Number Of Positive Lyapunov Exponents Research Articles

Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers

This paper explores the synchronization and implementation of a novel hyperchaotic system using an adaptive observer. Hyperchaotic systems, known for possessing a greater number of positive Lyapunov exponents compared to chaotic systems, present unique challenges and opportunities in control and synchronization. In this study, we introduce a novel hyperchaotic system, thoroughly examining its dynamic properties and conducting a comprehensive phase space analysis. The proposed hyperchaotic system undergoes validation through circuit simulation to confirm its behavior. Introducing an adaptive observer synchronization technique, we successfully synchronize the dynamics of the novel hyperchaotic system with an identical counterpart. Importantly, we extend the application of this synchronization method to the domain of secure communication, showcasing its practical usage. Simulation outcomes validate the effectiveness of our methodology, demonstrating favorable results in the realm of adaptive observer-based synchronization. This research contributes significantly to the understanding and application of hyperchaotic systems, offering insights into both the theoretical aspects and practical implementation. Our findings suggest potential advancements in the field of chaotic systems, particularly in their applications within secure communication systems. By presenting motivations, methods, results, conclusions and the significance of our work in a more appealing manner, we aim to engage readers and highlight the innovative contributions of this study.

A universal method for constructing n-dimensional polynomial hyperchaotic systems with any desired positive Lyapunov exponents

Most existing chaotic maps have many defects in engineering applications, such as discontinuous parameter range, uneven output of chaotic sequences and dynamic degradation. Based on this, a generalized n-dimensional polynomial chaotic map is proposed in this paper. By setting the coefficient of the linear term and the order of the highest order term of the polynomial, a series of n-dimensional polynomial chaotic maps of specific Lyapunov exponents can be obtained. The system solves the defects of the above system well, in addition, one can get the desired number of positive Lyapunov exponents, and one can get the desired value of positive Lyapunov exponents. Then, the effectiveness of the map is verified by a specific numerical example, and its dynamic analysis shows that the map has complex dynamic behavior. Finally, the map is applied to secure communication technology. Compared with other chaotic maps of the same dimension, the maps can obtain a smaller bit error rate, indicating that the chaotic map is more suitable for chaotic secure communication applications.

A Sine-coupled model for constructing N-dimensional non-degenerate discrete hyperchaotic map

The non-degenerate hyperchaotic systems with the maximum number of positive Lyapunov exponents (LEs) typically have better ergodicity, pseudo randomness, and stronger anti-degeneration property. Therefore, designing non-degenerate hyperchaotic maps with complex dynamics has attracted increasing attention from various research fields in recent years. By introducing the sine function, this paper proposes a construction model of N-dimensional non-degenerate discrete hyperchaotic map. To verify the effectiveness of this model, we provide three sub-maps of different dimensions based on this model as illustrative examples, and the dynamic behavior is explored using multiple numerical measures. The results demonstrate that the sub-maps with concise symmetric structures have complex dynamics, such as ultra-wide non-degenerate hyperchaotic parameter range, state transition phenomenon, and multistability. In particular, coexisting symmetric attractors and quasi-periodic curves switch periodically with the change of initial value. Furthermore, the hyperchaotic sequences generated by the three sub-maps have excellent performance, and the NIST test also further verifies the super randomness and unpredictability of these sequences. Finally, through the DSP hardware platform, the physical realizability of the sub-maps is verified successfully.

An Image Encryption Transmission Scheme Based on a Polynomial Chaotic Map.

Most existing chaotic systems have many drawbacks in engineering applications, such as the discontinuous range of chaotic parameters, weak chaotic properties, uneven chaotic sequence outputs, and dynamic degradation. Therefore, based on the above, this paper proposes a new method for the design of a three-dimensional chaotic map. One can obtain the desired number of positive Lyapunov exponents, and can also obtain the desired value of positive Lyapunov exponents. Simulation results show that the proposed system has complex chaotic behavior and high complexity. Finally, the method is implemented into an image encryption transmission scheme and experimental results show that the proposed image encryption scheme can resist brute force attacks, correlation attacks, and differential attacks, so it has a higher security.

Design and geometric control of polynomial chaotic maps with any desired positive Lyapunov exponents

Digital chaotic maps are severely hampered by the finite calculation accuracy of the hardware device that is used to implement them, and their applications in cryptography and information assurance are seriously degraded. To resolve this issue, we put forward a universal iterative model to construct non-degenerate polynomial chaotic maps with any desired number of positive Lyapunov exponents. In addition, we innovatively propose the geometric control methods of polynomial chaotic maps, including amplitude control, offset boosting, plane rotation, shape control, and combined regulation. Furthermore, to assess the effectiveness and feasibility of the proposed method, a microcontroller-based platform was developed to demonstrate the hardware implementation and geometric control of the proposed polynomial chaotic map. Finally, a PRNG is constructed by interval quantization. Numerical experiments are performed to verify the desirable statistical properties of the PRNG in terms of local weak random test, discrete Fourier transform test, linear complexity and NIST SP800-22 test.

A universal method for constructing non-degenerate hyperchaotic systems with any desired number of positive Lyapunov exponents

Due to the limited machine word length of hardware devices, the dynamics of digital chaotic systems will degenerate. To combat this issue, we proposed a universal method that is based on singular value decomposition (SVD), which can reversely construct non-degenerate hyperchaotic systems with any desired number of positive Lyapunov exponents by controlling pre-specified singular values. To assess the practicability and effectiveness of the method, we construct a 6-dimensional non-degenerate hyperchaotic system as an example. Furthermore, based on the hyperchaotic system, a pseudorandom number generator (PRNG) with desirable statistical characteristics is designed for image encryption. Numerical simulations were performed to evaluate the security of the image encryption algorithm in terms of histogram, information entropy, differential attack test, etc. The proposed non-degenerate hyperchaotic system can be effectively applied in the field of multimedia data encryption and information security.

Design of Higher-Dimensional Hyperchaotic System based on Combined Control and its Encryption Application

According to the anti-control principle of chaos, a combined control method is proposed based on a class of asymptotically stable linear systems with multiple controllers. A higher-dimensional hyperchaotic system is investigated by the Lyapunov exponents method and equilibrium points analysis, and it exists the largest number of positive Lyapunov exponents. The chaotic pseudo-random sequences of the higher-dimensional hyperchaotic system can pass all NIST tests after preprocess-ing, and behave better chaotic characteristics. Meanwhile, a new encryption algorithm of image information with position scrambling, sequential diffusion and reverse diffusion is designed based on the chaotic pseudo-random sequences. The experiments of image information are given to verify the effectiveness and feasibility of the encryption algorithm. Finally, the security analyses are also discussed by the key sensitivity, differential attack and statistical analysis. It is shown that the encryption algorithm has large enough key space and can be applied to secure communication.

Hyperchaos and its control in two-level quantum oscillators lattice

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.

Hyperchaos in Wilson-Cowan oscillator circuits.

The Wilson-Cowan equations were originally shown to produce limit cycle oscillations for a range of parameters. Others subsequently showed that two coupled Wilson-Cowan oscillators could produce chaos, especially if the oscillator coupling was from inhibitory interneurons of one oscillator to excitatory neurons of the other. Here this is extended to show that chains, grids, and sparse networks of Wilson-Cowan oscillators generate hyperchaos with linearly increasing complexity as the number of oscillators increases. As there is now evidence that humans can voluntarily generate hyperchaotic visuomotor sequences, these results are particularly relevant to the unpredictability of a range of human behaviors. These also include incipient senescence in aging, effects of concussive brain injuries, autism, and perhaps also intelligence and creativity.NEW & NOTEWORTHY This paper represents an exploration of hyperchaos in coupled Wilson-Cowan equations. Results show that hyperchaos (number of positive Lyapunov exponents) grows linearly with the number of oscillators in the array and leads to high levels of unpredictability in the neural response.

Construction of Higher-Dimensional Hyperchaotic Systems with a Maximum Number of Positive Lyapunov Exponents under Average Eigenvalue Criteria

Over the last 40 years, the design of [Formula: see text]-dimensional hyperchaotic systems with a maximum number ([Formula: see text]) of positive Lyapunov exponents has been an open problem for research. Nowadays it is not difficult to design [Formula: see text]-dimensional hyperchaotic systems with less than ([Formula: see text]) positive Lyapunov exponents, but it is still extremely difficult to design an [Formula: see text]-dimensional hyperchaotic system with the maximum number ([Formula: see text]) of positive Lyapunov exponents. This paper aims to resolve this challenging problem by developing a chaotification approach using average eigenvalue criteria. The approach consists of four steps: (i) a globally bounded controlled system is designed based on an asymptotically stable nominal system with a uniformly bounded controller; (ii) a closed-loop pole assignment technique is utilized to ensure that the numbers of eigenvalues with positive real parts of the controlled system be equal to ([Formula: see text]) and ([Formula: see text]), respectively, at two saddle-focus equilibrium points; (iii) the number of average eigenvalues with positive real parts is ensured to be equal to ([Formula: see text]) for the controlled system over a given control period; (iv) the smallest value of the positive real parts of the average eigenvalues is ensured to be greater than a given threshold value. Finally, the paper is closed with some typical examples which illustrate the feasibility and performance of the proposed design methodology.

Specificities of Generalized Synchronization in Delayed Systems

Specificities of the generalized synchronization mode in unidirectionally coupled delayed oscillators are studied. Four different cases of the interaction of systems characterized by different numbers of positive Lyapunov exponents are considered. It is established that the threshold for the rise of a synchronous mode strongly depends on the degree of chaoticity of interacting systems and the coupling of systems characterized by a different number of positive Lyapunov exponents gives rise to additional synchronization regions.

Design and Implementation of Autonomous and Non-Autonomous Time-Delay Chaotic System Based on Field Programmable Analog Array.

Time-delay chaotic systems can have hyperchaotic attractors with large numbers of positive Lyapunov exponents, and can generate highly stochastic and unpredictable time series with simple structures, which is very suitable as a secured chaotic source in chaotic secure communications. But time-delay chaotic systems are generally designed and implemented by using analog circuit design techniques. Analog implementations require a variety of electronic components and can be difficult and time consuming. At this stage, we can now solve this question by using FPAA (Field-Programmable Analog Array). FPAA is a programmable device for implementing multiple analog functions via dynamic reconfiguration. In this paper, we will introduce two FPAA-based design examples: An autonomous Ikeda system and a non-autonomous Duffing system, to show how a FPAA device is used to design programmable analog time-delay chaotic systems and analyze Shannon entropy and Lyapunov exponents of time series output by circuit and simulation systems.

Особенности обобщенной синхронизации в системах с запаздыванием

Peculiarities of the generalized synchronization regime in unidirectionally coupled time-delayed generators are studied. Four different cases of interaction between systems characterized by different number of positive Lyapunov exponents are considered. The thresholds of the generalized synchronization regime onset is found to depend sufficiently on the degree of chaotic states of interacting systems, whereas the coupling between systems characterized by different number of positive Lyapunov exponents results in appearance of additional synchronous fields.

A new method for generating chaotic system with arbitrary shaped distributed attractors.

In this paper, a new method for generating a chaotic system with arbitrary shaped (including heart-shaped, oval, circle, piecewise-linear, and cuboid) distributed attractors is proposed. In this article, a simple four-wing chaotic attractor is first presented by using a periodic piecewise function instead of a constant parameter in the Lorenz system, on the basis of which the chaotic system with arbitrary shaped distributed attractors in the plane can be constructed. This means that the distributed chaotic attractors can be arranged in an arbitrary shape in the plane. The chaotic system can generate any quantity of distributed chaotic attractors, and simulation results show that any desired number of positive Lyapunov exponents can be obtained. Therefore, the chaotic system will have more complicated dynamic characteristics. The dynamical mechanisms of this chaotic system are further investigated, and theoretical analysis and numerical simulation are in accordance with each other, which verifies the effectiveness of the approach. Lastly, the proposed chaotic system is used for image encryption. Numerical results show that the proposed scheme has an excellent performance.

Constructing Discrete Chaotic Systems with Positive Lyapunov Exponents

The chaotic system is widely used in chaotic cryptosystem and chaotic secure communication. In this paper, a universal method for designing the discrete chaotic system with any desired number of positive Lyapunov exponents is proposed to meet the needs of hyperchaotic systems in chaotic cryptosystem and chaotic secure communication, and three examples of eight-dimensional discrete system with chaotic attractors, eight-dimensional discrete system with fixed point attractors and eight-dimensional discrete system with periodic attractors are given to illustrate how the proposed methods control the Lyapunov exponents. Compared to the previous methods, the positive Lyapunov exponents are used to reconstruct a hyperchaotic system.

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity). A general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems is discussed here. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed.

Constructing Higher-Dimensional Nondegenerate Hyperchaotic Systems with Multiple Controllers

This paper proposes a new approach for constructing higher-dimensional nondegenerate hyperchaotic system with multiple controllers. Here, the so-called higher-dimensional nondegenerate hyperchaotic system means that it can be provided with a maximum number of positive Lyapunov exponents, which has been an open problem for research in recent years. The details of design are given by three steps as follows: (i) Design an [Formula: see text]-dimensional nominal matrix and similarity transformation matrix, and get an asymptotic stable nominal system; (ii) Add a master controller for the nominal matrix and get the controlled system. Then, find suitable control positions such that the controlled system satisfies the average eigenvalue criterion, i.e. the number of average eigenvalues with positive real parts of all Jacobi matrices over a given period of time is equal to ([Formula: see text]), and the maximum value of average eigenvalues with positive real parts is greater than a given threshold [Formula: see text]; (iii) Add nonmaster controllers, and the control positions are fixed and parameters are given in advance. So it can generate nondegenerate hyperchaotic systems with ([Formula: see text]) positive Lyapunov exponents. Finally, two typical examples are given to show the feasibility and effectiveness of the proposed method.

A Time-Delayed Hyperchaotic System Composed of Multiscroll Attractors With Multiple Positive Lyapunov Exponents

The paper proposes a time-delayed hyperchaotic system composed of multiscroll attractors with multiple positive Lyapunov exponents (LEs), which are described by a three-order nonlinear retarded type delay differential equation (DDE). The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyperchaotic attractors with multiscroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we setup electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.

Designing Hyperchaotic Cat Maps With Any Desired Number of Positive Lyapunov Exponents.

Generating chaotic maps with expected dynamics of users is a challenging topic. Utilizing the inherent relation between the Lyapunov exponents (LEs) of the Cat map and its associated Cat matrix, this paper proposes a simple but efficient method to construct an -dimensional ( -D) hyperchaotic Cat map (HCM) with any desired number of positive LEs. The method first generates two basic -D Cat matrices iteratively and then constructs the final -D Cat matrix by performing similarity transformation on one basic -D Cat matrix by the other. Given any number of positive LEs, it can generate an -D HCM with desired hyperchaotic complexity. Two illustrative examples of -D HCMs were constructed to show the effectiveness of the proposed method, and to verify the inherent relation between the LEs and Cat matrix. Theoretical analysis proves that the parameter space of the generated HCM is very large. Performance evaluations show that, compared with existing methods, the proposed method can construct -D HCMs with lower computation complexity and their outputs demonstrate strong randomness and complex ergodicity.

Design of fractional-order hyperchaotic systems with maximum number of positive lyapunov exponents and their antisynchronisation using adaptive control

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.