Constructing Discrete Chaotic Systems with Positive Lyapunov Exponents

ABSTRACTA systematic design procedure for generating fractional-order hyperchaotic systems (FOHSs) with a desired number of positive Lyapunov Exponents (LEs) remains an open problem. This paper puts forward a simple step-wise algorithm to tackle the above problem for a fractional-order system (FOS) to generate hyperchaos with effective dimension nd > 3 onwards; unlike the integer-order hyperchaotic systems where the net dimension must be nd ⩾ 4. The problem is a significant one as a lower-dimensional system with higher number of positive LEs has crucial potentiality in secure communication. Two proposals of controller design are put forward. The first one is to design an FOHS with maximum possible positive LEs from a stable system. The second is an adaptive control scheme in fractional dynamics to perform antisynchronisation between the generated FOHS with itself considering unknown parameters. Two representative examples are presented to validate the two control proposals. The superior effectiveness in generating FOHSs by following the proposed procedure is established in comparison with the existing hit-and-trial-methods.

Hyperchaotic systems are defined as chaotic systems with more than one positive Lyapunov exponent. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure boundedness of the solution, the minimal dimension for a continuous hyperchaotic system is four. The hyperchaotic systems are known to have important applications in secure communications and cryptosystems. First, this work describes an eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities. The qualitative properties of the novel hyperchaotic system are described in detail. The Lyapunov exponents of the system are obtained as \( L_{1} = 0.7781,L_{2} = 0.2299,L_{3} = 0 \) and \( L_{4} = - 12.5062 \). The maximal Lyapunov exponent of the system (MLE) is \( L_{1} = 0.7781 \). The Lyapunov dimension of the novel hyperchaotic system is obtained as \( D_{L} = 3.0806 \). Next, the work describes an adaptive controller design for the global chaos control of the novel hyperchaotic system. The main result for the adaptive controller design has been proved using Lyapunov stability theory. MATLAB simulations are described in detail for all the main results derived in this work for the eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities.

Discrete chaotic system, as a pseudo-random signal source, plays a very important role in realizing secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an <i>n</i>-dimensional discrete hyperchaotic system with a simple structure. Firstly, the <i>n</i>-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed based on Jacobian matrix method that the system can have the positive Lyapunov exponents. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with those of the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, and the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed. The results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. According to this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resist various common attacks. It is also validated that the proposed system can be effectively applied to the field of audio encryption.

The behavior of the square lattices of coupled two-level quantum oscillators is investigated. As quantum oscillators, we use the model of Rydberg atoms obtained using the approximation of a fully factorized density matrix. To investigate the behavior of the system the Lyapunov exponents spectra are calculated. Chaos and hyperchaos in the systems are revealed. It was shown that the number of positive Lyapunov exponents almost linearly depends on the number of atoms in the system, at a rate suggesting that adding three atoms leads to the appearance of an additional positive Lyapunov exponent. Using an external parametric effect and continuous feedback is suggested to control the complicated dynamics in the system. Using continuous feedback allows reducing the number of positive Lyapunov exponents from 3 to only 2 while introducing external parametric influence into the system allows reducing their number down to 0 and completely suppresses hyperchaos.

A dual-parameter hyperchaotic system with constant Lyapunov exponent and its circuit emulation

Though numerous hyperchaotic systems are reported, few of them can generate hyperchaotic attractors with more than three positive Lyapunov exponents. A new hyperchaotic system, constructed by adding the approximate time‐delayed state feedback to Chen system, is presented in this paper. This system has ring architecture with Chen system and several all‐pass filters coupled. With the increase of the number of all‐pass filters used in this system, the number of positive Lyapunov exponents also increases steadily. Hyperchaos of order 11 can be generated by this system. The resulting system is demonstrated by computer simulations and confirmed by an electronic circuit. Copyright © 2009 John Wiley & Sons, Ltd.

This paper reports a novel five Dimensional (5-D) Hyper-chaotic system with two positive Lyapunov exponents which is generated by adding two other dimensions to the 3-D chaotic system which was presented in 2016, by Tuna and Can. It modified generalized Lorenz system. This hyper-chaotic system has very simple algebraic structure but can show a very complex behavior. It has just one equilibrium point. In five dimensional hyper-chaotic systems obtaining a single equilibrium point very rare and interesting. Furthermore, in this letter presents a new numerical calculation method for the Lyapunov exponents of a hyper-chaotic system. After all of them in this research, designed a controller for the above system using the optimal generalized Back-Stepping method (OGBM). This method is an extension of the traditional Back-Stepping method and it optimized with the Shuffled Frog Leaping Algorithm (SFLA).

In recent years, a large amount of work on chaos-based cryptosystems has been published (Kocarev (2001); Millerioux et al. (2008)). A general methodology for designing chaotic and hyperchaotic cryptosystems has been developed using the control systems theory (Grassi et al. (1999); Liao et al. (1999); Yang et al. (1997a;b)). The chaotic communication system is closely related to the concept of chaos synchronization. An overview of chaotic secure communication systems can be found in (Yang (2004)). He classified the continuous-time chaotic secure communication systems into four generations. In the third generation, the combination of the classical cryptographic technique and chaotic synchronization is used to enhance the degree of security. Specifically, Yang et al. proposed a new chaos-based secure communication scheme in an attempt to thwart the attacks (Yang et al. (1997a;b)). They have combined both conventional cryptographic method and synchronization of chaotic systems. Their cryptographic method consists of an encryption function (the multi-shift cipher), a decryption function (the inverse of the encryption function), a chaotic encrypter that generates the key signal for the encryption function, and a decrypter that estimates the key signal. The approach has a limitation since the cryptosystem design may fail if different chaotic circuits are utilized. So far, this generation has the highest security in all the chaotic communication systems had been proposed and has not yet broken. From the control theoretic perspective, the transmitter and the receiver in the chaotic communication system can be considered as the nonlinear plant and its observer, respectively. Grassi et al. proposed a nonlinear-observer-based decrypter to reconstruct the state of the encrypter (Grassi et al. (1999); Liao et al. (1999)). They extended the Chua’s oscillator to the observer-based decrypter. The cryptosystem does not require initial conditions of the encrypter and the decrypter belonging to the same basin of attraction. If we can design a decrypter without the knowledge of the parameters of the encrypter, the chaos-based secure communication systems are not secure, because the parameters of the encrypter is selected as static secret keys in the cryptosystem. Parameter identification and adaptive synchronization methods may be effective for intruders in building reconstruction mechanisms, even when a synchronizing system is not available. Therefore, it is important for secure issues to investigate whether adaptive identifiers without the system information of encrypter can be constructed or not. We have recently designed an observer-based chaotic communication system combining the cryptosystems proposed by Grassi et al. (Grassi et al. (1999)) and by Liao et al. (Liao et al. (1999)) that allows us to assign the relative degree and the zeros of its encrypter system (Matsuo et al. (2004)). Specifically, we constructed three cryptosystems based on a Chua’s circuit by assigning its relative degree and zeros. The cryptosystem consists of 20

A hyperchaotic system is often used to generate secure keys or carrier wave for secure communication and the realistic hyperchaotic circuit often is made of capacitor, nonlinear resistor unit and induction coil. Parameters are often fixed in these hyperchaotic circuits and the hyperchaotic property of the system can be estimated by using a scheme of synchronization and time series analysis. In this paper, a time-varying hyperchaotic system is proposed by introducing changeable electric power source into the circuit; the changeable electric power source is combined with induction coil or capacitor in series to generate changeable output signals to excite the system. The diagrams of improved circuit are illustrated and critical parameters in experimental circuits are presented; the Lyapunov exponent spectrum vs. external applied electric power source is calculated. It is confirmed that the improved circuit always holds two positive Lyapunov exponents when the external electric power source works, and the chaotic attractors are much too different from the original one; thus, a more changeable hyperchaotic system is constructed in experiment.

The analysis of potentially chaotic behavior in biological and biomedical phenomena has attracted increased attention in recent years. Chaotic systems exhibit a trademark “sensitive dependence on initial conditions” (SDIC), meaning that very small perturbations will cause the resulting trajectories to separate exponentially fast. Lyapunov exponents directly measure SDIC by quantifying the exponential rates at which neighboring orbits on an attractor diverge (or converge) as the system evolves in time. Dissipative deterministic systems that exhibit at least one positive Lyapunov exponent are by definition “chaotic.” Hence, Lyapunov exponents are considered one of the most useful diagnostic tools available for analyzing potentially chaotic dynamical systems. However, Lyapunov exponents are also notoriously difficult to estimate reliably from experimental data and one should always be very skeptical about accepting any claims of “chaos” based solely on findings of positive Lyapunov exponents alone. This article reviews the basic definitions of Lyapunov exponents and outlines several of the more widely disseminated algorithms available to estimate them from experimental time series data. Critical issues related to finding spurious exponents, analyzing noisy signals, and validating against surrogate data are reviewed. A brief discussion of how Lyapunov exponents have been applied to study a wide variety of biomedical and biological phenomena is also presented. Finally, some thoughts on appropriate interpretations of such findings and the future directions for using Lyapunov exponents to study potentially chaotic systems are presented.

A novel universal approach for design and implementation of chaotic digital communication system based on IEEE-754 standard and field programmable gate array technology (FPGA) is proposed, combing chaos encryption with traditional cipher realization. By using the Euler algorithm and appropriate discrete processing, the continuous chaotic system is converted to a discrete chaotic system. Using FPGA hardware design system, digital chaotic sequence is generated as the key. Scrambling and expansion encryption algorithm is realized and analyzed. Driven and response of secure communication system is designed by constructing a loop including the signal and achieving chaotic synchronization between sender and receiver. Taking grid Chua chaotic system as an example, the secure communication system using FPGA hardware platform is implemented. The technical development process, algorithm flow chart, hardware design and realization result are also given.

In this work, we describe the model of a new 5-D hyperchaotic system with three positive Lyapunov exponents. Since the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve, the new hyperchaotic system is highly hyperchaotic. We also show that the new 5-D hyperchaotic system exhibits multistability with coexisting attractors. Using Multisim, we design an electronic circuit for the new 5-D hyperchaotic system. The hardware implementation of the new 5-D hyperchaotic system is done by applying two numerical methods. From the experimental results of the FPGA-based implementation, we show that the attractors observed in a Lecroy oscilloscope are in good agreement with numerical simulations. To prove the reliability of the proposed system for cybersecurity purposes, we present a new image cryptosystem using our hyperchaotic system. Experimental outcomes show the efficiency and the reliability of our cryptosystem based on the proposed hyperchaotic system.

Alternant-channel chaotic laser secure communication system and its chaos alternant coding (CAC) are presented and deeply studied via two coupled lasers synchronizing or desynchronizing with another single laser. The coupled lasers are used as two chaotic transmitters and transmit two chaotic carriers to mask two information signals, respectively. One conventional encoding scheme is that the single laser is used as a receiver and obtain synchronization with one of two chaotic transmitters for chaos decoding. And another CAC scheme as a non-conventional coding is presented and its coding principle is defined for secure communication. CAC coding and scheme is that one laser of chaotic transmitters transmits a chaotic carrier to mask an information signal, where the carrier is modulated by an information signal, while the decoding of CAC is performed via synchronization or de-synchronization between another laser of two chaotic transmitters and a receiver. A physical model of the synchronous system is presented using a drive-feedback technique while a chaotic synchronization is obtained between a chaotic transmitter and a receiver. And alternant-channel chaotic secure communication and CAC are successfully implemented while a novel CAC technique is validated. The system has the characteristics of high dimension, many degrees of freedom in phase-space, a high degree of nonlinearity, a nonconventional CAC, and many secret keys. So it has high security and can transmit securely two chaotic carries masking two information signals, respectively.

In this paper, a proposed novel seven dimensions (7D) hyperchaotic system with twelve positive parameters is presented. A chaotic behavior for the proposed system is proved through zero-one test, waveform analysis, Sensitivity Dependent on Initial Condition (SDIC) analysis and Lyapunov exponent. It is well known that one of the chaotic definitions is the proposed system is chaotic if it has positive Lyapunov exponent or it satisfies SDIC on its domain and it is known that the waveform analysis is an indication of the chaotic novel 7D system. It was approved that a little change in the initial values creates a high sensitivity in the chaotic behavior when the period time becomes large because the distance between initial conditions will become large. The dynamics of the novel 7D hyperchaotic simulated using the Mathematica program. Testing results show that the proposed system is hyperchaotic because it has three Lyapunov positive exponent, the time-domain waveform has non-cyclical characteristics and it has a better sensitivity dependent on initial condition on initial values.

This paper reports a new four-dimensional hyperchaotic system obtained by adding a controller to a three-dimensional autonomous chaotic system. The new system has two parameters and each equation of the system has one quadratic cross-product term. Some basic properties of the new system are analysed. The different dynamic behaviours of the new system are studied when the system parameter a or b is varied. The system is hyperchaotic in several different regions of the parameter b. Especially, the two positive Lyapunov exponents are both larger and the hyperchaotic region is also larger when this system is hyperchaotic in the case of varying a. The hyperchaotic system is analysed by Lyapunov-exponents spectrum, bifurcation diagrams and Poincaré sections.