Generalized synchronization of commensurate fractional-order chaotic systems: Applications in secure information transmission

doi:10.1016/j.dsp.2022.103494

Digital Signal Processing

Volume 126, 30 June 2022, 103494

https://doi.org/10.1016/j.dsp.2022.103494 Get rights and content

Abstract

In this work, a class of chaotic nonlinear fractional systems of commensurate order called Liouvillian systems is considered to solve the problem of generalized synchronization. To solve this problem, the master and the slave systems are expressed in the Fractional Generalized Observability Canonical Form (FGOCF), then a fractional-order dynamical control law is designed to achieve the generalized synchronization. The encryption of color images is presented as an application to the proposed synchronization method, the encryption algorithm allows to decrypt data without loss. The synchronization and its applications are then illustrated with numerical examples.

Introduction

The synchronization of chaotic systems has received many advances since its introduction. There are various types of synchronization, one of the most important is Generalized Synchronization which is the focus of this paper, generalized synchronization refers to the coupling of two systems, one called master and the other slave, both systems have different trajectories and different attractors, then it is said that the systems are synchronized when the trajectories of the slave system converge to the trajectories of the master system making them have the same at tractor. The coupling is usually a control law that uses data from the master to drive the states of the slave into convergence through a functional mapping [1], [2], [3].

Initially synchronization of chaotic systems was studied in integer-order dynamical systems [4], [5], [6], [7], as time passed several applications where found ([8], [9], [10], [11], [12]), then it was noticed that the behavior of the phenomenon in which synchronization was applied, could be better model by using fractional-order calculus. Some studies support the behavior of the fractional derivative for the modeling of systems with memory as well as the design of high-performance controllers, inducing properties such as stability or robustness that improve the performance of automatic closed-loop systems. Fractional-order systems are modeled using physical laws using the integer derivative, taking first variations. Consequently many advances in synchronization of fractional-order chaotic systems began to appear, including mathematical modeling, stability analysis and new forms of synchronization such as synchronization based on state observers, along with all the techniques, benefits and peculiarities that observers involve [13], [14], [15], [16], [17], [18], [19]. Works such as [20], [21], [22], [23], [24] are examples of this, the first one is about an observer using sliding modes, the second is an observer that does not need full knowledge of the master system to make the error converge close to zero and the last one is an application of the synchronization of fractional-order chaotic systems by state observers.

Using a state observer for synchronization requires that the system is observable, when the analysis of the synchronization is done from the perspective of differential algebra and differential geometry, the analogous of the observability is then called algebraic observability. A system is said to be algebraically observable if its states can be expressed as a function of the output and the derivatives of the output. There are several chaotic systems that do not satisfy the definition of algebraic observability, this motivates to use an alternative definition that allows to synchronize this systems even if they are not algebraically observable: Liouvillian systems make possible to perform the same task that an observer would, even if the system is not algebraically observable, and in some cases, outperform state observers [25], [26].

This work considers synchronization under a master-slave configuration for non-identical chaotic systems, the process of synchronization is done by finding an Fractional Generalized Observability Canonical form (FGOCF) for both the master and slave system, this canonical form originates from a differential primitive element (DPE) and allows to apply a nonlinear control law to achieve generalized synchronization (GS) of the master and slave systems.

One of the more representative applications of synchronization is data encryption, it has been extensively explored in integer-order chaotic systems [8], [27], [28], [29] and there are several new advances in fractional-order chaotic systems [30], [31], [32], [33], [34], [35], [36]. Even if various forms of synchronization are tested, very little care is put in the encryption, many only use a very basic masking which can easily be defeated, rendering the encryption useless and not showing the advantages of using chaotic systems, in this paper a different method for encryption is presented. It relies on a cryptographic function for encryption and uses the characteristics of chaotic systems to induce variation in the encryption values when different messages are ciphered, it also makes use of a hash function to make a key validation for enhance the resilience to cryptanalysis, finally it employs a binary signal which is masked by the dynamic of the fractional-order chaotic system, the signal is similar to the ones found in chaos shift keying, to allow the recovery of messages without data loss even in the presence of error or noise in the synchronization of the master and slave system. This paper's main focus is fractional-order chaotic systems, but the theoretical results also consider the integer order system as a particular case of the fractional-order system, thus it is possible to cipher messages using the integer-order system. Numerical results for both the fractional-order system and the integer order systems are given. Another key aspect of encryption is the security that the algorithm provides, so various test are performed to show the viability of the proposed encryption algorithm, these test include cryptographic analysis such as known plaintext attacks, chosen plaintext attacks and cryptanalysis involving robust parametric estimation via state observers, all security test are performed to the proposed encryption algorithm and, for comparison purposes, to other current proposals on encryption with chaotic systems in both, fractional and integer order chaotic systems. Along with the data security tests noise resilience and key sensitivity are also verified.

The rest of the paper is organized as follows: Important mathematical concepts about fractional calculus are stated in section 2. In section 3 the synchronization of fractional-order systems is presented along with fractional-order Liouvillian systems and the method for their synchronization. In section 4 an encryption algorithm that relies on chaotic systems synchronization is introduced. Some numerical results and the application to image encryption can be found in section 5. In addition, in section 6 the security analysis is presented, and finally the section 7 contains concluding remarks.

Section snippets

Preliminaries

Fractional calculus is about the generalization of the usual theory of differentiation and integration of integer order n [37], [38], [39]. The first generalizations are based on the $n- fold$ iterative integral [40], [41] ${}_{a}I_{t}^{n} x (t) : = \frac{1}{(n - 1)!} \int_{a}^{t} {(t - τ)}^{n - 1} x (τ) d τ,$ whit $n \in N$ and $t > a$ . If the value n changes to an arbitrary order $α \in R^{+}$ , then the Riemann-Liouville fractional integral is defined as follows [40], [41]: ${}^{R L}I_{a +}^{α} x (t) = \frac{1}{Γ (α)} \int_{a}^{t} {(t - τ)}^{α - 1} x (τ) d τ,$ where $Γ (α) = \int_{0}^{\infty} t^{α - 1} e^{- t} d t$ is the Gamma function and the

Problem formulation and main result

In this section the synchronization problem for a class of fractional-order nonlinear systems is solved. The notion of algebraic observability for fractional-order systems and the definition of Liouvillian systems are introduced, as well as some necessary results for the solution of the generalized synchronization problem.

A variable $x_{i} \in R$ of the system (8) satisfies the fractional algebraic observability (FAO) condition, if $x_{i}$ is a function of the first $r_{1}, r_{2} \in N$ sequential fractional derivatives

Encryption based on fractional-order Liouvillian systems synchronization

One of the best known applications of synchronization of chaotic systems is encryption. A very characteristic feature of chaotic systems is that a small change on the initial conditions makes the systems trajectories change significantly, this is highly valued in encryption, since this feature resembles the behavior a key must have: a small change in the key or in the information to be encrypted must make entirely different values used for encryption. To show this advantage and the application

Numerical examples

In this section, three examples of generalized synchronization with the Chua-Hartley, Arneodo, and Rossler systems of fractional-order are shown. The first example considers two different systems, while in the second, synchronization is done between a Liouvillian and a non-Liouvillian system. The third example shows the generalized synchronization between two Chua-Hartley systems and the application to data encryption.

Security analysis

A security analysis is performed to the encryption algorithm, for the analysis a known plaintext attack and a chosen plaintext attack are attempted, the purpose of these attacks is to generate an equivalent key that allows to decrypt message without knowing the key. These type of attack is particularly effective when applied to chaotic masking, in many cases needing only a few messages to obtain an equivalent key.

A known plaintext attack is performed when an unauthorized observer has access to

Conclusions

In this work the states variables of Liouvillian systems are represented through the Riemann-Liouville fractional integral of the available output. The problem of generalized synchronization is solved for chaotic Liouvillian systems of fractional-order expressed in their fractional generalized observability canonical form (FGOCF), coupled with a fractional dynamical control law. The FGOCF is obtained from a coordinate transformation matrix whose first element is called the differential

CRediT authorship contribution statement

Oscar Martínez-Fuentes: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Software, Writing – original draft. Juan Javier Montesinos-García: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Writing – review & editing. José Francisco Gómez-Aguilar: Methodology, Supervision, Validation, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Oscar Martínez-Fuentes acknowledges the support provided by MEng. Héctor Octavio Fanghanel Córdova (Academic Dean of the School of Engineering at Universidad Anáhuac Veracruz, campus Xalapa). José Francisco Gómez-Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014.

Oscar Martínez-Fuentes received the M.Sc. and Ph.D. degrees from the Department of Automatic Control at the Center for Research and Advanced Studies of IPN (CINVESTAV), Mexico city, Mexico, in 2015 and 2019, respectively. Currently, he is a postdoctoral researcher at the Department of Electronics in the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE). His research interests include control and stability analysis of fractional-order nonlinear systems, nonlinear observers,

References (59)

S. Boccaletti et al.
The synchronization of chaotic systems
Phys. Rep.
(2002)
Y.-J. Chen et al.
A polynomial-fuzzy-model-based synchronization methodology for the multi-scroll Chen chaotic secure communication system
Eng. Appl. Artif. Intell.
(2020)
J. Luo et al.
Synchronization of memristor-based chaotic systems by a simplified control and its application to image en-/decryption using DNA encoding
Chin. J. Phys.
(2019)
X. Chu et al.
Exponential quasi-synchronization of conformable fractional-order complex dynamical networks
Chaos Solitons Fractals
(2020)
A.G. Tene et al.
Generalized synchronization of regulate seizures dynamics in partial epilepsy with fractional-order derivatives
Chaos Solitons Fractals
(2020)
N. Yu et al.
Event-triggered impulsive chaotic synchronization of fractional-order differential systems
Appl. Math. Comput.
(2021)
X. Wu et al.
Finite time complete synchronization for fractional-order multiplex networks
Appl. Math. Comput.
(2020)
F. Du et al.
New criterion for finite-time synchronization of fractional order memristor-based neural networks with time delay
Appl. Math. Comput.
(2021)
X. Liu et al.
Chaotic vibration, bifurcation, stabilization and synchronization control for fractional discrete-time systems
Appl. Math. Comput.
(2020)
H. Jahanshahi et al.
A new fractional-order hyperchaotic memristor oscillator: dynamic analysis, robust adaptive synchronization, and its application to voice encryption
Appl. Math. Comput.
(2020)

R. Martínez-Guerra et al.

Secure communications via synchronization of Liouvillian chaotic systems

J. Franklin Inst.

(2016)

A. Gupta et al.

Generalized framework for the design of adaptive fractional-order masks for image denoising

Digit. Signal Process.

(2022)

A. Khan et al.

Secure communication: using parallel synchronization technique on novel fractional order chaotic system

IFAC-PapersOnLine

(2020)

W.S. Sayed et al.

Generalized switched synchronization and dependent image encryption using dynamically rotating fractional-order chaotic systems

AEÜ, Int. J. Electron. Commun.

(2020)

A. Coronel-Escamilla et al.

Design of a state observer to approximate signals by using the concept of fractional variable-order derivative

Digit. Signal Process.

(2017)

J.-S. Lin et al.

Design and implementation of digital secure communication based on synchronized chaotic systems

Digit. Signal Process.

(2010)

J.-F. Chang et al.

Controlling chaos of the family of Rössler systems using sliding mode control

Chaos Solitons Fractals

(2008)

C. Li et al.

Chaos and hyperchaos in the fractional-order Rössler equations

Physica A

(2004)

S. Moon et al.

Chaos synchronization in generalized Lorenz systems and an application to image encryption

Commun. Nonlinear Sci. Numer. Simul.

(2021)

A.C. Luo et al.

Complex Systems: Fractionality, Time-Delay and Synchronization

(2011)

A. Balanov et al.

Synchronization: from Simple to Complex

(2008)

A.C. Luo

Dynamical System Synchronization

(2013)

L. Kocarev et al.

General approach for chaotic synchronization with applications to communication

Phys. Rev. Lett.

(1995)

U. Parlitz et al.

Transmission of digital signals by chaotic synchronization

Int. J. Bifurc. Chaos Appl. Sci. Eng.

(1992)

Z. Zheng et al.

Generalized synchronization versus phase synchronization

Phys. Rev. E

(2000)

T. Azizi et al.

Chaos synchronization in discrete-time dynamical systems with application in population dynamics

J. Appl. Math. Phys.

(2020)

S. Nobukawa et al.

Synchronization of chaos in neural systems

Front. Appl. Math. Stat.

(2020)

C. Li et al.

Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability

Int. J. Syst. Sci.

(2016)

F. Wang et al.

Quasi-synchronization of heterogenous fractional-order dynamical networks with time-varying delay via distributed impulsive control

Chaos Solitons Fractals

(2020)

Cited by (16)

Principles of fractional signal processing
2024, Digital Signal Processing: A Review Journal
A general framework for fractional signal processing is described and used to derive several interesting formulations. This scheme is based on the Liouville approach that gave rise to the classic Riemann-Liouville and Liouville-Caputo derivatives, here dismissed. Liouville's idea consisted of fractionalizing the transfer function of the basic definition of derivative. Various coherent formulations are introduced from suitable derivative definitions and the corresponding ARMA-type linear systems are obtained. In particular, the Euler discrete-time, classic continuous-time, bilinear (Tustin), and scale-invariant systems are introduced and studied as applications of the proposed scheme. The two-sided derivatives and systems are also considered.
A finite-time sliding mode control technique for synchronization chaotic fractional-order laser systems with application on encryption of color images
2023, Optik
In recent years, fractional order Laser chaotic systems have gained popularity in both theory and applications, and various classes of these systems have been introduced. This paper presents a dynamic-free sliding mode control (SMC) methodology to synchronize a class of unknown fractional order (FO) Laser chaotic systems with input saturation. The proposed method uses a defined continuous function instead of the discrete sign function and is based on the FO version of the Lyapunov stability theorem. The result is a novel finite-time SMC (FTSMC) methodology that effectively suppresses chaotic behavior in FO Laser chaotic systems without undesirable chattering. This approach is designed to take advantage of the boundedness feature of the FO chaotic system. The efficacy of the FTSMC is demonstrated by applying the method to a chaotic FO Laser system at two different non-integer orders, and its practical applicability is demonstrated by using it to encrypt/decrypt color pictures. The suggested encryption/decryption methodology uses an adaptation of the pre-diffusion permutation-diffusion structure to increase security. Performance and security analyses, including histogram analysis, neighboring pixel correlation analysis, and information entropy analysis, provide further support for the suggested encryption system's superiority.
Synchronization of fractional-order chaotic networks in Presnov form via homogeneous controllers
2023, Integration
The Presnov decomposition for continuously differential vector fields consists of a gradient of a scalar field (dissipative term) and a vector field orthogonal to a radial vector of the origin (circulative or non-dissipative term). This paper uses the Presnov decomposition associated with a fractional-order Lorenz family to design algorithms that solve the multi-synchronization problem. Assuming that the whole state is known, two active control solutions are proposed to achieve the desired synchronization. Both control algorithms take advantage of homogeneous functions to have a powerful and simple Lyapunov analysis to show Mittag-Leffler stabilization in the first algorithm. In contrast, the second one achieves synchronization before a predefined time defined by the user. The combination of homogeneous functions and Riemann–Liouville integral in the controllers improve the behavior of the dynamic systems due to the impact on the robustness properties in the control scheme. Numerical simulations are presented to show the reliability of the proposed methods.
Event-triggered adaptive fuzzy tracking control for a class of fractional-order uncertain nonlinear systems with external disturbance
2022, Chaos, Solitons and Fractals
Citation Excerpt :
Fractional-order calculus, due to its memory and heritability and high accuracy in characterizing many practical physical phenomena [1], is gradually showing broad application prospects in many fields, such as chaotic system [2], neural networks [3–6], mechanical systems [7], electrical circuit system [8], secure information transmission [9] and so on.
In this paper, we are dedicated to addressing the event-triggered adaptive fuzzy tracking control problem for a class of fractional-order nonlinear systems (FONSs) with uncertainty and external disturbance. A novel condition is presented for estimating the fractional derivative of event-triggered strategy measurement error signal. To avoid the “expansion of complexity” problem caused by repeatedly solving the fractional derivatives of the virtual control signals in the backstepping method, an event-triggered-based adaptive fuzzy controller is constructed by taking advantage of the fractional dynamic surface control (DSC) scheme and fractional Lyapunov direct method. The designed controller can not only guarantee the tracking error finally converges to a small neighborhood around the origin, but also ensure all signals included in closed-loop FONSs are semiglobally uniformly ultimate bounded (SUUB) and the Zeno behavior is favorably avoided. In addition, the fuzzy logic systems (FLSs) tend to be applied in estimating the unknown nonlinear function, and an auxiliary function is also adopted to compensate for the approximation error of FLSs and the external disturbance. Finally, a numerical example indicates that the event-triggered adaptive fuzzy controller designed in this paper for FONSs is feasible and effective.
Sliding Mode Control on Finite-Time Synchronization of Nonlinear Hyper-mechanical Fractional Systems
2024, Arabian Journal for Science and Engineering
Synchronization of time-varying fractional-order chaotic systems in the presence of unknown disturbance using a fast disturbance identification
2024, Transactions of the Institute of Measurement and Control

View all citing articles on Scopus

Juan Javier Montesinos-García received the B.S. degree in mechatronic engineering from Papaloapan University, Oaxaca, Mexico in 2014. He received the M. Sc. and Ph.D. degrees in automatic control both from CINVESTAV, Mexico city, Mexico in 2015 and 2019 respectively. Currently he is a postdoctoral researcher at Technological University of Mixteca (UTM). His current research areas of interest include robust state observers, control of fractional-order systems and secure communications.

José Francisco Gómez-Aguilar received the B.S. and M.Eng. degrees in electrical engineering from Guanajuato University, Guanajuato, Mexico, in 2005 and 2007, respectively, and the Ph.D. degree in physics from División de Ciencias e Ingenierías, Guanajuato University, in 2012. He is currently a Full Research Professor with the Electronics Engineering Department commissioned for the CONACyT in the Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET), Tecnológico Nacional de México, Cuernavaca, México. His current research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbations methods, image and signal processing, and control theory.

View full text

Generalized synchronization of commensurate fractional-order chaotic systems: Applications in secure information transmission

Abstract

Introduction

Section snippets

Preliminaries

Problem formulation and main result

Encryption based on fractional-order Liouvillian systems synchronization

Numerical examples

Security analysis

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Phys. Rep.

Eng. Appl. Artif. Intell.

Chin. J. Phys.

Chaos Solitons Fractals

Chaos Solitons Fractals

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

J. Franklin Inst.

Digit. Signal Process.

IFAC-PapersOnLine

AEÜ, Int. J. Electron. Commun.

Digit. Signal Process.

Digit. Signal Process.

Chaos Solitons Fractals

Physica A

Commun. Nonlinear Sci. Numer. Simul.

Complex Systems: Fractionality, Time-Delay and Synchronization

Synchronization: from Simple to Complex

Dynamical System Synchronization

General approach for chaotic synchronization with applications to communication

Phys. Rev. Lett.

Transmission of digital signals by chaotic synchronization

Int. J. Bifurc. Chaos Appl. Sci. Eng.

Generalized synchronization versus phase synchronization

Phys. Rev. E

Chaos synchronization in discrete-time dynamical systems with application in population dynamics

J. Appl. Math. Phys.

Synchronization of chaos in neural systems

Front. Appl. Math. Stat.

Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability

Int. J. Syst. Sci.

Quasi-synchronization of heterogenous fractional-order dynamical networks with time-varying delay via distributed impulsive control

Chaos Solitons Fractals