Adams Bashforth Moulton Algorithm Research Articles

Hidden hyperchaotic attractors in a new 4D fractional order system and its synchronization.

The research of finding hidden attractors in nonlinear dynamical systems has attracted much consideration because of its practical and theoretical importance. A new fractional order four-dimensional system, which can exhibit some hidden hyperchaotic attractors, is proposed in this paper. The predictor-corrector method of the Adams-Bashforth-Moulton algorithm and the parameter switching algorithm are used to numerically study this system. It is interesting that three different kinds of hidden hyperchaotic attractors with two positive Lyapunov exponents are found, and the fractional order system can have a line of equilibria, no equilibrium point, or only one stable equilibrium point. Moreover, a self-excited attractor is also recognized with the change of its parameters. Finally, the synchronization behavior is studied by using a linear feedback control method.

Dynamical Analysis of Fractional Order Model for Computer Virus Propagation with Kill Signals

Abstract The kill signals are alert about possible viruses that infect computer network to decrease the danger of virus propagation. In this work, we focus on a fractional-order SEIR-KS model in the sense of Caputo derivative to analyze the effects of kill signal nodes on the virus propagation. For this purpose, we first prove the existence and uniqueness of the model and give qualitative analysis. Then, we obtain the numerical solution of the model by using the Adams–Bashforth–Moulton algorithm. Finally, the effects of model parameters are demonstrated with graphics drawn by MATLAB program.

Bifurcation analysis of a fractional order time-delay Chua’s circuit based on coupled memristors

In this paper, a fractional-order (and an integer-order) chaotic system, obtained from Chua’s circuit by substituting Chua’s diode with two active coupled memristors (MRs) characterized by quadratic nonlinearity, is introduced to probe the memristive coupling effect. Two MRs connected in parallel are coupled by the flux. For the integer-order memristive system, the dynamical characteristics depending on the coupling strength coefficient between MRs without changing the circuit parameters are illustrated theoretically and numerically by using phase portraits, time domain diagram, bifurcation diagram and the Lyapunov diagram. Then based on the Adams–Bashforth–Moulton algorithm, the study of dynamic behavior of the fractional-order memristive system containing the time-delay reveals that appropriately setting the coupling strength between MRs generates more interesting attractors that differ from its integer-order counterpart. Besides, the effects of the order and the time-delay on dynamics are discussed briefly. Finally, the simulation results verify the validity of the theoretical analysis.

Accurate real-time truck simulation via semirecursive formulation and Adams–Bashforth–Moulton algorithm

In this paper, a tailored four-step Adams–Bashforth–Moulton (ABM) algorithm is applied to a semirecursive formulation to perform a real-time simulation of a semitrailer truck. In the ABM algorithm, each integration step involves two function evaluations, namely predictor and corrector. This is fundamentally different when compared to the classic fourth-order Runge–Kutta (RK) integrator approach that contains four function evaluations. A semitrailer truck under investigation is modeled in term of a semirecursive method and simulated by using the presented ABM algorithm. The results show that the four-step ABM method can reduce CPU time almost 50% for solving the truck dynamics with very similar accuracy, in comparison to the fourth-order RK method. The presented ABM algorithm could be used in the semirecursive formulation to carry out accurate real-time simulation of medium-large vehicle systems.

Finite-Time Stability of a Time-Delay Fractional-Order Hydraulic Turbine Regulating System

Finite-time terminal sliding mode control of a time-delay fractional-order hydraulic turbine regulating system (HTRS) is studied. First, an improved Adams–Bashforth–Moulton algorithm is introduced to solve the fractional-order nonlinear system with a time delay. Then, given the unique advantage of fractional calculus and the great influence that time delay has on system stability, a time-delay fractional-order HTRS is introduced. Moreover, by means of a frequency distribution model, the transformation of the fractional-order HTRS is realized. To stabilize the system under the influence of a random disturbance, a novel terminal sliding surface and a controller are proposed, and the detailed mathematical deduction for system stability is given. Finally, the simulation results, compared with traditional proportional-integral-derivative control and the conventional sliding mode control in the existing literature, demonstrate the validity and significant advantages of the proposed finite-time control scheme.

Numerical solution of fractional-order time-varying delayed differential systems using Lagrange interpolation

In this paper, a numerical solution of fractional-order time-varying delayed differential systems using Lagrange interpolation is investigated. Based on Lagrange interpolation method, the Adams–Bashforth–Moulton algorithm has been extended to solve fractional-order time-varying delayed differential systems. Furthermore, a detailed error analysis of this algorithm is presented. A fractional-order time-varying delayed Hopfield neural network as numerical example is given. In addition, the different parameters in the fractional-order time-varying delayed neural network are considered. Finally, some simple and direct numerical methods which are compared with Lagrange interpolation method in the fractional-order time-varying delayed neural network are discussed. The example with numerical simulation clearly illustrated that the present method is reliable.

Adomian Decomposition Algorithm for Studying Incommensurate Fractional-Order Memristor-Based Chua’s System

Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical approximate solution of the incommensurate fractional-order memristor-based Chua’s system is investigated. Dynamical characteristics of the proposed system are studied by using phase diagram, bifurcation analysis and power spectrum. Research results show that compared with the Adams–Bashforth–Moulton algorithm, the Adomian decomposition algorithm yields more accurate results and its solution generally converges more rapidly. Compared with 3.776 achieved by the Adams–Bashforth–Moulton algorithm, the minimum order of the incommensurate fractional-order memristor-based Chua’s system solved by using Adomian decomposition algorithm is 1.76, which is much smaller. A reliable and efficient binary test for chaos, called “0–1 test”, is utilized to detect the presence of chaotic attractors in the system dynamics.

A new 5D hyperchaotic system with stable equilibrium point, transient chaotic behaviour and its fractional-order form

Hidden attractors with the family of stable equilibrium points in higher-dimensional systems are more interesting and difficult discover compared to other families of hidden attractors. In this paper, a new 5D hyperchaotic system is reported. The proposed system has only one stable equilibrium point. Hence, the new system belongs to the category of hidden attractors. Although some lower-dimensional chaotic systems with stable equilibrium points are available in the literature, but very few hyperchaotic systems with stable equilibrium points are reported. The new system is simple considering the number of terms, compared with the existing similar type of systems. The proposed system exhibits multistability and transient chaotic behaviour. The fractional-order counterpart of the proposed system is analysed using Adams–Bashforth–Moulton algorithm and the chaotic nature is validated by bifurcation diagram. The simulation results confirm the claims made in the paper.

The slow-fast dynamical behaviors of a hydro-turbine governing system under periodic excitations

This paper studies the dynamic evolution behaviors of the hydro-turbine governing system by using Adams–Bashforth–Moulton algorithm. Based on the non-autonomous dynamic model of the hydro-turbine governing system, the effects of the frequency and intensity of periodic excitation on the dynamic characteristics of the hydro-turbine governing system are analyzed in detail. Due to the different scales between the natural frequency and the excitation frequency, the fast-slow effect is obviously found on the behavior of the system under different motion modes. Furthermore, the influence rules of the fast-slow effect for the dynamic behavior of the hydro-turbine governing system are given. The results of the study can contribute to the optimization analysis and control of the hydro-turbine governing system in practical process.

Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system

This paper present Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Firstly, regarding the time delay τ as a bifurcation parameter, we investigate the stability and bifurcation behaviors of this fractional order delayed memristor-based chaotic circuit system. Some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. Secondly, corresponding to different system parameters, the complex dynamics behaviors of this system are discussed by using the bifurcation diagrams, the Max Lyapunov exponents (MLEs) diagram, the time domain waveforms, the phase portraits and the power spectrums. Thirdly, we study the influence of the two parameters (time delay τ and fractional order q) on the chaotic behavior, and it is found when time delay τ and fractional order q increases, the transitions from period one to period two and period four to chaos are observed in this memristor–based system. Meanwhile, corresponding critical values of time delay τ and fractional order q, the lowest orders q and the minimum time delay τ for generating chaos in the fractional order delayed memristor–based system are determined, respectively. Also, when the system occurs period one, the corresponding frequency is verified theoretically and experimentally. Finally, numerical simulations are provided to demonstrate the validity of theoretical analysis using the improved Adams–Bashforth–Moulton algorithm.

Finite-time stability of a class of nonlinear fractional-order system with the discrete time delay

ABSTRACTThis paper investigates the finite-time stability problem of a class of nonlinear fractional-order system with the discrete time delay. Employing the Laplace transform, the Mittag-Leffler function and the generalised Gronwall inequality, the new criterions are derived to guarantee the finite-time stability of the system with the fractional-order 0 < α < 1. Further, we propose the sufficient conditions for ensuring the finite-time stability of the system with the fractional-order 1 < α < 2. Finally, based on the modified Adams–Bashforth–Moulton algorithm for solving fractional-order differential equations with the time delay, we carry out the numerical simulations to demonstrate the effectiveness of the proposed results, and calculate the estimated time of the finite-time stability.

Complex nonlinear dynamics in fractional and integer order memristor-based systems

In this paper, a fractional-order (and an integer-order) memristor-based system with the flux-controlled memristor characterized by smooth quadratic nonlinearity is proposed and detailed dynamical analysis is carried out by means of theoretical and numerical methods. To be more specific, stability of each equilibrium point in the equilibrium set is analyzed for the integer-order memristive system. Meanwhile, dynamical behavior depending on the initial states of the memristor is investigated and dynamical bifurcation depending on the slope of the memductance function is also considered. The bifurcation analysis is verified by numerical methods, including phase portraits, bifurcation diagrams, Lyapunov exponents spectrum, and Poincaré mappings. For the fractional-order case, based on the fractional-order stability theory, stability analysis is carried out just for a certain equilibrium point. Moreover, bifurcation behavior depending on the incommensurate order is discussed by virtue of numerical methods based on the Adams–Bashforth–Moulton algorithm. This paper indicates how the fractional order model and the initial state of the memristor extend the dynamical behaviors of the traditional chaotic systems.

Chaotic Behavior and Its Control in a Fractional-Order Energy Demand–Supply System

In this paper, we first propose a fractional-order energy demand–supply system, with the background of the energy resources demand in the eastern regions of China and the energy resources supply in the western regions of China. Then, we confirm the energy resource attractor with a necessary condition about the existence of chaotic behaviors. By employing an improved version of Adams Bashforth Moulton algorithm, we use three cases with different fractional values to verify the necessary condition. Finally, chaos control of fractional-order energy demand–supply system is investigated by two different control strategies: a linear feedback control and an adaptive switching control strategy via a single control input. Numerical simulations show that the energy demand and import in Eastern China and energy supply in Western China are self-feedback controlled around the system’s equilibrium point.

Numerical simulation for SI model with variable-order fractional

In this paper numerical studies for the variable-order fractional delay differential equations are presented. Adams-BashforthMoulton algorithm has been extended to study this problem, where the derivative is defined in the Caputo variable-order fractional sense. Numerical test examples are presented to demonstrate utility of the method. Chaotic behaviors are observed in variable-order one dimensional delayed systems.

Chaos in fractional and integer order NSG systems

The nuclear spin generator (NSG) is a high-frequency oscillator that generates and controls the oscillations of the precessional motion of a nuclear magnetization vector in a magnetic field. This nonlinear system was first described by S. Sherman in 1963, and exhibits a wide variety of chaotic behavior, but it is not as well studied as the classic Lorenz chaotic system. In this paper, chaos in the integer order nuclear spin generator system is reviewed. In addition, using fractional order stability analysis, the chaotic behavior of the fractional order NSG (FNSG) is studied. The numerical results are obtained using the Adams–Bashforth–Moulton algorithm encoded in the fde12 Matlab function. In order to confirm the numerically demonstrated chaotic behavior in the nuclear spin generator, we prepared a bifurcation diagram. The phase portrait of the FNSG is also depicted for different fractional orders to show the overall chaotic behavior of the system. These results are also verified using bifurcation analysis. Our results demonstrate a modulating effect on chaos as the fractional order decreases, which could be used to improve the design of the controller in the NSG model. This work also demonstrates how the fractional order model extends the dynamic behavior of the NSG system.