Fractional-order Dynamical Systems Research Articles

Chaotic Control in a Fractional-Order Modified Van Der Pol Oscillator

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator. We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4. Finally, a simple, but effective, linear feedback controller is also designed to stabilize the fractional order chaotic van der Pol oscillator.

Inverse synchronization of coupled fractional-order systems through open-plus-closed-loop control

In this paper, the inverse synchronization problem of fractional-order dynamical systems is investigated. A general explicit coupling via an open-plus-closed-loop control for inverse synchronization of two arbitrary unidirectionally or bidirectionally coupled fractional-order systems is proposed. The inverse synchronization is proved analytically based on the stability theorem of the fractional differential equations. A key feature of this proposed scheme is that it can be applied not only to nonchaotic but also to chaotic fractional-order systems whenever they exhibit regular or irregular oscillations. Feasibility of the proposed inverse synchronization scheme is illustrated through numerical simulations.

Chaos in a Fractional-Order Modified Van Der Pol Oscillator

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator . We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4.

Complexity evolvement of a chaotic fractional-orderfinancial system

A novel science branch, econophysics, is set up because various physical theories and methods are applied to economic and financial fields. More and more researchers are fascinated by the complex dynamical behavior of the fractional-order dynamical system. The paper analyzes the stability of the fractional-order financial system, and then simulates the generalized model complexity with Adams-Bashforth-Moulton predictor-corrector scheme by using bifurcation diagram, phase portrait and history time-series.

Generations and mechanisms of multi-stripe chaotic attractors of fractional order dynamic system

In this paper, the generations of multi-stripe chaotic attractors of fractional order system are considered. The original fractional order chaotic attractors can be turned into a pattern with multiple “parallel” or “ rectangular” stripes by employing certain simple periodic nonlinear functions. The relationships between the parameters relate to the periodic functions and the shape of the generated attractors are analyzed. Theoretical investigations about the underlying mechanisms of the parallel striped attractors of fractional order system are presented, with the fractional order Lorenz, Rössler and Chua’s systems as examples. Moreover, the periodic doubling striped route to chaos of fractional order Rössler system and maximum Lyaponov exponent calculations are also given.

Can We Obtain a Fractional Lorenz System from a Physical Problem?

A new fractional-order Lorenz system is obtained from the convection of fractional Maxwell fluids in a circular loop. This is the first fractional-order dynamical system derived from an actual physical problem, and rich dynamical properties are observed. In the case of short fluid relaxation time, with the decreasing effective dimension Σ, we find a critical value of the effective dimension Σcr1, at which the solution of the system undergoes a transition from the chaotic motion to the periodic motion and another critical value Σcr2(Σcr2 < Σcr1) at which the regular dynamics of the system returns to the chaotic one. In the case of long relaxation time, the phenomenon of overstability is observed and the decrease of Σ is found to delay the onset of it.

The 0-1 test algorithm for chaos and its applications

To determine whether a given deterministic nonlinear dynamic system is chaotic or periodic, a novel test approach named zero-one (0-1) test has been proposed recently. In this approach, the regular and chaotic motions can be decided by calculating the parameter K approaching asymptotically to zero or one. In this study, we focus on the 0-1 test algorithm and illustrate the selection of parameters of this algorithm by numerical experiments. To validate the reliability and the universality of this algorithm, it is applied to typical nonlinear dynamic systems, including fractional-order dynamic system.

Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System

This paper deals with implementation of the multistage Adomian decomposition method (MADM) to solve a class of nonlinear programming (NLP) problems, which are reformulated with a nonlinear system of fractional differential equations. The multistage strategy is used to investigate the relation between an equilibrium point of the fractional order dynamical system and an optimal solution of the NLP problem. The preference of the method lies in the fact that the multistage strategy gives this relation in an arbitrary longtime interval, while the Adomian decomposition method (ADM) gives the optimal solution just only in the neighborhood of the initial time. The numerical results taken by the fractional order MADM show that these results are compatible with the solution of NLP problem rather than the ADM. Furthermore, in some cases the fractional order MADM can perform more rapid convergency to the optimal solution of optimization problem than the integer order ones.

Delayed feedback control of fractional-order chaotic systems

We study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional-order dynamical systems by the time-delayed feedback method. By performing a linear stability analysis, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parameterized by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. We demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain, providing that the time delay is chosen to coincide with the period of the target orbit. In addition, it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of steady states in comparison to the classical time-delayed feedback scheme with a constant delay.

BIFURCATIONS AND CHAOS IN FRACTIONAL-ORDER SIMPLIFIED LORENZ SYSTEM

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.

Comparison between two approximation methods of state space fractional systems

A large number of real dynamic systems are better characterized using a non-integer order dynamic model based on fractional order differential equations. Nevertheless, their analytical solutions are rather complicated. In this paper we present two alternative methods of simulation of the fractional order dynamical system in state space. We particularly present a comparison between two approximation methods. The first one is based on the differentiation operator approximation and thus uses the usual form of the state space representation D ( α ) ( x ) = Ax + Bu . The second method is based on the approximation of the integration operator and uses, in this case, the new representation w = AI ( α ) ( w ) + Bu . The comparison between these two methods is made on the basis of initial and steady state approximation state and output errors which are characterized in this present paper. Numerical examples are also given to support this comparison.

Load frequency control and automatic generation control using fractional-order controllers

Recently, fractional calculus has received extensive attention and research. Accordingly, there is an increasing interest in fractional-order (FO) dynamic systems and controllers. The widely used classical integer-order proportional-integral controller and proportional-integral-derivative controller are usually adopted in the load frequency control (LFC) and automatic generation control (AGC) to improve the dynamic response and to eliminate or reduce steady-state errors. This paper utilizes the FO controllers to improve stability and response of LFC and AGC system. The paper uses the integral of the time-weighted absolute error performance index for optimal controller design. The paper investigates LFC and AGC for both isolated and interconnected power systems and shows that FO controllers perform better than classical integer-order controllers in theses systems.

Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag–Leffler stability and generalized Mittag–Leffler stability notions. With the definitions of Mittag–Leffler stability and generalized Mittag–Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.

A partly deterministic and partly stochastic scheme for the identification of fractional-order processes

This contribution proposes a simple yet elegant scheme for identifying a fractional-order dynamic system based on its observed response to a standard excitation. If the fractional powers in the transfer function are precisely known, a set of simultaneous linear equations connecting the unknown coefficient values is obtained and then solved deterministically to yield the desired estimates. In case the fractional powers are uncertain, these are estimated by formulating an optimisation problem and employing a stochastic search algorithm. Results show that the proposed method offers a high degree of accuracy even for data that are intentionally corrupted to simulate real-life conditions.

Fractional-order sinusoidal oscillators: Design procedure and practical examples

Sinusoidal oscillators are known to be realized using dynamical systems of second-order or higher. Here we derive the Barhkausen condition for a linear noninteger-order (fractional-order) dynamical system to oscillate. We show that the oscillation condition and oscillation frequency of some famous integer-order sinusoidal oscillators can be obtained as special cases from general equations governing their fractional-order counterparts. Examples including fractional-order Wien oscillators, Colpitts oscillator, phase-shift oscillator and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</i> tank resonator are given supported by numerical and PSpice simulations.

On stability, persistence, and Hopf bifurcation in fractional order dynamical systems

This is a preliminary study about the bifurcation phenomenon in fractional order dynamical systems. Persistence of some continuous time fractional order differential equations is studied. A numerical example for Hopf-type bifurcation in a fractional order system is given, hence we propose a modification of the conditions of Hopf bifurcation. Local stability of some biologically motivated functional equations is investigated.

Continued fractions approximation of the impulse response of fractional-order dynamic systems

Noninteger (fractional)-order controllers are recognised to guarantee better closed-loop performance and robustness with respect to conventional integer order controllers. But irrational transfer functions make time-domain analysis and simulation much difficult. A systematic approach is proposed here for inverting the transfer function of fractional systems of commensurate order. The approach is based on the continued fractions expansion to approximate irrational transfer functions by minimum-phase, stable, rational functions, which can be easily transformed in the time-domain. The accuracy of the designed low-order approximation is shown by simulation results.

Complete identification of dynamic physical systems using a five parameter model

System identification is important in science and engineering. This contribution proposes a simple yet elegant scheme for identifying a fractional order dynamic system based on its observed response to a standard excitation. Our approach consists in obtaining and solving a set of simultaneous linear equations connecting the parameters to yield the desired estimates. To minimise the effect of noise, we have performed an in-depth analysis of the Grunwald-Letnikov definition and, as far as we know, are the first ones to do so. Results show that the proposed method offers a very high degree of accuracy even for erroneous data.

THREE SCHEMES TO SYNCHRONIZE CHAOTIC FRACTIONAL-ORDER RUCKLIDGE SYSTEMS

The synchronization problem of chaotic fractional-order Rucklidge systems is studied both theoretically and numerically. Three different synchronization schemes based on the Pecora–Carroll principle, the linear feedback control and the bidirectional coupling are proposed to realize chaotic synchronization. It is shown that such schemes can achieve the same aim for the same set of system parameter values (including fractional orders). This provides an alternate choice for applications of fractional-order dynamical systems in engineering fields.

Numerical simulation of the fractional-order control system

Multi-term fractional differential equations have been used to simulate fractional-order control system. It has been demonstrated the necessity of the such controllers for the more efficient control of fractionalorder dynamical system. In this paper, the multi-term fractional ordinary differential equations are transferred into equivalent a system of equations. The existence and uniqueness of the new system are proved. A fractional order difference approximation is constructed by a decoupled technique and fractional-order numerical techniques. The consistence, convergence and stability of the numerical approximation are proved. Finally, some numerical results are presented to demonstrate that the numerical approximation is a computationally efficient method. The new method can be applied to solve the fractional-order control system.