Discrete Fractional Order Research Articles

A discrete fractional order cournot duopoly game model with relative profit delegation: Stability, bifurcation, chaos, 0-1 testing and control

Due to the memory effect, fractional order dynamical systems provide more realistic results compared with ordinary counterparts. In this study, we consider a Cournot-duopoly game model with relative profit delegation in the sense of Caputo fractional derivative. To describe richer dynamical behavior such as chaos in the model, a discrete dynamical system is needed. As a result of the discretization method based on the use of piecewise constant arguments, we obtain a two dimensional system of difference equations. The stability conditions of all equilibrium points of the discrete dynamical system are given comprehensively. The existence of the flip bifurcation in the system has been demonstrated theoretically. Lyapunov exponents and 0–1 test chaos imply that chaotic structures are formed as a result of this bifurcation. In addition, we present the chaos control technique such as Pyragas method to eliminate chaos in the model. All theoretical results dealing with the stability, bifurcation and chaos in the model are stimulated by numerical simulations.

Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.

Multiple Bifurcations and Chaos Control in a Coupled Network of Discrete Fractional Order Predator–Prey System

Multiple Bifurcations and Chaos Control in a Coupled Network of Discrete Fractional Order Predator–Prey System

Discrete Second Order Sliding Mode Control for Fractional Order Hammerstein System

This study explores the control of discrete fractional order Hammerstein systems using first-order sliding mode control (FOSMC) and improves its performance by integrating second-order sliding mode control (2SMC). Two simulation examples are presented to assess the effectiveness of these techniques. Results indicate that 2SMC performs better than FOSMC, demonstrating its superiority in controlling these complex systems and offering promising advancements in control strategies.

Discrete fractional order chaotic systems synchronization based on the variable structure control with a new discrete reaching-law

In this paper, we directly derive a new discrete state space expression of the fractional order chaotic system based on the fractional order Grnwald-Letnikow (G-L) definition and design a variable structure controller with a new faster reaching-law. The new reaching-law has the advantages of weakening the high frequency shake. Firstly, the condition of the discrete sliding mode surface is demonstrated. Then a multi-parametric function for sliding mode surface is constructed for weakening the high frequency shake through improving the Gao discrete reaching-law. Finally, the newly designed variable structure controller is applied to realize the synchronization of two different order discrete fractional chaotic systems. The simulation results show that the designed controller in this paper is effective, as it can achieve the synchronization of the discrete fractional order chaotic systems with external disturbances. Theoretical analysis and simulation results prove the effectiveness and robustness of this control method.

A study on qualitative analysis of a discrete fractional order prey-predator model with intraspecific competition

A study on qualitative analysis of a discrete fractional order prey-predator model with intraspecific competition

Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus

Abstract A pathological study in the definition of uncertain numbers is carried out, and some solutions are proposed. Fundamental theorems for uncertain discrete fractional and integer order calculus are established. The concept of maximal solution for obtaining a unique uncertain solution is introduced. The solutions of uncertain discrete relaxation equations for the integer and the fractional order are obtained. Various numerical examples are accompanied to clarify the theoretical results and study of uncertain system behavior.

On existence results of coupled pantograph discrete fractional order difference equations with numerical application

The goal of this article is to study the existence and unique solutions of coupled pantograph discrete fractional order difference equations. The Banach contraction principal method and fixed point theorems are used to prove the existence and uniqueness criterion of the suggested coupled system associated with a discrete fractional operator in the Caputo sense. In addition, stability analysis for the considered discrete pantograph fractional order coupled system is investigated by Hyers–Ulam theoretical approach. Finally, as an application, an example is presented to illustrate the obtained results.

Synchronization of discrete time fractional order neuromuscular models in the presence and absence of stimulus

This article proposes a fractional order discrete-time neuromuscular model incorporating calcium kinetics to study the essential role of the neuromuscular system in facilitating rapid information transmission between nerve cells and muscles for bodily functions. The study conducts a dynamical analysis to investigate the stability and periodic oscillations of the system in the presence and absence of tetanus stimuli from neurons. Bifurcation diagrams are generated for different fractional orders and rate constants, highlighting the occurrence of chaotic dynamics. The region of chaos is determined using the largest Lyapunov exponents and the Jacobian matrix method. The study also achieves synchronization of the systems by employing nonlinear control functions with feedback gains. Emphasizing the significance of constructing the biological model using fractional order operators, the article provides a chemical interpretation of the analytical findings. Moreover, the article employs dynamic plots to visually depict the changing system variables over time, allowing for an examination of the concurrent presence of multiple attractors through bifurcation diagrams and phase plane portraits. By studying how chemical kinetics and muscle activation are interrelated, the article uncovers captivating revelations that could guide the formulation of strategies to combat muscle disorders. The article also underscores the potential utility of synchronization in managing neuromuscular conditions, thus emphasizing its practical implications in disease control.

Stability Analysis for Discrete Fractional Order Steady-State Heat Equation with Neumann Boundary Conditions

Stability Analysis for Discrete Fractional Order Steady-State Heat Equation with Neumann Boundary Conditions

On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population

This research investigates the discrete prey–predator model by including harvesting on the predator population, in the sense of Caputo fractional derivative. We define the topological categories of the model fixed points. We demonstrate mathematically that, under certain parametric conditions, a fractional order prey-predator model undergoes both a Neimark–Sacker (NS) and a Period-doubling (PD) bifurcations. Using the central manifold and bifurcation theory, we present proof for NS and PD bifurcations. It has been discovered that the fractional order prey-predator model’s dynamical behavior is significantly influenced by the parameter values and the initial conditions. Two chaos control techniques have been used to eliminate the chaos in the model. In order to support our theoretical and analytical results and to illustrate complex and chaotic behavior, numerical simulations have been shown.

Finite Time Stability Results for Neural Networks Described by Variable-Order Fractional Difference Equations

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied.

The dynamic analysis of discrete fractional-order two-gene map

The evolutionary processes are based on information transmission by nervous systems and inheritance by genes in DNA. Various continuous and discrete mathematical models have been presented for genes. Discrete gene models are particularly interesting due to their simple analysis and low computational costs. It is imperative to create genetic factors based on gene models that depend on the past. This paper proposes a discrete fractional-order two-gene map model. At first, the gene map is evaluated using the phase plane, bifurcation diagram, and Lyapunov exponent, and the periodic and chaotic behaviors of the system are shown. Then, the fractional-order gene map model is introduced. The system’s dynamic behaviors are investigated using bifurcation diagrams according to system parameters and derivative order. It is shown that increasing the value of the fractional order increases complexity, leading to chaotic behavior in the model. While decreasing the fractional derivative order mostly changes the dynamics to periodic. Finally, the synchronization of two two-gene maps with discrete fractional order is investigated using the electrical connection. The results show that in contrast to the integer-order model, the fractional-order model can reach synchronization.

Modelling discrete time fractional Rucklidge system with complex state variables and its synchronization

This article proposes a discrete time fractional order three dimensional Rucklidge system with complex state variables. The dynamical nature and chaotic behavior exhibited by the Rucklidge system with real state variables and higher dimensional system derived from complex state variables are compared. The stability of the proposed system is analyzed at their equilibrium states by obtaining eigenvalues numerically. Chaotic dynamics exhibited by the system with real and complex variables are investigated employing different methods like bifurcation analysis and maximum Lyapunov exponents via the Jacobian matrix method. Nonlinear controllers are introduced for chaos synchronization of the subsystems of the proposed discrete time Rucklidge system with fractional order. The article further discusses the coexisting behavior of the attractors for the Rucklidge system of real state variables with coexisting bifurcation diagrams. The impact of the parameters on the system dynamics is demonstrated with a sequence of bifurcation diagrams for the simultaneous variation of two parameters.

Human gait tracking for rehabilitation exoskeleton: adaptive fractional order sliding mode control approach

To improve the rehabilitation training effect of hemiplegic patients, in this paper, a discrete adaptive fractional order fast terminal sliding mode control approach is proposed for the lower limb exoskeleton system to implement high-precision human gait tracking tasks. Firstly, a discrete dynamic model is established based on the Lagrange system discretization criterion for the lower limb exoskeleton robot. Then, in order to design a discrete adaptive fractional order fast terminal sliding mode controller, the Grünwald–Letnikov fractional order operator is introduced to combine with fast terminal attractor to construct a fractional order fast terminal sliding surface. An adaptive parameter adjustment strategy is proposed for the reaching law of sliding mode control, which drives the sliding mode to the stable region dynamically. Moreover, the stability of the control system is proved in the sense of Lyapunov, and the guidelines for selecting the control parameters are given. Finally, the simulations are tested on the MATLAB-Opensim co-simulation platform. Compared with the conventional discrete sliding mode control and discrete fast terminal sliding mode control, the results verify the superiority of the proposed method in improving lower limb rehabilitation training.

Dynamic analysis of the discrete fractional-order Rulkov neuron map.

Human evolution is carried out by two genetic systems based on DNA and another based on the transmission of information through the functions of the nervous system. In computational neuroscience, mathematical neural models are used to describe the biological function of the brain. Discrete-time neural models have received particular attention due to their simple analysis and low computational costs. From the concept of neuroscience, discrete fractional order neuron models incorporate the memory in a dynamic model. This paper introduces the fractional order discrete Rulkov neuron map. The presented model is analyzed dynamically and also in terms of synchronization ability. First, the Rulkov neuron map is examined in terms of phase plane, bifurcation diagram, and Lyapunov exponent. The biological behaviors of the Rulkov neuron map, such as silence, bursting, and chaotic firing, also exist in its discrete fractional-order version. The bifurcation diagrams of the proposed model are investigated under the effect of the neuron model's parameters and the fractional order. The stability regions of the system are theoretically and numerically obtained, and it is shown that increasing the order of the fractional order decreases the stable areas. Finally, the synchronization behavior of two fractional-order models is investigated. The results represent that the fractional-order systems cannot reach complete synchronization.

Discrete fractional distributed Halanay inequality and applications in discrete fractional order neural network systems

In this paper, the discrete fractional-order Halanay inequality with distributed delays is introduced. Then, based on the generalized discrete fractional Halanay inequality and Lyapunov functional method, several novel Mittag-Leffler stability and synchronization conditions of discrete fractional-order neural network systems with distributed delays are derived. Finally, an example is given to illustrated one of the results.

Finite-time stability in measure for nabla uncertain discrete linear fractional order systems

Finite-time stability in measure for nabla uncertain discrete linear fractional order systems

On Chaos of Discrete Time Fractional Order Host-Immune-Tumor Cells Interaction Model

On Chaos of Discrete Time Fractional Order Host-Immune-Tumor Cells Interaction Model