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ABSTRACTIn the development of Type 1 diabetes (T1D), there are critical states just before drastic changes, and identifying these pre-disease states may predict T1D or provide crucial early-warning signals. Unlike gene expression data, gut microbiome data can be collected noninvasively from stool samples. Gut microbiome sequencing data contain different levels of phylogenetic information that can be utilized to detect the tipping point or critical state in a reliable manner, thereby providing accurate and effective early-warning signals. However, it is still difficult to detect the critical state of T1D based on gut microbiome data due to generally non-significant differences between healthy and critical states. To address this problem, we proposed a new method – microbiome network flow entropy (mNFE) based on a single sample from each individual – for detecting the critical state before seroconversion and abrupt transitions of T1D at various taxonomic levels. The numerical simulation validated the robustness of mNFE under different noise levels. Furthermore, based on real datasets, mNFE successfully identified the critical states and their dynamic network biomarkers (DNBs) at different taxonomic levels. In addition, we found some high-frequency species, which are closely related to the unique clinical characteristics of autoantibodies at the four levels, and identified some non-differential ‘dark species’ play important roles during the T1D progression. mNFE can robustly and effectively detect the pre-disease states at various taxonomic levels and identify the corresponding DNBs with only a single sample for each individual. Therefore, our mNFE method provides a new approach not only for T1D pre-disease diagnosis or preventative treatment but also for preventative medicine of other diseases by gut microbiome.

Abstract Background There exists a critical transition or tipping point during the complex biological process. Such critical transition is usually accompanied by the catastrophic consequences. Therefore, hunting for the tipping point or critical state is of significant importance to prevent or delay the occurrence of catastrophic consequences. However, predicting critical state based on the high-dimensional small sample data is a difficult problem, especially for single-cell expression data. Results In this study, we propose the comprehensive neighbourhood-based perturbed mutual information (CPMI) method to detect the critical states of complex biological processes. The CPMI method takes into account the relationship between genes and neighbours, so as to reduce the noise and enhance the robustness. This method is applied to a simulated dataset and six real datasets, including an influenza dataset, two single-cell expression datasets and three bulk datasets. The method can not only successfully detect the tipping points, but also identify their dynamic network biomarkers (DNBs). In addition, the discovery of transcription factors (TFs) which can regulate DNB genes and nondifferential ‘dark genes’ validates the effectiveness of our method. The numerical simulation verifies that the CPMI method is robust under different noise strengths and is superior to the existing methods on identifying the critical states. Conclusions In conclusion, we propose a robust computational method, i.e., CPMI, which is applicable in both the bulk and single cell datasets. The CPMI method holds great potential in providing the early warning signals for complex biological processes and enabling early disease diagnosis.

Abstract Complex biological systems often undergo sudden qualitative changes during their dynamic evolution. These critical transitions are typically characterized by a catastrophic progression of the system. Identifying the critical point is critical to uncovering the underlying mechanisms of complex biological systems. However, the system may exhibit minimal changes in its state until the critical point is reached, and in the face of high throughput and strong noise data, traditional biomarkers may not be effective in distinguishing the critical state. In this study, we propose a novel approach, mutual information weighted entropy (MIWE), which uses mutual information between genes to build networks and identifies critical states by quantifying molecular dynamic differences at each stage through weighted differential entropy. The method is applied to one numerical simulation dataset and four real datasets, including bulk and single-cell expression datasets. The critical states of the system can be recognized and the robustness of MIWE method is verified by numerical simulation under the influence of different noises. Moreover, we identify two key transcription factors (TFs), CREB1 and CREB3, that regulate downstream signaling genes to coordinate cell fate commitment. The dark genes in the single-cell expression datasets are mined to reveal the potential pathway regulation mechanism.

Abstract Background In the field of computational personalized medicine, drug response prediction (DRP) is a critical issue. However, existing studies often characterize drugs as strings, a representation that does not align with the natural description of molecules. Additionally, they ignore gene pathway-specific combinatorial implication. Results In this study, we propose drug Graph and gene Pathway based Drug response prediction method (GPDRP), a new multimodal deep learning model for predicting drug responses based on drug molecular graphs and gene pathway activity. In GPDRP, drugs are represented by molecular graphs, while cell lines are described by gene pathway activity scores. The model separately learns these two types of data using Graph Neural Networks (GNN) with Graph Transformers and deep neural networks. Predictions are subsequently made through fully connected layers. Conclusions Our results indicate that Graph Transformer-based model delivers superior performance. We apply GPDRP on hundreds of cancer cell lines’ bulk RNA-sequencing data, and it outperforms some recently published models. Furthermore, the generalizability and applicability of GPDRP are demonstrated through its predictions on unknown drug-cell line pairs and xenografts. This underscores the interpretability achieved by incorporating gene pathways.

Abstract To explore the potential network markers and related signaling pathways of human B cells infected by COVID-19, we performed standardized integration and analysis of single-cell sequencing data to construct conditional cell-specific networks (CCSN) for each cell. Then the peripheral blood cells were clustered and annotated based on the conditional network degree matrix (CNDM) and gene expression matrix (GEM), respectively, and B cells were selected for further analysis. Besides, based on the CNDM of B cells, the hub genes and ‘dark’ genes (a gene has a significant difference between case and control samples not in a gene expression level but in a conditional network degree level) closely related to COVID-19 were revealed. Interestingly, some of the ‘dark’ genes and differential degree genes (DDGs) encoded key proteins in the JAK-STAT pathway, which had antiviral effects. The protein p21 encoded by the ‘dark’ gene CDKN1A was a key regulator for the COVID-19 infection-related signaling pathway. Elevated levels of proteins encoded by some DDGs were directly related to disease severity of patients with COVID-19. In short, the proteins encoded by ‘dark’ genes complement some missing links in COVID-19 and these signaling pathways played an important role in the growth and activation of B cells.

Abstract The traditional complete dual-branch structure is effective for semantic segmentation tasks. However, it is redundant in some sense. Moreover, the simple additive fusion of the features from the two branches may not achieve the satisfactory performance. To alleviate these two problems, in this paper we propose an efficient compact interactive dual-branch network (CIDNet) for real-time semantic segmentation. Specifically, we first build a compact interactive dual-branch structure by constructing a compact detail branch and a semantic branch. Furthermore, we build a detail-semantic interactive module to fuse several specific stages of the two branches in the backbone network with the corresponding stages of the detail resolution branch. Finally, we propose a dual-branch contextual attention fusion module to deeply fuse the extracted features and predict the final segmentation result. Extensive experiments on Cityscapes and CamVid dataset demonstrate that the proposed CIDNet achieve satisfactory trade-off between segmentation accuracy and inference speed, and outperforms 20 representative real-time semantic segmentation methods.

In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.

All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis system accompanying delay. Utilizing fixed point theorem, inequality tactics and an appropriate function, we gain the sufficient criteria on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated delayed Lotka–Volterra commensal symbiosis system. Making use of stability and bifurcation theory of delayed differential equation, we focus on the emergence of bifurcation behavior and stability nature of the formulated delayed Lotka–Volterra commensal symbiosis system. A new delay-independent stability and bifurcation conditions on the model are presented. By constructing a positive definite function, we explore the global stability. By constructing two diverse hybrid delayed feedback controllers, we can adjusted the domain of stability and time of appearance of Hopf bifurcation of the delayed Lotka–Volterra commensal symbiosis system. The effect of time delay on the domain of stability and time of appearance of Hopf bifurcation of the model is given. Matlab experiment diagrams are provided to sustain the acquired key outcomes.

Spatially resolved transcriptomics (SRT) is capable of comprehensively characterizing gene expression patterns and providing an unbiased image of spatial composition. To fully understand the organizational complexity and tumor immune escape mechanism, we propose stMGATF, a multiview graph attention fusion model that integrates gene expression, histological images, spatial location, and gene association. To better extract information, stMGATF exploits SimCLRv2 for visual feature exaction and employs edge feature enhanced graph attention networks for the learning potential embedding of each view. A global attention mechanism is used to adaptively integrate 3 views to obtain low-dimensional representation. Applied to diverse SRT datasets, stMGATF is robust and outperforms other methods in detecting spatial domains and denoising data even with different resolutions and platforms. In particular, stMGATF contributes to the elucidation of tissue heterogeneity and extraction of 3-dimensional expression domains. Importantly, considering the associations between genes in tumors, stMGATF can identify the spatial dark genes ignored by traditional methods, which can be used to predict tumor-driving transcription factors and reveal tumor immune escape mechanisms, providing theoretical evidence for the development of new immunotherapeutic strategies.

Recent years have seen an increase in scientific interest in the El Nio/La Nia Southern Oscillation (ENSO), a quasiperiodic climate phenomenon that takes place throughout the tropical Pacific Ocean over five years and causes significant harm. It is associated with the warm oceanic stage known as El Nio and the cold oceanic stage known as La Nia. In this research, the ENSO model is considered under a fractional operator, which is defined via a nonsingular and nonlocal kernel. Some theoretical features, such as equilibrium points and their stability, bifurcation maps, the existence of a unique solution via the Picard–Lindelof approach, and the stability of the solution via the Ulam–Hyres stability approach, are deliberated for the proposed ENSO model. The Adams–Bashforth numerical method, associated with Lagrangian interpolation, is used to obtain a numerical solution for the considered ENSO model. The complex dynamics of the ENSO model are displayed for a few fractional orders via MATLAB-18.

In the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks.

In order to maximize benefits, oligopolistic competition often occurs in contemporary society. Establishing the mathematical models to reveal the law of market competition has become a vital topic. In the current study, on the basis of the earlier publications, we propose a new fractional-order Bertrand duopoly game model incorporating both nonidentical time delays. The dynamics involving existence and uniqueness, non-negativeness, and boundedness of solution to the considered fractional-order Bertrand duopoly game model are systematacially analyzed via the Banach fixed point theorem, mathematical analysis technique, and construction of an appropriate function. Making use of different delays as bifurcation parameters, several sets of new stability and bifurcation conditions ensuring the stability and the creation of Hopf bifurcation of the established fractional-order Bertrand duopoly game model are acquired. By virtue of a proper definite function, we set up a new sufficient condition that ensures globally asymptotically stability of the considered fractional-order Bertrand duopoly game model. The work reveals the impact of different types of delays on the stability and Hopf bifurcation of the proposed fractional-order Bertrand duopoly game model. The study shows that we can adjust the delay to achieve price balance of different products. To confirm the validity of the derived criteria, we put computer simulation into effect. The derived conclusions in this article are wholly new and have great theoretical value in administering companies.

Abstract In this paper, we propose a new fractional-order financial model which is a generalized version of the financial model reported in the previous publications. By applying a suitable time-delayed feedback controller, we have control for the chaotic behavior of the fractional-order financial model. We investigate the stability and the existence of a Hopf bifurcation of the fractional-order financial model. A new sufficient condition that guarantees the stability and the existence of a Hopf bifurcation for a fractional-order delayed financial model is presented by regarding the delay as bifurcation parameter. The investigation shows that the delay and the fractional order have an important effect on the stability and Hopf bifurcation of involved model. Some simulations justifying the validity of the derived analytical results are given. The obtained results of this article are innovative and are of great significance in handling the financial issues.

Abstract We study the weighted pseudo almost periodic solutions of a Lasota–Wazewska system. With the aid of fixed point theory and differential inequality strategies, we give a set of new sufficient criteria that guarantee the existence and global exponential stability of weighted pseudo almost periodic solutions to a Lasota–Wazewska system. The obtained results of this manuscript are completely innovative and complement the work of Shao (Appl. Math. Lett. 43:90–95, 2015) to some degree. So far, no scholars have investigated this aspect.

Abstract In the present paper, the existence of nontrivial solutions of impulsive fractional differential equations with Dirichlet boundary conditions is studied. We apply Morse theory coupled with local linking arguments to solve the topic, and we prove the existence of at least one nontrivial solution for the impulsive fractional differential equations.

In this current study, we formulate a kind of new fractional BAM neural network model concerning five neurons and time delays. First, we explore the existence and uniqueness of the solution of the formulated fractional delay BAM neural network models via the Lipschitz condition. Second, we study the boundedness of the solution to the formulated fractional delayed BAM neural network models using a proper function. Third, we set up a novel sufficient criterion on the onset of the Hopf bifurcation stability of the formulated fractional BAM neural network models by virtue of the stability criterion and bifurcation principle of fractional delayed dynamical systems. Fourth, a delayed feedback controller is applied to command the time of occurrence of the bifurcation and stability domain of the formulated fractional delayed BAM neural network models. Lastly, software simulation figures are provided to verify the key outcomes. The theoretical outcomes obtained through this exploration can play a vital role in controlling and devising networks.

Abstract We deal with Nicholson’s blowflies model with proportional delays. Employing the differential inequality theory, we give a new sufficient condition that guarantees the exponential convergence of all solutions of Nicholson’s blowflies model with proportional delays. Numerical simulations are put into effect to examine our theoretical findings. The derived results of this manuscript are innovative and complement some known investigations.

Abstract In this paper, we deal with a class of cellular neural networks with time-varying delays. Applying differential inequality strategies without assuming the boundedness conditions on the activation functions, we obtain a new sufficient condition that ensures that all solutions of the considered neural networks converge exponentially to the zero equilibrium point. We give an example to illustrate the effectiveness of the theoretical results. The results obtained in this paper are completely new and complement the previously known studies of Tang (Appl. Math. Lett. 21:872–876, 2008).

In this paper, a delayed Nicholsonos blowflies model with a linear harvesting term is investigated. By transforming the model into an equivalent integral equation, and applying a fixed point theorem inc ones, we establish a sufficient condition which ensure the existence of positive almost periodic solutions for the Nicholsonos blowflies model. The results of this paper are completely new and complement those of the previous studies. The approach is new.

Abstract In this paper, we put up and study a competition and cooperation model of two enterprises with multiple delays and feedback controls. A set of sufficient conditions which ensure the existence of periodic solution of the two-enterprise competition and cooperation model with multiple delays and feedback controls are established by applying the fixed point theorem of strict-set-contraction. An example is given to check the obtained theoretical findings. The derived results are completely new and complement earlier publications.

In this paper, we deal with a class of BAM neural networks with time-varying leakage delays. By applying the exponential dichotomy of linear differential equations, fixed point theorems and differential inequality techniques, we obtain some sufficient conditions which guarantee the existence and exponential stability of almost periodic solutions for such class of BAM neural networks. An example is provided to illustrate the effectiveness of the theoretical predictions. The results obtained in this paper are completely new and complement the previously known publications.

Over the past several decades, numerous scholars have studied the stability and Hopf bifurcation problem of integer-order delayed neural networks. However, the fruits about the stability and Hopf bifurcation for fractional-order delayed neural networks are very scarce. In this paper, we will consider the stability and the existence of Hopf bifurcation of fractional-order bidirectional associative memory (BAM) neural networks with four delays. A set of sufficient criteria to ensure the stability and the existence of Hopf bifurcation for the fractional-order BAM neural networks with four delays are established by choosing the sum of two different delays as a bifurcation parameter. This paper manifests that the delay has an important influence on the stability and Hopf bifurcation of involved networks. An example is displayed to test the rationality of the derived theoretical findings. The derived results of this paper are new and play a key role in optimizing networks and improving human life.

In this paper, we are concerned with a non-autonomous competing model with time delays and feedback controls. Applying the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. An example with its numerical simulations is given to illustrate the feasibility of our results.

Type 2 diabetes mellitus (T2DM) is a metabolic disease caused by multiple etiologies, the development of which can be divided into three states: normal state, critical state/pre-disease state, and disease state. To avoid irreversible development, it is important to detect the early warning signals before the onset of T2DM. However, detecting critical states of complex diseases based on high-throughput and strongly noisy data remains a challenging task. In this study, we developed a new method, i.e., degree matrix network entropy (DMNE), to detect the critical states of T2DM based on a sample-specific network (SSN). By applying the method to the datasets of three different tissues for experiments involving T2DM in rats, the critical states were detected, and the dynamic network biomarkers (DNBs) were successfully identified. Specifically, for liver and muscle, the critical transitions occur at 4 and 16 weeks. For adipose, the critical transition is at 8 weeks. In addition, we found some “dark genes” that did not exhibit differential expression but displayed sensitivity in terms of their DMNE score, which is closely related to the progression of T2DM. The information uncovered in our study not only provides further evidence regarding the molecular mechanisms of T2DM but may also assist in the development of strategies to prevent this disease.

MotivationThe evolution of complex diseases can be modeled as a time-dependent nonlinear dynamic system, and its progression can be divided into three states, i.e., the normal state, the pre-disease state and the disease state. The sudden deterioration of the disease can be regarded as the state transition of the dynamic system at the critical state or pre-disease state. How to detect the critical state of an individual before the disease state based on single-sample data has attracted many researchers’ attention.MethodsIn this study, we proposed a novel approach, i.e., single-sample-based Jensen-Shannon Divergence (sJSD) method to detect the early-warning signals of complex diseases before critical transitions based on individual single-sample data. The method aims to construct score index based on sJSD, namely, inconsistency index (ICI).ResultsThis method is applied to five real datasets, including prostate cancer, bladder urothelial carcinoma, influenza virus infection, cervical squamous cell carcinoma and endocervical adenocarcinoma and pancreatic adenocarcinoma. The critical states of 5 datasets with their corresponding sJSD signal biomarkers are successfully identified to diagnose and predict each individual sample, and some “dark genes” that without differential expressions but are sensitive to ICI score were revealed. This method is a data-driven and model-free method, which can be applied to not only disease prediction on individuals but also targeted drug design of each disease. At the same time, the identification of sJSD signal biomarkers is also of great significance for studying the molecular mechanism of disease progression from a dynamic perspective.

In this current manuscript, we study a fractional-order modificatory hybrid optical model (FOMHO model). Experiments manifest that under appropriate parameter conditions, the fractional-order modificatory hybrid optical model will generate chaotic behavior. In order to eliminate the chaotic phenomenon of the (FOMHO model), we devise two different control techniques. First of all, a suitable delayed feedback controller is designed to control chaos in the (FOMHO model). A sufficient condition ensuring the stability and the occurrence of Hopf bifurcation of the fractional-order controlled modificatory hybrid optical model is set up. Next, a suitable delayed mixed controller which includes state feedback and parameter perturbation is designed to suppress chaos in the (FOMHO model). A sufficient criterion guaranteeing the stability and the onset of Hopf bifurcation of the fractional-order controlled modificatory hybrid optical model is derived. In the end, software simulations are implemented to verify the accuracy of the devised controllers. The acquired results of this manuscript are completely new and have extremely vital significance in suppressing chaos in physics. Furthermore, the exploration idea can also be utilized to control chaos in many other differential chaotic dynamical models.

Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies.

In the present work, we mainly focus on shunting inhibitory cellular neural networks (SICNNs) involving leakage delays and proportional delays. By applying the inequality technique, a novel sufficient criterion to ascertain the convergence of every solution of SICNNs with leakage delays and proportional delays is derived. Simulation results are delineated to substantiate the correctness of our theoretical findings. Up to now, very few scholars deal with the neural networks with leakage delays and proportional delays. The derived conclusion of this work is a novelty and supplements several earlier works.

In this paper, a discrete ratio-dependent food-chain system with delay is investigated. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, a set of sufficient conditions for the existence of positive periodic solutions and global asymptotic stability of the model are established.

The multiplicity of classical solutions for impulsive fractional differential equations has been studied by many scholars. Using Morse theory, Brezis and Nirenberg’s Linking Theorem, and Clark theorem, we aim to solve this kind of problems. By this way, we obtain the existence of at least three classical solutions and k distinct pairs of classical solutions. Finally, an example is presented to illustrate the feasibility of the main results in this paper.

In this paper, cellular neural networks (CNNs) with neutral type delays and time-varying leakage delays are investigated. By applying the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, a set of sufficient conditions which ensure the existence and exponential stability of almost automorphic solutions of the model are obtained. An example with its numerical simulations is given to support the theoretical findings.

In this manuscript, we principally probe into a class of fractional-order tri-neuron neural networks incorporating delays. Making use of fixed point theorem, we prove the existence and uniqueness of solution to the fractional-order tri-neuron neural networks incorporating delays. By virtue of a suitable function, we prove the uniformly boundedness of the solution to the fractional-order tri-neuron neural networks incorporating delays. With the aid of the stability theory and bifurcation knowledge of fractional-order differential equation, a new delay-independent condition to guarantee the stability and creation of Hopf bifurcation of the fractional-order tri-neuron neural networks incorporating delays is established. Taking advantage of the mixed controller that contains state feedback and parameter perturbation, the stability region and the time of onset of Hopf bifurcation of the fractional-order trineuron neural networks incorporating delays are successfully controlled. Software simulation plots are displayed to illustrate the established key results. The obtained conclusions in this article have important theoretical significance in designing and controlling neural networks.

Abstract In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results.

Abstract In this article, a kind of fuzzy cellular neural networks (FCNNs) with proportional delays and leakage delays are involved. Utilizing the differential inequality strategies, a sequence of sufficient criteria ensuring the global exponential convergence of involved model are presented. Computer simulations are performed to verify the analytic findings. The analytic findings of this article are innovative and complete several existing works.

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.

In recent years, with the rapid development of the Internet and the Internet of Things, network security is urgently needed. Malware becomes a major threat to network security. Thus, the study on malware propagation model plays an important role in network security. In the past few decades, numerous researchers put up various kinds of malware propagation models to analyze the dynamic interaction. However, many works are only concerned with the integer-order malware propagation models, while the investigation on fractional-order ones is very few. In this paper, based on the earlier works, we will put up a new fractional-order delayed malware propagation model. Letting the delay be bifurcation parameter and analyzing the corresponding characteristic equations of considered system, we will establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model. The study shows that the delay and the fractional order have important effect on the stability and Hopf bifurcation of considered system. To check the correctness of theoretical analyses, we carry out some computer simulations. At last, a simple conclusion is drawn. The derived results of this paper are completely innovative and play an important guiding role in network security.

In this paper, based on the earlier research, a new fractional-order chaotic Genesio-Tesi model is established. The chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedback controllers. With the aid of Laplace transform, we obtain the characteristic equation of the controlled chaotic Genesio-Tesi model. Then by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation, some new sufficient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model are derived. The research shows that when time delay remains in some interval, the equilibrium point of the controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a critical value. The effect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model is shown. At last, computer simulations check the rationalization of the obtained theoretical prediction. The derived key results in this paper play an important role in controlling the chaotic behavior of many other differential chaotic systems.

In this article, based on the previous works, a new fractional-order financial model is put up. The chaotic behavior of the fractional-order financial model is suppressed by designing an appropriate PDϑ controller. By choosing the delay as the bifurcation parameter, we establish the sufficient condition to guarantee the stability and the existence of Hopf bifurcation of fractional-order financial model. Also, the influence of the delay and the fractional order on the stability and the existence of Hopf bifurcation of fractional-order financial model is revealed. An example is given to confirm the effectiveness of the analysis results. The main findings of this article play an important role in maintaining economic stability.

This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings.

In this paper, stochastic fuzzy Cohen–Grossberg neural networks with discrete and distributed delays are investigated. By using Lyapunov function and the Ito differential formula, some sufficient conditions for the pth moment exponential stability of such stochastic fuzzy Cohen–Grossberg neural networks with discrete and distributed delays are established. An example is given to illustrate the feasibility of our main theoretical findings. Finally, the paper ends with a brief conclusion. Methodology and achieved results is to be presented.

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

We investigate the existence of mild solutions for fractional order differential equations with integral boundary conditions and not instantaneous impulses. By some fixed-point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, two interesting examples are given to illustrate our theory results.

We investigate the nonlocal boundary value problems of impulsive fractional differential equations. By Banach’s contraction mapping principle, Schaefer’s fixed point theorem, and the nonlinear alternative of Leray-Schauder type, some related new existence results are established via a new special hybrid singular type Gronwall inequality. At last, some examples are also given to illustrate the results.

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results.

A class of Beddington-DeAngelis functional response predator-prey model is considered. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are given.

A predator-prey model with disease in prey, Ivlev-type functional response, and impulsive effects is proposed. By using Floquet theory and small amplitude perturbation skill, sufficient conditions of the existence and global stability of susceptible pest-eradication periodic solution are obtained. By impulsive comparison theorem, conditions ensuring the permanence of the system are established. Examples and simulation are given to show the complex dynamics for the key parameters.

An asymptotically periodic predator-prey model with time delay is investigated. Some sufficient conditions for the uniformly strong persistence of the system are obtained. Our result is an important complementarity to the earlier results.