On chaos control and synchronization of the commensurate fractional order Liu system

doi:10.1016/j.cnsns.2012.09.026

Communications in Nonlinear Science and Numerical Simulation

Volume 18, Issue 5, May 2013, Pages 1193-1202

https://doi.org/10.1016/j.cnsns.2012.09.026 Get rights and content

Abstract

In this work, we study chaos control and synchronization of the commensurate fractional order Liu system. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate fractional order systems are discussed. The existence and uniqueness of solutions for a class of commensurate fractional order Liu systems are investigated. We also obtain the necessary condition for the existence of chaotic attractors in the commensurate fractional order Liu system. The effect of fractional order on chaos control of this system is revealed by showing that the commensurate fractional order Liu system is controllable just in the fractional order case when using a specific choice of controllers. Moreover, we achieve chaos synchronization between the commensurate fractional order Liu system and its integer order counterpart via function projective synchronization. Numerical simulations are used to verify the analytical results.

Highlights

► We study the conditions of local stability of nonlinear fractional order system. ► Existence and uniqueness of solutions of the fractional Liu systems are discussed. ► We derive the necessary condition for the existence of chaos in this system. ► We show the effect of fractional order on the chaos control of this system. ► A function projective synchronization technique has been applied to the system.

Introduction

The story of fractional calculus was initiated 300 years ago. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. Facts show that many systems can be elegantly described with the help of fractional derivatives in interdisciplinary fields, for example, electromagnetic waves [1], visco-elastic systems [2], diffusion waves [3] and nonlinear oscillation of earthquakes [4]. Furthermore, applications of fractional calculus have been reported in many areas such as physics [5], engineering [6], mathematical biology [7], [8], [9], finance [10], economy [11] and social sciences [12], [13].

There are many definitions of fractional order derivatives. The Riemann–Liouville definition [6] is one of these definitions and is given by $D_{*}^{α} f (t) = \frac{d^{l}}{{dt}^{l}} J^{l - α} f (t), α > 0,$ where J^θ is the θ-order Riemann–Liouville integral operator which is given as $J^{θ} u (t) = \frac{1}{Γ (θ)} \int_{0}^{t} (t - τ)^{θ - 1} u (τ) d τ, θ > 0 .$

Another one is the Caputo definition of fractional derivatives [14], which is often used in real applications: $D^{α} f (t) = J^{l - α} f^{(l)} (t), α > 0,$ where f⁽^l⁾ represents the l-order derivative of f(t) and l = [α], this means that l is the first integer which is not less than α. The operator D^α is called “Caputo differential operator of order α”.

On the other hand, chaos is an important dynamical phenomenon which has been extensively studied and developed by scientists since the work of Lorenz in 1963 [15]. The chaotic Liu system is one of the so called Lorenz family [16]. The Liu system has potential applications in secure communications [17] and its dynamical behaviors had been studied in [18]. Studying chaos in fractional order dynamical systems has become an interesting topic as well. Chaotic behaviors have been found in the fractional order systems of Lorenz [19], Chua [20], Chen [21], Rössler [22], Coullet [23], modified Van der Pol-Duffing [24] and Liu [25]. The fractional order Liu system has a circuit realization as shown in Ref. [26]. Also, the fractional order Liu system has been studied with time-delay in [27]. In Ref. [25], the authors have studied the dynamical behaviors of the fractional order Liu system with incommensurate orders using the classical stability theory of fractional order systems. It has been shown that, chaos in fractional order autonomous systems can occur for orders less than three and this cannot happen in their integer order counterparts according to the Poincaré-Bendixon theorem. Moreover, chaos control and synchronization are useful applications of chaos theory. Indeed, studying chaos control and synchronization in fractional order chaotic systems is just a recent focus of interest [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40] due to their potential applications in industry and secure communications. So, when a fractional order system is chaotic and how to control and synchronize this chaos have been three very important problems.

In the past, most studies used the stability theory of fractional order systems in their dynamical analysis. This paper applies the fractional Routh-Hurwitz criteria to study chaos control of the commensurate fractional order Liu system and obtain some useful results. Using these new stability criteria for chaos control in fractional order systems enables us to understand the effect of fractional order on chaos control of Liu system. Thus, our new results are to show that the commensurate fractional order Liu system is controllable just in the fractional order case when using a specific choice of controllers. Moreover, a function projective synchronization technique (FPS) is used to achieve chaos synchronization between the integer order Liu system and its commensurate fractional order form. The FPS is the most general definition of projective synchronization (PS) and could additionally be used to enhance the security of communication.

The rest of the paper is organized as follows: In Section 2, a review on fractional calculus is given. In Section 3, the commensurate fractional order Liu system is presented then the existence and uniqueness of solutions for a class of this system are investigated. A necessary condition for the existence of chaotic attractor in the commensurate fractional order Liu system is shown in Section 4. Section 5 is devoted to show the effect of fractional order on chaos control of this system. In Section 6, synchronization between the integer order Liu system and its commensurate fractional order form is achieved. Finally, in Section 7, conclusions are drawn.

Section snippets

Preliminaries

Consider the initial value problem: $D^{α} x (t) = f (t, x (t)), 0 ⩽ t ⩽ T, x^{(k)} (0) = x_{0}^{(k)}, k = 0, 1, . . ., l - 1 .$

Theorem 1

Existence [41]

Assume that $E : = [0, χ^{*}] \times [x_{0}^{(0)} - ε, x_{0}^{(0)} + ε]$ with some χ^∗ > 0 and some ɛ > 0, and let the function f:E → R be continuous. In addition, let χ: = min {χ^∗,(ɛΓ(α + 1)/||f||_∞)^1/^α}. Then, there exists a function x:[0, χ] → R solving the initial value problem (4).

Theorem 2

Uniqueness [41]

Assume that $E : = [0, χ^{*}] \times [x_{0}^{(0)} - ε, x_{0}^{(0)} + ε]$ with some χ^∗ > 0 and some ɛ > 0. Moreover, let the function f:E → R be bounded on E and satisfy a Lipschitz condition with respect to the

The commensurate fractional order Liu system

The commensurate fractional order Liu system is given as follows: $D^{α} x_{1} = a (x_{2} - x_{1}), D^{α} x_{2} = {bx}_{1} - {kx}_{1} x_{3}, D^{α} x_{3} = - {cx}_{3} + σ x_{1}^{2},$ where α is the fractional order and $α \in (0, 1]$ . The parameters a, c, k, σ are all positive real parameters and b ϵ R. When α = 1, system (12) becomes the original integer order Liu system which exhibits chaotic behaviors at the parameter values a = 10, b = 40, c = 2.5, σ = 4 and k = 1.

To evaluate the equilibrium points, let $D^{α} x_{1} (t) = 0, D^{α} x_{2} (t) = 0$ and D^αx₃(t) = 0, then $E_{0} = (0, 0, 0), E_{1} = (\sqrt{bc / σ k}, \sqrt{bc / σ k}, b / k)$ and $E_{2} = (-$

Chaos in the commensurate fractional order Liu system

An efficient method for solving fractional order differential equations is the predictor–correctors scheme or more precisely, PECE (Predict, Evaluate, Correct, Evaluate) [41], [44], which represents a generalization of Adams–Bashforth-Moulton algorithm. This method is described as follows:

Set $h = T / N, t_{n} = nh, n = 0, 1, . . ., N \in Z^{+}$ . Then Eq. (5) can be discretized as follows: $x_{h} (t_{n + 1}) = \sum_{k = 0}^{[α] - 1} \frac{t_{n + 1}^{k}}{k!} x_{0}^{(k)} + \frac{h^{α}}{Γ (α + 2)} f (t_{n + 1}, x_{h}^{p} (t_{n + 1})) + \frac{h^{α}}{Γ (α + 2)} \sum μ_{j, n + 1} f (t_{j}, x_{h} (t_{j})),$ where $μ_{j, n + 1} = \{\begin{matrix} n^{α + 1} - (n - α) (n + 1)^{α}, j = 0, \\ (n - j + 2)^{α + 1} + (n - j)^{α} \end{matrix}$

Controlling chaos

A three-dimensional commensurate fractional order chaotic system is described as $\frac{d^{α} x}{{dt}^{α}} = f (x),$ where x = (x₁, x₂, x₃). The controlled system is given as $\frac{d^{α} x}{{dt}^{α}} = f (x) - K (x - \bar{x}),$ where $K = diag (κ_{1}, κ_{2}, κ_{3}), κ_{1}, κ_{2}, κ_{3} ⩾ 0$ and $\bar{x} = ({\bar{x}}_{1}, {\bar{x}}_{2}, {\bar{x}}_{3})$ is an equilibrium point of (16). Now, if one selects the appropriate feedback control gains $κ_{1}, κ_{2}, κ_{3}$ which then make the eigenvalues of the linearized equation of the controlled system (17) satisfy the above-mentioned stability conditions (i) and (ii), then the trajectories of

The Function projective synchronization between the commensurate fractional order Liu system and its integer order form

Here, our goal is to achieve chaos synchronization between the commensurate fractional order Liu system and its integer order form using function projective synchronization technique (FPS) which means that the drive and response systems could be synchronized up to a scaling function, but not a constant. Thus, FPS is considered to be more general definition of projective synchronization (PS), and it is used to obtain higher unpredictability of the error dynamical system which can enhance the

Conclusions

We have studied chaos control and synchronization of the commensurate fractional order Liu system. The existence and uniqueness of solutions of this system have been investigated. We have shown that chaos exists in this system with order less than three. Also, the effect of fractional order on chaos control has been shown; the commensurate fractional order Liu system is controlled to the equilibrium points E₁ and E₂ unlike its integer order counterpart when using specific choice of linear

Acknowledgements

The authors would like to thank the anonymous reviewers for their useful comments and suggestions which help to improve the style of this work.

References (46)

A.E.M. El-Misiery et al.
On a fractional model for earthquakes
Appl. Math. Comput.
(2006)
E. Ahmed et al.
On fractional order differential equations model for nonlocal epidemics
Phys. A
(2007)
A.M.A. El-Sayed et al.
On the fractional-order logistic equation
Appl. Math. Lett.
(2007)
E. Ahmed et al.
Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models
J. Math. Anal. Appl.
(2007)
N. Laskin
Fractional market dynamics
Phys. A
(2000)
B. Xin et al.
Projective synchronization of chaotic fractional-order energy resources demand-supply systems via linear control
Commun. Nonlinear Sci. Numer. Simulat.
(2011)
W.M. Ahmad et al.
Fractional-order dynamical models of love
Chaos Solitons Fract.
(2007)
L. Song et al.
Dynamical models of happiness with fractional order
Commun. Nonlinear Sci. Numer. Simulat.
(2010)
C. Liu et al.
A new chaotic attractor
Chaos Solitons Fract.
(2004)
X.Y. Wang et al.
A chaotic secure communication scheme based on observer
Commun. Nonlinear Sci. Numer. Simulat.
(2009)

A.E. Matouk

Dynamical analysis, feedback control and synchronization of Liu dynamical system

Nonlinear Anal. Theory

(2008)

C.P. Li et al.

Chaos in Chen’s system with a fractional order

Chaos Solitons Fract.

(2004)

C.G. Li et al.

Chaos and hyperchaos in the fractional-order Rössler equations

Phys. A

(2004)

M. Shahiri et al.

Chaotic fractional-order Coul- let system: synchronization and control approach

Commun. Nonlinear Sci. Numer. Simulat.

(2010)

A.E. Matouk

Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit

Commun. Nonlinear Sci. Numer. Simulat.

(2011)

S. Bhalekar et al.

Fractional ordered Liu system with time-delay

Commun. Nonlinear Sci. Numer. Simulat.

(2010)

W.M. Ahmad et al.

On nonlinear control design for autonomous chaotic systems of integer and fractional orders

Chaos Solitons Fract.

(2003)

M.S. Tavazoei et al.

Synchronization of chaotic fractional-order systems via active sliding mode controller

Phys. A

(2008)

G.H. Erjaee et al.

Phase synchronization in fractional differential chaotic systems

Phys. Lett. A

(2008)

A.E. Matouk

Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system

Phys. Lett. A

(2009)

S. Bhalekar et al.

Synchronization of different fractional order chaotic systems using active control

Commun. Nonlinear Sci. Numer. Simulat.

(2010)

P. Zhou et al.

Function projective synchronization for fractional-order chaotic systems

Nonlinear Anal. Real

(2011)

M.M. Asheghan et al.

Robust synchronization of perturbed Chen’s fractional-order chaotic systems

Commun. Nonlinear Sci. Numer. Simulat.

(2011)

Cited by (110)

Parametric fractional-order analysis of Arneodo chaotic system and microcontroller-based secure communication implementation
2024, AEU - International Journal of Electronics and Communications
Chaotic systems are often used in cryptology applications thanks to their complex dynamic structures. By increasing the dynamic diversity of chaotic systems through fractional-order analysis, more complex systems for data security can be obtained. In this study, fractional calculus analyses of the Arneodo chaotic jerk system with cubic nonlinearity and its microcontroller-based chaotic masking application are presented. Numerical analyses such as bifurcation diagrams, spectral entropy complexity, and Lyapunov spectra are performed to examine the dynamic characteristics of the fractional-order system. The parametric analysis indicates the presence of chaotic states in the system for nearly all fractional-order values ranging from 0.55 to 1. In this manner, the study focuses on investigating the minimum applicable fractional-order value for use in secure communication applications, constituting the main contribution of this research. Consequently, the dynamic diversity and unpredictability of the cubic system are greatly increased. Moreover, two fractional-order Arneodo chaotic (FOAC) systems with distinct initial conditions are synchronized using a single-state fractional-order sliding mode controller (FOSMC). Finally, an innovative secure communication system based on a microcontroller is designed and implemented by employing the chaotic masking method. As anticipated, the voltage outputs of the microcontroller-based secure communication application demonstrated good agreement with the numerical simulations.
Studying changes in the dynamical patterns in two physical models involving new Caputo operator
2024, Journal of Advanced Research
Studying phase changes in physical models enables researchers to adequately describe the dynamics arising from such systems.
This work investigates the effect of a new Caputo fractional operator on the dynamics in two physical models.
The corresponding Riemann-Liouville fractional integral is presented. The main properties of the new Riemann-Liouville fractional integral are mentioned. In the case of the new Caputo fractional operator, new lemmas for derivatives of constant function, power function and its effect on the new fractional integral fractional integral are proved. In addition, an interpolation property is presented and proved. The new operator has a higher degree of freedom (new fractional parameter), which is used to produce changes in the dynamical patterns in some dynamical models. For example, the fractional modified autonomous Van der Pol-Duffing system exhibits variety of complex attractors such as one-band, double-band chaos and approximate periodic cycles as varying the new fractional parameter. In addition, the fractional Liu system shows variety of complex dynamics such as existences of double scroll, self-excited and hidden chaotic attractors.
Both of the systems involving the new fractional operators exhibit variety of chaotic dynamics. It is also found in both systems that the new operator’s parameter arises chaotic attractors, while non-chaotic states are found in the systems’ counterparts with the classic fractional operator.
Using particle swarm optimization and genetic algorithms for optimal control of non-linear fractional-order chaotic system of cancer cells
2023, Mathematics and Computers in Simulation
The purpose of this paper is to investigate optimal control the non-linear fractional-order chaotic system of cancer cells by use of particle swarm optimization and genetic algorithms. The chaotic behavior of cancer cell growth and the tumor growth model were expressed as a system of differential equations. Then, optimal control of cancer cells using drugs was presented. Particle swarm optimization method and genetic algorithms were used to solve the problem of optimal control of cancer cell growth. The results of the control applied to the model can control the cancer cell growth. The application of control results in decreasing the number of cancer cells to zero. When the controller is applied from the beginning, the Results of genetic algorithm method are excellent. All the results obtained for the particle swarm optimization method show that this method has also been very successful and have results very close to the genetic algorithm method.
Chaotic attractors that exist only in fractional-order case
2023, Journal of Advanced Research
Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena.
Phase transitions between the fractional- and integer-order cases is one of the main problems that have been extensively examined by scientists, economists, and engineers. This paper reports the existence of chaotic attractors that exist only in the fractional-order case when using the specific selection of parameter values in a new hyperchaotic (Matouk’s) system.
This paper discusses stability analysis of the steady-state solutions, existence of hidden chaotic attractors and self-excited chaotic attractors. The results are supported by computing basin sets of attractions, bifurcation diagrams and the Lyapunov exponent spectrum. These tools verify the existence of chaotic dynamics in the fractional-order case; however, the corresponding integer-order counterpart exhibits quasi-periodic dynamics when using the same choice of initial conditions and parameter set. Projective synchronization via non-linear controllers is also achieved between drive and response states of the hidden chaotic attractors of the fractional Matouk’s system.
Dynamical analysis and computer simulation results verify that the chaotic attractors exist only in the fractional-order case when using the specific selection of parameter values in the Matouk’s hyperchaotic system.
An example of the existence of hidden and self-excited chaotic attractors that appears only in the fractional-order case is discussed. So, the obtained results give the first example that shows chaotic states are not necessarily transmitted between fractional- and integer-order dynamical systems when using a specific selection of parameter values. Chaos synchronization using the hidden attractors’ manifolds provides new challenges in chaos-based applications to technology and industrial fields.
Novel fractional-order decentralized control for nonlinear fractional-order composite systems with time delays
2022, ISA Transactions
A novel decentralized non-integer order controller applied on nonlinear fractional-order composite system(NFOCS) is proposed. In addition, some novel results for the asymptotic stabilization are shown with fractional parameter $α \in (0, 1) .$ First, we derive certain novel results useful for the Mittag-Leffler function. Then, we design a new decentralized fractional-order controller for the NFOCS according to the novel results applied to Mittag-Leffler function. Next, this novel asymptotic stabilization condition has been proposed. Compared with other controllers our controller has wider control gain range and weaker requirements. Moreover, we solve the asymptotic stabilization problem of the NFOCS with time delays via the novel controller. In the end, four general examples are performed to show the progressiveness of the new fractional-order decentralized controller.
Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel
2020, Journal of Advanced Research
Fractional calculus (FC) is widely used in many interdisciplinary branches of science due to its effectiveness in describing and investigating complicated phenomena. In this work, nonlinear dynamics for a new physical model using nonlocal fractional differential operator with singular kernel is introduced. New Routh-Hurwitz stability conditions are derived for the fractional case as the order lies in [0,2). The new and basic Routh-Hurwitz conditions are applied to the commensurate case. The local stability of the incommensurate orders is also discussed. A sufficient condition is used to prove that the solution of the proposed system exists and is unique in a specific region. Conditions for the approximating periodic solution in this model via Hopf bifurcation theory are discussed. Chaotic dynamics are found in the commensurate system for a wide range of fractional orders. The Lyapunov exponents and Lyapunov spectrum of the model are provided. Suppressing chaos in this system is also achieved via two different methods.

View all citing articles on Scopus

View full text

On chaos control and synchronization of the commensurate fractional order Liu system

Abstract

Highlights

Introduction

Section snippets

Preliminaries

Existence [41]

Uniqueness [41]

The commensurate fractional order Liu system

Chaos in the commensurate fractional order Liu system

Controlling chaos

The Function projective synchronization between the commensurate fractional order Liu system and its integer order form

Conclusions

Acknowledgements

Appl. Math. Comput.

Phys. A

Appl. Math. Lett.

J. Math. Anal. Appl.

Phys. A

Commun. Nonlinear Sci. Numer. Simulat.

Chaos Solitons Fract.

Commun. Nonlinear Sci. Numer. Simulat.

Chaos Solitons Fract.

Commun. Nonlinear Sci. Numer. Simulat.

Nonlinear Anal. Theory

Chaos Solitons Fract.

Phys. A

Commun. Nonlinear Sci. Numer. Simulat.

Commun. Nonlinear Sci. Numer. Simulat.

Commun. Nonlinear Sci. Numer. Simulat.

Chaos Solitons Fract.

Phys. A

Phys. Lett. A

Phys. Lett. A

Commun. Nonlinear Sci. Numer. Simulat.

Nonlinear Anal. Real

Commun. Nonlinear Sci. Numer. Simulat.