Chaos-based Communication Research Articles

Static Casimir effect induced optical chaos in an optomechanical system

Static Casimir effect as one of the most interesting purely quantum effects is very general and may bring about novel physical phenomena and find applications in various fields of physics. Here, we investigate theoretically the generation of optical chaos induced by the static Casimir effect in an optomechanical system. We show that the dynamic behavior of the cavity field experiences the conversion from period to chaos motion under the condition of properly adjusting the static Casimir effect, even if the driven optical fields are comparatively weak. Furthermore, our numerical calculation with experimentally attainable parameters also shows that compared with the standard bare optomechanical system, the optical threshold of chaos is reduced about one order when the static Casimir effect is applied. Beyond the significance in broadening the scope of cavity optomechanics, this finding provides a different perspective to the study of optical chaos and lays a foundation for using static Casimir effect to comprehend the nonlinear characteristics of the optomechanical system. In particular, our results may find potential applications in chaos-based communication.

Differential Permutation Index DCSK Modulation for Chaotic Communication System

This letter proposes a differential permutation index differential chaos shift keying (DPI-DCSK) modulation for wireless communications. The conventional PI-DCSK has the disadvantage of wasting transmission energy and the loss of system performance. In $M$ -ary DPI-DCSK, one chaotic sequence to be sent represents current modulated ${M}$ -ary symbol as well as the reference of next modulation symbol in a data frame, which can fully utilize the transmission energy with the same hardware implementation as PI-DCSK. The bit error performance of the proposed DPI-DCSK over the multipath Rayleigh fading channel is theoretically analyzed. The simulation results show that the proposed DPI-DCSK achieves a performance gain of more than 3 dB over PI-DCSK, especially in high-delay multipath fading channels. The performance advantage of the proposed scheme is also illustrated in practical ultra-wideband (UWB) channels. Thus, DPI-DCSK is an efficient alternative transmission scheme for chaos-based wireless communication, such as indoor applications of transmitted-reference UWB communications.

Secure chaotic communication based on extreme multistability

Extreme multistability is the coexistence of a large number of attractors which can be reached by varying initial conditions. In this paper we show how this fascinating phenomenon can be used for secure communication. The main advantage of the communication system based on extreme multistability over a conventional chaos-based communication system is its exceptionally high security. The proposed system consists of two identical six-order oscillators; one in the transmitter and another one in the receiver, each exhibiting the coexistence of a large number of chaotic attractors. The oscillators are synchronized using a private channel through one of the system variables, while the information is transmitted via a public channel through another variable. The information is encrypted by varying the initial condition of one of the state variables in the transmitter using a chaotic map, adhering message packages in a staggered form to the coexisting attractors within the same time series of another state variable, which leads to switching among the coexisting chaotic attractors. To ensure communication security, the duration of the packages is shorter than synchronization time, so that synchronization attacks are ineffective.

Artificially Time-Varying Differential MIMO for Achieving Practical Physical Layer Security

In this paper, we propose a differential multiple-input multiple-output (MIMO) scheme based on the novel concept of chaos-based time-varying unitary matrices to demonstrate—for the first time in the literature—the ability of differential encoding in achieving practical physical layer security even without the need for using channel estimation. In the proposed scheme, an erroneous secret key, which is extracted from the wireless nature, is used to initialize a chaos sequence that is responsible for generating artificially time-varying unitary matrices capable of obfuscating the transmitted data symbols from illegitimate eavesdroppers. Contrary to conventional studies, the key agreement ratio in this study is assumed to be imperfect, which is often true and very realistic in high-mobility scenarios. Following this, we conceive a new calibration algorithm for reconciling the chaotic sequence generated at the legitimate parties, thus making this calibration algorithm a unique, novel solution to the key sharing problem of conventional chaos-based communication techniques, which has been overlooked over the past few decades. It is found out that differential encoding obviates additional complexity and insecurity in dealing with channel estimation, whereas an eavesdropper must tackle the complicated differentially encoded patterns, which have an exponentially increasing complexity order. In addition, the obtained simulation results demonstrate that the proposed scheme can outperform conventional chaos-based MIMO schemes that assume perfect channel knowledge.

Design and analysis of an OFDM-based orthogonal multilevel code-shifted differential chaos shift keying

In this paper, a new non-coherent <span class="il">chaos</span>-<span class="il">based</span> communication that combined <span class="il">Orthogonal</span> <span class="il">Multilevel</span> <span class="il">Code</span>-<span class="il">Shifted</span> <span class="il">Differential</span> <span class="il">Chaos</span> <span class="il">Shift</span> <span class="il">Keying</span> with <span class="il">orthogonal</span> frequency division multiplexing named (<span class="il">OFDM</span>-OMCS-DCSK) modulation system. New <span class="il">orthogonal</span> chaotic signal sets are generated by hybrid Gram-Schmidt algorithm <span class="il">and</span> Walsh <span class="il">code</span> function <span class="il">and</span> used it to carry high data rate in the same time slot. The Bandwidth efficiency <span class="il">of</span> the <span class="il">OFDM</span>-OMCS-DCSK system is comparing with conventional system. Also, the BER analytic expressions are derived under (AWGN) <span class="il">and</span> multipath Rayleigh fading channels <span class="il">and</span> comparing with the simulation results. Furthermore, the BER performance is compared with <span class="il">OFDM</span>-DCSK to found that the proposed system has the best BER performance with high data rate feature.

Finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity and uncertain terms

In this article, the finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity in the presence of uncertain stochastic parameters has been studied. The finite-time synchronization is a popular topic from a practical point of view in engineering, which has many applications in the areas like chaos-based communication, image and voice encryptions. The present chaotic system scrolls (up to five) are incorporated with the help of hyperbolic tangent functions (sigmoid non-linearity), but in real applications, there is no limit of multi-scroll in a chaotic system. Finite-time synchronization is achieved with the help of Lyapunov stability using some lemmas and definitions. The Lyapunov exponents are also calculated for multi-scroll chaotic systems with uncertainties and it confirms that the multi-scroll system is chaotic with uncertainties. A drive has been taken to achieve finite-time synchronization of multi-scroll chaotic systems with uncertain parameters, where the multi-scrolls are generated by nonlinear functions.

Artificial switched chaotic system used as transmitter in chaos-based secure communication

In this paper, a new idea of chaos synchronization and chaos-based secure communication is developed. First, instead of considering one single chaotic system, several uncertain chaotic systems are combined to form an uncertain switched chaotic system artificially. And then the switched chaotic system is used as transmitter in chaos-based secure communication consideration. Second, a drive signal is constructed and the information message is then embedded into the drive signal to form a transmitted signal for secure communication purpose. Third, an augmented switched chaotic system is constructed and a robust sliding model observer is proposed to sever as response system. Since the response system can not only estimate the transmitter’s states but also recover the embedded information message, it can be regarded as a receiver for chaos-based secure communication. Besides, we discuss the stability problem in detail based on the stability theory of the switched system. Finally, two simulation examples are introduced to illustrate the effectiveness and the advantages of our methods.

Dynamics editing based on offset boosting.

Multistability in a dynamical system has attracted great attention recently for its complex and unexpected states. Since in most chaotic systems coexisting attractors reside in their own individual basin of attraction with a fractal structure, it becomes a challenge to choose correct initial conditions to obtain desired dynamics. Selecting typical dynamics as the basic components in a dynamical sequence and then arranging them in the phase space in a desired order make the multistability transparent and controllable in the domain of initial conditions; thereafter, one can identify an attractor according to its initial sequence. Dynamics editing provides an effective technique to select typical attractors under different system parameters to form a flexible sequence in the phase space, which shows great potential for chaos-based secure communications.

Guest editorial: Chaos-based secure communications

With the massive increase of wirelessly connected devices and the rapid development of the Internet, information is transmitted more and more frequently. Since the information transmissions over these networks are greatly threatened not only by the imperfection of channels but also by the potential attackers or eavesdroppers, security and reliability become some of the crucial requirements for our future networks. Due to the complex dynamic behavior, ergodicity, wide spectrum and sensitivity to initial values, chaos signals show most required attributes of good cryptosystems and great potentials in optical fiber communications for desirable spatial accuracy and distance. In this context, it has become imperative to investigate and apply chaos theories to solve emerging secure communication problems in various networks. This includes leveraging chaos systems and signals to address a wide range of secure communication challenges in optical fiber communications, chaos cipher, image encryption, etc.

Energy Conservation, Singular Orbits, and FPGA Implementation of Two New Hamiltonian Chaotic Systems

Since the conservative chaotic system (CCS) has no general attractors, conservative chaotic flows are more suitable for the chaos-based secure communication than the chaotic attractors. In this paper, two Hamiltonian conservative chaotic systems (HCCSs) are constructed based on the 4D Euler equations and a proposed construction method. These two new systems are investigated by equilibrium points, dynamical evolution map, Hamilton energy, and Casimir energy. They look similar, but it is found that one can be explained using Casimir power and another cannot be explained in terms of the mechanism of chaos. Furthermore, a pseudorandom signal generator is developed based on these proposed systems, which are tested based on NIST tests and implemented by using the field programmable gate array (FPGA) technique.

Generation and synchronization of wideband chaos in semiconductor lasers subject to constant-amplitude self-phase-modulated optical injection.

We propose a novel wideband chaos generation scheme by using an external-cavity semiconductor laser (ECSL) subject to optical-electronic hybrid feedback. In this scheme, the output of ECSL is photo-detected and used to modulate the output of a continuous wave laser by an electro-optical phase modulator, the constant-amplitude self-phase-modulated light is then injected back into the ECSL. The experimental results indicate that, compared with the chaos generation with conventional optical feedback (COF), significant bandwidth enhancement is achieved in the proposed scheme. The effective bandwidth of generated chaos is increased from a few GHz to over 20 GHz, and moreover, the spectrum flatness and the complexity of generated chaos are also considerably improved. Furthermore, we propose a wideband chaos synchronization system based on the proposed chaos generation scheme. It is experimentally demonstrated that high-quality synchronization between two wideband chaos signals with an effective bandwidth greater than 20 GHz is achieved. This work simultaneously achieves the generation and the synchronization of wideband chaos, which shows valuable potential in chaos-based secure communication, such as enhancing the transmission capacity and improving the security.

Performance analysis for DCSK system over land mobile satellite channel

In chaos-based digital communications, differential chaos-shift keying (DCSK) system has been widely studied thanks to its non-coherent demodulation without chaotic sequence synchronisation. In respect of the performance evaluation, bit error rate (BER) of the DCSK system was investigated over various transmission environments such as noise, fading, and multi-path-affected channels which are radio-propagation models from simple to complex. In this paper, the idea of applying DCSK for operation in land mobile satellite environments is proposed. A land mobile satellite channel, i.e., Lutz model with two states, is used to evaluate the system performance. Discrete-time models of transmitter and receiver over the studied channel are described. BER analysis is performed by means of Gaussian approximation and discrete integration. The analysed expressions are verified by Monte Carlo simulation. Performance comparison between different DCSK-based schemes over the studied channel is carried out. Obtained results point out that DCSK system can be used as a physical-layer modulation method using chaos for land mobile satellite applications.

Chaotic Dynamics and FPGA Implementation of a Fractional-Order Chaotic System With Time Delay

This article proposes a numerical solution approach and Field Programmable Gate Array implementation of a delayed fractional-order system. The proposed method is amenable to a sufficiently efficient hardware realization. The system’s numerical solution and hardware realization have two requirements. First, the delay terms are implemented by employing LookUp Tables to keep the already required delayed samples in the dynamical equations. Second, the fractional derivative is numerically approximated using Grunwald-Letnikov approximation with a memory window size, $L$ , according to the short memory principle such that it balances between accuracy and efficiency. Bifurcation diagrams and spectral entropy validate the chaotic behaviour of the system for commensurate and incommensurate orders. Additionally, the dynamic behaviour of the system is studied versus the delay parameter, $\tau $ , and the window size, $L$ . The system is realized on Nexys 4 Artix-7 FPGA XC7A100T with throughput 1.2 Gbit/s and hardware resources utilization 15% from the total LookUp Tables and 4% from the slice registers. Oscilloscope experimental results verify the numerical solution of the delayed fractional-order system. The amenability to digital hardware realization, which is experimentally validated in this article, is added to the system’s advantages and encourages its utilization in future digital applications such as chaos control and synchronization and chaos-based communication applications.

Hopf bifurcation in three-dimensional based on chaos entanglement function

Chaotic entanglement is a new method used to deliver chaotic physical process, as suggested in this work. Primary rationale is to entangle more than two mathematical product stationery linear schemes by means of entanglement functions to make a chaotic system that develops in a chaotic manner.Existence of Hopf bifurcation is looked into by selecting the set aside bifurcation parameter. More accurately, we consider the stableness and bifurcations of sense of equilibrium in the modern chaotic system. In addition, there is involvement of chaos in mathematical systems that have one positive Lyapunov exponent. Furthermore, there are four requirements that are needed to achieve chaos entanglement. In that way through dissimilar linear schemes and dissimilar entanglement functions, a collection of fresh chaotic attractors has been created and abundant coordination compound dynamics are exhibited. The breakthrough suggests that it is not difficult any longer to construct new obviously planned chaotic systems/networks for applied science practical application such as chaos-based secure communication.

Generating Multi-Scroll Chua’s Attractors via Simplified Piecewise-Linear Chua’s Diode

High implementation complexity of multi-scroll circuit is a bottleneck problem in real chaos-based communication. Especially, in multi-scroll Chua's circuit, the simplified implementation of piecewise-linear resistors with multiple segments is difficult due to their intricate irregular breakpoints and slopes. To solve the challenge, this paper presents a systematic scheme for synthesizing a Chua's diode with multi-segment piecewise-linearity, which is achieved by cascading even-numbered passive nonlinear resistors with odd-numbered ones via a negative impedance converter. The traditional voltage mode op-amps are used to implement nonlinear resistors. As no extra DC bias voltage is employed, the scheme can be implemented by much simpler circuits. The voltage-current characteristics of the obtained Chua's diode are analyzed theoretically and verified by numerical simulations. Using the Chua's diode and a second-order active Sallen-Key high-pass filter, a new inductor-free Chua's circuit is then constructed to generate multi-scroll chaotic attractors. Different number of scrolls can be generated by changing the number of passive nonlinear resistor cells or adjusting two coupling parameters. Besides, the system can be scaled by using different power supplies, satisfying the low-voltage low-power requirement of integrated circuit design. The circuit simulations and hardware experiments both confirmed the feasibility of the designed system.

Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression.

The isochronous cluster synchronization with time delay (TD) signature suppression in delay-coupled vertical-cavity surface-emitting laser (VCSEL) networks subject to variable-polarization optical injection (VPOI) is theoretically and numerically studied. Based on the inherent symmetries of network topology, parameter spaces for stable cluster synchronization are presented, and zero-lag synchronization are achieved for VCSELs in same clusters. Additionally, the TD signature reduction for the dynamics of VCSELs in the stable clusters are systematically discussed. It is shown that both moderate polarizer angle and frequency detuning between different clusters have strengthen the effect of TD signature suppression. Moreover, the isochronous cluster synchronization with TD signature concealment is also verified in another VPOI-VCSEL network with different topology, indicating the generality of proposed results. Our results shed a new light on the research of chaos synchronization and chaos-based secure communications in VCSEL networks.

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.

Impulsive observer with predetermined finite convergence time for synchronization of fractional-order chaotic systems based on Takagi–Sugeno fuzzy model

This paper is devoted to the problem of observer design for synchronization of nonlinear fractional-order chaotic systems described by the Takagi–Sugeno fuzzy model. We propose a new method of designing an observer that converges in finite time. The novelty of the proposed observer compared to those developed in the literature is that the estimated state exactly converges to the true state in a finite time which can be chosen beforehand. This is made possible by updating the estimated state at a defined time instant. The abrupt update shows up an impulsive behavior of the observer’s dynamics. Two cases are considered. In the first case, the system is not affected by an unknown input. The second case considers the system subjected to the action of an unknown input. Finite-time convergence of the two proposed observers for both cases is established using linear matrix inequality formulation. Simulation results on the synchronization of fractional-order chaotic systems clearly illustrate the impulsive behavior of the proposed observer and the predefined finite-time synchronization. The advantage of predefined finite-time synchronization is highlighted by the ability of recovering an encrypted message without loss of information in a fractional-order chaos-based secure communication application.

Multi‐carrier chaotic communication scheme for underwater acoustic communications

Multi-carrier (MC) and spread-spectrum (SS) are two good technologies for underwater acoustic (UWA) communications. In this study, the authors propose a novel MC chaos modulation scheme, which combines code-shifted differential chaos shift keying (CS-DCSK) with orthogonal frequency division multiplexing (OFDM), namely MC-CS-DCSK. The proposed scheme exploits the orthogonal characteristic of the Walsh code to superimpose the chaotic reference chips and the information bearing chips in the same time/frequency unit. Besides, a cyclic-shift interleaver is used on chips to harvest frequency diversity. To the in-depth understanding of the proposed system, the bit error rate (BER) of the proposed system under Gaussian channels and multipath Rayleigh fading channels are derived and verified by simulation. Furthermore, the spectral efficiency of the proposed system is analysed and compared with some existing chaos-based communication systems. Finally, they study BER performance of the system under UWA channels and compare it with traditional MC direct sequence SS and OFDM-based MC-DCSK systems. Simulation results show that the system has good robustness under time-varying UWA channels.

Multiscroll chaotic system with sigmoid nonlinearity and its fractional order form with synchronization application

In this paper, a multiscroll snap oscillator with hyperbolic tangent function is proposed. There is no limitation in the number of scrolls and it can be increased by proper choice of a specific function. The Lyapunov exponents of the proposed system are obtained to testify the chaotic behavior of the system. Fractional order multiscroll system is derived from its integer order model by using the Adams–Bashforth–Moulton algorithm. A new scheme is applied in order to investigate the synchronization of the multiscroll systems. The main objective of the paper is to propose a multiscroll attractor and show that the number of scrolls can be controlled by the only nonlinear function. Such systems are less investigated in the literatures and has many real time applications like image and voice encryption, random number generators, chaos based communication systems and so on.