Fractional-order Duffing System Research Articles

Numerical simulation of fractional-order Duffing system with extended Mittag-Leffler derivatives

In this paper, we studied the dynamics of a nonlinear fractional-order Duffing system combined with Mittag-Leffler derivatives in order to provide dynamic behaviors different from existing ones. The Mittag-Leffler derivative is a generalized version of the exponential kernel derivative. To achieve this goal, we introduced a modified extension to higher-order Mittag-Leffler derivatives to overcome the initialization problem. Moreover, we discussed some properties and relationships of the studied derivatives. Then we presented numerical schemes to handle fractional extensions of the considered oscillatory system including the Mittag-Leffler and the Caputo derivatives. Numerical simulations are carried out and the resulting simulation dynamics of the studied fractional oscillatory system are compared.

Weak Signal Detection Application Based on Incommensurate Fractional-Order Duffing System

Weak Signal Detection Application Based on Incommensurate Fractional-Order Duffing System

Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation

The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation are discussed in detail, and it was found that Hopf bifurcation only happens while the fractional order is bigger than 1. On the other hand, the influence of the amplitude of parametric excitation on cluster oscillation models is discussed. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The point–point cluster oscillation only relates to pitchfork bifurcation. The point–cycle cluster oscillation includes pitchfork bifurcation and Hopf bifurcation. The point–cycle–cycle cluster oscillation involves three kinds of bifurcation, i.e., the pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation. The larger the amplitude, the more bifurcation types are involved. The research results of cluster oscillation and its generation mechanism will provide valuable theoretical basis for mechanical manufacturing and engineering practice.

Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales

The dynamical behavior on fractional-order Duffing system with two time scales is investigated, and the point-cycle coupling type cluster oscillation is firstly observed herein. When taking the fractional order as bifurcation parameter, the dynamics of the autonomous Duffing system will become more complex than the corresponding integer-order one, and some typical phenomenon exist only in the fractional-order one. Different attractors exist in various parameter space, and Hopf bifurcation only happens while fractional order is bigger than 1 under certain parameter condition. Moreover, the bifurcation behavior of the autonomous system may regulate dynamical phenomenon of the periodic excited system. It results into the point-cycle coupling type cluster oscillation when the fractional order is bigger than 1. The related generation mechanism based on slow-fast analysis method is that the slow variation of periodic excitation makes the system periodically visit different attractors and critical points of different bifurcations of the autonomous system.

Novel fractional-order chaotic systems of different order and multiswitching synchronisation

This paper gives multiswitching synchronisation scheme for a class of fractional-order chaotic systems by combining active and adaptive control theories. Adaptive controllers have been designed by using different laws of switching and fractional-order Lyapunov stability theory. We have also constructed a new fractional-order Duffing system. The fractional-order Duffing system and fractional-order Rabinovich–Fabrikant system have been taken as the drive system and the response system respectively. Applications have been demonstrated. Theoretical analysis and numerical simulations are also given to verify the robustness of the proposed controllers.

Finite-Time Neural Network Backstepping Control of an Uncertain Fractional-Order Duffing System With Input Saturation

In this paper, neural network (NN) control of fractional-order Duffing system (FODS) by suing backstepping method within finite-time in the presence of input saturation is investigated. A fractional-order filter with order lying on the interval (1,2) is used to estimate the virtual input together with its fractional derivative, and the estimation error tends to a small region in some finite time. Fractional-order law is designed for the parameter of NN, and an adaptive NN controller is given. The proposed method drives the tracking error tending to an arbitrary small region within finite time. The simulation results verify the validity of the proposed method.

Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

Sequential Parameter Identification of Fractional-Order Duffing System Based on Differential Evolution Algorithm

Using the dynamic properties of fractional-order Duffing system, a sequential parameter identification method based on differential evolution optimization algorithm is proposed for the fractional-order Duffing system. Due to the step by step parameter identification method, the dimension of parameter identification of each step is greatly reduced and the search capability of the differential evolution algorithm has been greatly improved. The simulation results show that the proposed method has higher convergence reliability and accuracy of identification and also has high robustness in the presence of measurement noise.

Global bifurcations in fractional-order chaotic systems with an extended generalized cell mapping method.

Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.

Crises in a non-autonomous fractional-order Duffing system

In this paper, the crises in a non-autonomous fractional-order Duffing system are investigated. Firstly, based on the short memory principle of fractional derivative, a global numerical method called an extended generalized cell mapping (EGCM), which combines the generalized cell mapping with the improved predictor-corrector algorithm, is proposed for fractional-order nonlinear systems. The one-step transition probability matrix of Markov chain of the EGCM is generated by the improved predictor-corrector approach for fractional-order systems. The one-step mapping time of the proposed method is evaluated with the help of the short memory principle for fractional derivative to deal with its non-local property and to properly define a bound of the truncation error by considering the features of cell mapping. On the basis of the characteristics of the cell state space, the bound of the truncation error is defined to ensure that the truncation error is less than half a cell size. For a fractional-order Duffing system, boundary and interior crises with varying the derivative order and the intensity of external excitation are determined by the EGCM method. A boundary crisis results from the collision of a chaotic (or regular) saddle in the fractal (or smooth) basin boundary with a periodic (or chaotic) attractor. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes a chaotic attractor to occur, and simultaneously the previous attractor and the unstable chaotic set are converted into a part of the chaotic attractor. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause the chaotic set to have a sudden discontinuous change. Here the chaotic set involves three different kinds of chaotic basic sets: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. The results further reveal that the EGCM is a powerful tool to determine the global dynamics of fractional-order systems.

Nonlinear dynamics of fractional order Duffing system

In this paper, we analyze the nonlinear dynamics of fractional order Duffing system. First, we present the fractional order Duffing system and the numerical algorithm. Second, nonlinear dynamic behaviors of Duffing system with a fixed fractional order is studied by using bifurcation diagrams, phase portraits, Poincare maps and time domain waveforms. The fractional order Duffing system shows some interesting dynamical behaviors. Third, a series of Duffing systems with different fractional orders are analyzed by using bifurcation diagrams. The impacts of fractional orders on the tendency of dynamical motion, the periodic windows in chaos, the bifurcation points and the distance between the first and the last bifurcation points are respectively studied, in which some basic laws are discovered and summarized. This paper reflects that the integer order system and the fractional order one have close relationship and an integer order system is a special case of fractional order ones.

Undamped oscillations in fractional-order Duffing oscillator

This paper studies undamped oscillations of fractional-order Duffing system. Stability theorems for fractional order systems are used to determine the characteristic polynomial of the system in order to find the parametric ranges for undamped oscillations in this system. We also derive relations for estimating the frequency and the amplitude of the oscillations in this system using a describing function method. Finally numerical simulation results are provided to justify the analysis.

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lu system, and fractional-order Duffing system.

EFFECTS OF FRACTIONAL DIFFERENTIATION ORDERS ON A COUPLED DUFFING CIRCUIT

Classical electrical circuits consist of resistors and capacitors and are governed by integer-order model. Circuits may have so-called fractance which represents an electrical element with fractional order impedance. Therefore, fractional order derivative is important to study the dynamical behaviors of circuits. This paper extends the classical coupled Duffing circuits to cover a new fractional order Duffing system consisting of two identical periodic forced circuits coupled by a linear resistor. The fundamental resonance responses under various paired and unpaired fractional orders are investigated in detail using the harmonic balance in combination with polynomial homotopy continuation. The approximate solutions having a high degree of accuracy in the steady state response are sought. There exist different shapes of frequency versus response and excitation amplitude versus response curves under various fractional orders. Multiple-valued solutions and nonclassical bifurcations are observed analytically and verified numerically. The influence of coupling intensity on the fundamental resonance response is also examined. New contributions include the innovative introduction of fractional order to the coupled Duffing circuits, the explicit integration of fractional order and the linear consideration of higher harmonics to improve the nonlinear solutions of the lower harmonics without increasing the computational complexity.

Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system

We develop an analytical technique to investigate the phenomenon of vibrational resonance in a fractional-order Duffing system with linear time delay feedback and driven by both low frequency and high frequency periodic signals. At first, the theoretical predication of the response amplitude at the low-frequency is obtained by the method of direct separation of slow and fast motions. Then, the bifurcation analysis is carried out based on three kinds of resonance behaviors. Further, influences of the high frequency signal, the fractional-order damping and the delay parameter on the vibrational resonance are discussed by both theoretical and numerical simulations. If the value of the fractional-order is a controllable parameter, the monotonicity of the response amplitude versus the value of the fractional-order depends on the amplitude of the high-frequency signal. If the delay parameter is a controllable parameter, the response amplitude with respect to the delay parameter presents periodic or quasi-periodic pattern, and it is similar to that in the integer-order differential system with linear time delay feedback. The good agreement between the analytical and numerical results indicates the validity of the theoretical predications.

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology, the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research. In this paper, the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated. With the stability criterion of linear fractional systems, the synchronization of a fractional non-autonomous system is obtained. Specifically, an effective singly active control is proposed and used to synchronize a fractional order Duffing system. The numerical results demonstrate the effectiveness of the proposed methods.