sabotaged under different leakage delays. It degrades the conservatism of the previous results (Huang and Cao, 2018, Yuan et al., 2019). (3) It perceives that fractional order plays an essential role in controlling the emergence of bifurcations of the developed FONNs. The generation of bifurcation can be efficiently advanced (handicapped) by selecting an appropriate fractional order parameter.

The configuration of the paper is excogitated as follows: Section 2 addresses the fundamental analysis tools on the definitions regarding fractional calculus and stability theory of linear fractional-order systems. Section 3 formulates the investigative mathematical model. Section 4 deduces the bifurcation conditions by use of different delays as bifurcation parameters. Section 5 affirms the usefulness and efficacy of the developed theory by experimental simulations. Section 6 ultimately epitomizes this paper.