Chaotic Systems Research Articles

An image encryption scheme based on an improved memristive neuron chaotic system

With fast-developing Internet and communication, the security transmission of image in network has become a research highlight. So far, researchers have designed a lot of image encryption methods based on chaotic models, some of them are not secure enough. To enhance security of image transmission on the network, in this paper, an image encryption method is developed from a 3D memeristive FitzHugh-Nagumo (FHN) neuron. First, a 3D memeristive FHN model is obtained by connecting a memeristor into a 2D FHN model, and dynamics for 3D FHN model are estimated by applying phase diagrams, bifurcation and Lyapunov exponent. Then, an image encryption algorithm is proposed by using this 3D model. Finally, security of encryption algorithm is estimated. Simulation results confirm the effectiveness of encryption scheme.

Construction algorithm of non-degenerate complex domain chaotic system with application on PRNG

Construction algorithm of non-degenerate complex domain chaotic system with application on PRNG

State Observer Synchronization of Three-dimensional Chaotic Oscillatory Systems Based on DNA Strand Displacement.

Currently, DNA strand displacement (DSD) as the theoretical basis of DNA chemical reaction networks (CRNs) has promoted the development of chaotic synchronization technique. This paper introduces the synchronization technology of two isomorphic three-dimensional chaotic systems based on DNA strand displacement under state observer. By studying the theoretical knowledge of DNA molecules, multiple DSD reactions are used to construct three-dimensional chaotic system. Based on two isomorphic chaotic systems, the linear transformation system and the state observer system are designed according to the theory of state observer construction. In addition, in order to realize the synchronization of chaotic systems, a coupling controller is designed between the drive system and the linear transformation system, and a soft variable-structure controller is designed between the state observer system and the response system. Through multiple DSD reactions, the chemical reaction networks of four chaotic systems and two controllers are constructed, and they are cascaded to realize the synchronization of two isomorphic three-dimensional chaotic systems. Numerical simulations verify the effectiveness and robustness of the scheme. Our work will extend and provide a reference for new methods to achieve synchronization of chaotic systems using DSD.

Application of Fractal Dimension for Cardiac Arrhythmias Classification

Fractal analysis is crucial for understanding complex, irregular patterns found in nature, finance, and various scientific fields. It helps to reveal self-similarity, where structures repeat at different scales, providing insights into chaotic systems like weather patterns, stock markets, and biological growth. By applying fractal analysis, researchers can model phenomena that traditional geometric methods cannot easily describe, enabling better predictions and deeper comprehension of dynamic systems. The Fractals are a fascinating mathematical tool for modeling the roughness of nature and understanding structure of such complex objects. They are considered a tool for understanding the world. In general, fractal objects are characterized by the fractal dimension. The application of fractal geometry to the analysis of ECG time series data is examined in this paper. A method based on the assessment of the Fractal Dimension (FD) of ECG recordings is suggested for the identification of cardiac diseases. In this work, and in order to exploit the fractal dimension to analyze fractal signals, the notion of fractal dimension is defined by presenting methods for calculating this dimension such as Higuchi algorithm, Katz method, regularization, box-counting etc… Each of them has its own advantages and disadvantages. This study has shown that the electrocardiogram (ECG) is a fractal signal. This allows to classify heartbeats founded on the concept of fractals. The main aim is to develop a digital technique to analyze ECG signals in order to make an accurate diagnosis of cardiovascular diseases.

FPGA based implementation of a perturbed Chen oscillator for secure embedded cryptosystems

This paper introduces an enhancement to the Chen chaotic system by incorporating a constant perturbation term d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d$$\\end{document} to one of the state variables, aiming to improve the performance of pseudo-random number generators (PRNGs). The perturbation significantly enhances the system's chaotic properties, resulting in superior randomness and increased security. An FPGA-based realization of a perturbed Chen oscillator (PCO)-derived PRNG is presented, tailored for embedded cryptosystems and implemented on a Nexys 4 FPGA card featuring the XILINX Artix-7 XC7A100T-1CSG324C integrated chip. The Xilinx-based system generator (XSG) tool is utilized to generate a digital version of the new oscillator, minimizing resource utilization. Experimental results demonstrate that the PCO-generated data successfully passes the NIST and TestU01 test suites. Additionally, statistical tests with key sensitivity are performed, validating the suitability of the designed PRNG for cryptographic applications. This establishes the PCO as a straightforward and efficient tool for multimedia security.

Singularity, Observability, and Independence: Unveiling Lorenz’s Cryptographic Potential

The key findings of this study include a detailed examination of the Lorenz system’s observability, revealing that it maintains high observability compared to other chaotic systems, thus supporting its potential use in cryptographic applications. We also investigated the singularity manifolds, identifying regions where observability might be compromised, but overall demonstrating that the system remains reliable across various states. Additionally, statistical tests confirm that the Lorenz system exhibits strong statistical independence in its outputs, further validating its suitability for encryption purposes. These findings collectively underscore the Lorenz system’s potential to enhance cryptographic security and contribute significantly to the field of secure communications. By providing a thorough analysis of its key properties, this study positions the Lorenz system as a promising candidate for advanced encryption technologies.

Chaotic systems based on higher-order oscillatory equations

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.

Science Studies as an educational discipline: experience of reconstruction

Introduction. The paper presents a reconsideration of the foundations of the discipline of Science Studies, taking into account post-non-classical ideas about science. This approach positions Science Studies not only as a scientific field but also as an educational discipline aimed at novice researchers. Aim. The present research aimed to assess the current state of science as a scientific and educational discipline in the Russian and Euro-American scientific traditions. It also aimed to present the project of teaching the discipline of Science Studies as a rigorous science, encompassing both its fundamental and applied aspects, and utilising quantitative and qualitative research methods. Methodology and research methods. The synthesis of universal evolutionism approaches (including deterministic chaos concepts) and the methods of quantitative and qualitative content analysis were employed. Results. A quantitative analysis of the state of science in Russia was conducted. The analysis revealed an increase in the interest of Russian researchers in scientific issues. However, it also highlighted a lack of clear understanding of its disciplinary parameters. Scientific novelty and practical significance. It is proved that Science Studies can be a significant incentive for the growth of scientific knowledge in Russia. The work has led to an understanding of the role of epistemological (cognitive) and social mechanisms in the development of science.

Design and FPGA implementation of nested grid multi-scroll chaotic system

Conventional multi-scroll chaotic systems are often constrained by the number of attractors and the complexity of generation, making it challenging to meet the increasing demands of communication and computation. This paper revolves around the modified Chua’s system. By modifying its differential equation and introducing traditional nonlinear functions, such as the step function sequence and sawtooth function sequence. A nested grid multi-scroll chaotic system (NGMSCS) can be established, capable of generating nested grid multi-scroll attractors. In contrast to conventional grid multi-scroll chaotic attractors, scroll-like phenomena can be initiated outside the grid structure, thereby revealing more complex dynamic behavior and topological features. Through the theoretical design and analysis of the equilibrium point of the system and its stability, the number of saddle-focused equilibrium points of index 2 is further expanded, which can generate (2 N+2) × M attractors, and the formation mechanism is elaborated and verified in detail. In addition, the generation of an arbitrary number of equilibrium points in the y-direction is achieved by transforming the x and y variables, which can generate M×(2 N+2) attractors, increasing the complexity of the system. The system’s dynamical properties are discussed in depth via time series plots, Lyapunov exponents, Poincaré cross sections, 0–1 tests, bifurcation diagrams, and attraction basins. The existence of attractors is confirmed through numerical simulations and FPGA-based hardware experiments.

A robust color image encryption scheme with complex whirl wind spiral chaotic system and quadrant-wise pixel permutation

This paper presents a novel encryption technique that uses a unique chaotic circuit design called as 3D Complex Whirl Wind Spiral chaotic system (CWWS). The major goal of this novel approach is to create an efficient 3D chaotic systems with increased randomness and multistability, specifically designed to encrypt multimedia data. The design incorporates the sine function sin(x) to introduce complexity and unpredictability in the chaotic circuit. The dynamic behaviour of the proposed scheme’s chaotic system is thoroughly evaluated using a variety of analyses, including KY dimension, dissipativity, Lyapunov exponent spectra, and bifurcation diagrams. There are two key stages to the encryption process: diffusion and confusion. The diffusion process is strengthened by the smooth integration of quadrant-wise pixel permutation (QWPP) algorithms, which eliminate correlations between neighbouring pixels. Following that, the image components are concealed using the chaotic sequence that was generated from the 3D CWWS chaotic system. The complete encrypted image is then created by combining these encrypted components. The simulation results of the proposed strategy are thoroughly investigated using statistical analysis, differential analysis, and brute force attacks. The system has optimal key space, entropy, UACI, and NPCR metric values of 2400, 7.99, 0.334, and 0.996, respectively. Furthermore, the experimental findings show robust resistance to statistical, differential, and brute force attacks for a single round of iteration.

True Random Number Generator Based on Chaotic Oscillation of a Tunable Double-Well MEMS Resonator.

Chaotic systems have aroused interest across various scientific disciplines such as physics, biology, chemistry, and meteorology. The deterministic but unpredictable nature of a chaotic system is an ideal feature for random number generation. Microelectromechanical systems (MEMS) are a promising technology that effectively harnesses chaos, offering advantages such as a compact footprint, scalability, and low power consumption. This paper presents a true random number generator (TRNG) based on a double-well MEMS resonator integrated with an actuator and position sensor. The potential energy landscape of the proposed MEMS resonator is actively tunable with a direct current voltage. Experimental demonstrations of tunable bistability and chaotic resonance are reported in this paper. A chaotic time sequence is generated through piezoresistive sensing of the position of the MEMS resonator once it is driven into the chaotic regime. Subsequently, the randomness of the bit sequence, achieved by applying the exclusive or function to a digital chaotic sequence and its delayed differential is confirmed to meet the National Institute of Standards and Technology specifications. Moreover, the throughput and energy efficiency of the proposed MEMS-based TRNG can be adjusted from 50 kb s-1 and 0.44 pJ per bit at a low energy barrier to 167 kb s-1 and 6.74 pJ per bit at a high energy barrier by changing the MEMS device's potential well. The tunability of the proposed double-well MEMS resonator not only offers continuous adjustments in the energy efficiency of TNRG but also unveils vast and diverse research opportunities in analog computing, encryption, and secure communications.

Quantum next generation reservoir computing: an efficient quantum algorithm for forecasting quantum dynamics

Next Generation Reservoir Computing (NG-RC) is a modern class of model-free machine learning that enables an accurate forecasting of time series data generated by dynamical systems. We demonstrate that NG-RC can accurately predict full many-body quantum dynamics in both integrable and chaotic systems. This is in contrast to the conventional application of reservoir computing that concentrates on the prediction of the dynamics of observables. In addition, we apply a technique which we refer to as skipping ahead to predict far future states accurately without the need to extract information about the intermediate states. However, adopting a classical NG-RC for many-body quantum dynamics prediction is computationally prohibitive due to the large Hilbert space of sample input data. In this work, we propose an end-to-end quantum algorithm for many-body quantum dynamics forecasting with a quantum computational speedup via the block-encoding technique. This proposal presents an efficient model-free quantum scheme to forecast quantum dynamics coherently, bypassing inductive biases incurred in a model-based approach.

Entanglement membrane in exactly solvable lattice models

Entanglement membrane theory is an effective coarse-grained description of entanglement dynamics and operator growth in chaotic quantum many-body systems. The fundamental quantity characterizing the membrane is the entanglement line tension. However, determining the entanglement line tension for microscopic models is in general exponentially difficult. We compute the entanglement line tension in a recently introduced class of exactly solvable yet chaotic unitary circuits, so-called generalized dual-unitary circuits, obtaining a nontrivial form that gives rise to a hierarchy of velocity scales with vE<vB. For the lowest level of the hierarchy, L¯2 circuits, the entanglement line tension can be computed entirely, while for the higher levels the solvability is reduced to certain regions in spacetime. This partial solvability enables us to place bounds on the entanglement velocity. We find that L¯2 circuits saturate certain bounds on entanglement growth that are also saturated in holographic models. Furthermore, we relate the entanglement line tension to temporal entanglement and correlation functions. We also develop methods of constructing generalized dual-unitary gates, including constructions based on complex Hadamard matrices that exhibit additional solvability properties and constructions that display behavior unique to local dimension greater than or equal to three. Our results shed light on entanglement membrane theory in microscopic Floquet lattice models and enable us to perform nontrivial checks on the validity of its predictions by comparison to exact and numerical calculations. Moreover, they demonstrate that generalized dual-unitary circuits display a more generic form of information dynamics than dual-unitary circuits. Published by the American Physical Society 2024

Synchronization of Bidirectionally Coupled Fractional-Order Chaotic Systems with Unknown Time-Varying Parameter Disturbance in Different Dimensions

In this article, the synchronization of bidirectionally coupled fractional-order chaotic systems with unknown time-varying parameter disturbance in different dimensions is investigated. The scale matrices are designed to address the problem of the synchronization for fractional-order chaotic systems across two different dimensions. Congelation of variables is used to deal with the unknown time-varying parameter disturbance. Based on Lyapunov’s stability theorem, the synchronization controllers in different dimensions are obtained. At the same time, adaptive laws of the unknown disturbance can be designed. Benefiting from the proposed methods, we verify all the synchronization errors can converge to zero as time approaches infinity, regardless of whether in n-D or m-D synchronization, simultaneously ensuring that both control and estimation signals are bounded. Finally, simulation studies based on fractional-order financial systems are carried out to validate the effectiveness of the proposed synchronization method.

A noncoherent chaotic communication system with multidimensional index modulation based on cyclic permutation grouping

With the surge in technological advancements and application demands, the study of high-data-rate communication systems has become increasingly significant. To enhance the data transmission rate and spectral efficiency of traditional DCSK systems, we have designed a differential chaos shift keying (DCSK) system with multidimensional index modulation based on cyclic permutation grouping (CPG-MIM-DCSK). Multidimensional index resources are efficiently combined by the CPG-MIM-DCSK system using the proposed cyclic permutation grouping method. In the CPG-MIM-DCSK system, both selected and unselected subcarriers, as well as selected and unselected time slots, are distinguished through different groups of cyclic permutations. Permutation, carrier, and time slot resources are fully utilized by this modulation method, thereby enhancing the data rate and spectral efficiency of traditional DCSK systems. The theoretical Bit Error Rate (BER) expressions of the CPG-MIM-DCSK system under Additive White Gaussian Noise (AWGN) and multipath Rayleigh fading channels are derived. The data rate, complexity, and spectral efficiency of the CPG-MIM-DCSK system are covered in further analysis. The simulation results highlight the superiority of the CPG-MIM-DCSK system in terms of data rate and spectral efficiency. Additionally, the proposed CPG-MIM-DCSK system exhibits better BER performance in both Gaussian and multipath Rayleigh fading channels compared to its comparative systems.

Soliton and lump and travelling wave solutions of the (3 + 1) dimensional KPB like equation with analysis of chaotic behaviors

Nonlinear evolution equations (NLEEs) have a wide range of applications in various fields, including physics, engineering, biology, economics, and more. These equations are used to describe complex systems that change over time, where the state or behavior of the system not only depends on the current situation, but may also be related to the past. A deep understanding of the solutions to NLEEs is of great significance for predicting and controlling the behavior of complex systems. The (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like (KPB-like) equation represents a partial differential equation that describes nonlinear wave phenomena in multi-dimensional spaces. This model is commonly employed in fields such as fluid dynamics, plasma physics, and nonlinear optics to study wave behaviors. This paper investigates the KPB-like equations, which have meaningful physical implications in describing nonlinear wave phenomena. Hirota bilinear method is employed to derive the bilinear form for the KPB-like equation and 1-soliton, 2-soliton, lump, and travelling wave solutions are studied to this kind of nonlinear model. In addition, the chaotic behaviors of soliton and lump solutions are explored via applying the Duffing chaotic system. To have a better understanding of dynamic structures, 3D, line, density and contour map plots of begotten results are displayed. Our results indicate the directness and effectiveness of the applied method for analyzing high-dimensional differential equations that arise in nonlinear science.

An encryption algorithm for color images based on an improved dual-chaotic system combined with DNA encoding

This study improves the Logistic chaotic system and combines it with the hyperchaotic Chen system to create a dual chaotic system. The algorithm encrypts images in three stages. In the first stage, a plaintext-related key generation scheme is designed to generate the parameters and initial values of the dual chaotic system. In the second stage, the chaotic sequences generated by the dual chaotic system are used for dynamic DNA encoding and computation. In the third stage, the chaotic sequences generated by the improved Logistic chaotic system are used to perform row-column permutations, completing the scrambling. The security analysis of the encrypted images shows that the algorithm described in this paper is robust and secure, capable of resisting most known attacks. The algorithm is fast in encryption, provides high-quality image reconstruction, and is suitable for scenarios with high comprehensive performance and image quality requirements.

A novel memristive chaotic jerk circuit and its microcontroller-based sliding mode control

Due to the experimental realization of memristor circuit elements, research on memristors and memristor-based circuits has surged. Because of their nonvolatile and nonlinear behavior, memristors can be easily applied to chaotic circuits. This study introduces a novel memristive 3D chaotic jerk system, comprising only seven terms, along with its electronic model and microcontroller-based control. The flux-controlled memristor-based jerk system exhibits complex dynamics, which were analyzed through various properties such as phase portraits, the Jacobian matrix, equilibria, eigenvalues, Lyapunov spectra, bifurcation diagrams, and transient chaos behavior. Three controllers, namely, nonlinear feedback, classical sliding mode, and integral sliding mode were designed to control the chaotic jerk oscillator. Lyapunov functions were used to synthesize the nonlinear feedback controller and ensure system stability with the sliding mode technique. Numerical tests under various performance criteria and disturbance conditions showed that the sliding mode controller outperforms the nonlinear feedback controller due to its single-state control structure. The chaotic jerk oscillator hardware circuit was designed and implemented, operating easily with initial conditions set to zero and low DC supply voltages, with all output voltages within ±6V. Both theoretical and simulation results demonstrate the system’s complexity and applicability, with experimental results aligning well with simulations. Consequently, effective microcontroller-based control was achieved using a single-state controller.

Spread complexity and quantum chaos for periodically driven spin chains.

The complexity of quantum states under dynamical evolution can be investigated by studying the spread with time of the state over a predefined basis. It is known that this complexity is minimized by choosing the Krylov basis, thus defining the spread complexity. We study the dynamics of spread complexity for quantum maps using the Arnoldi iterative procedure. The main illustrative quantum many-body model we use is the periodically kicked Ising spinchain with nonintegrable deformations, a chaotic system where we look at both local and nonlocal interactions. In the various cases, we find distinctive behavior of the Arnoldi coefficients and spread complexity for regular versus chaotic dynamics: suppressed fluctuations in the Arnoldi coefficients as well as larger saturation value in spread complexity in the chaotic case. We compare the behavior of the Krylov measures with that of standard spectral diagnostics of chaos. We also study the effect of changing the driving frequency on the complexity saturation.

A new hyperchaotic system: circuit realization, nonlinear analysis and synchronization control

This paper presents a novel seven-dimensional nonlinear hyperchaotic system characterized by a minimal number of nonlinear terms and variables, yet exhibiting high complexity. Standard nonlinear analysis is conducted to unveil the system’s intricacies, emphasizing its notable feature of possessing four to five Lyapunov exponents in certain intervals, signifying its volatility and complexity. Hyperchaotic synchronization is explored using a novel nonsingular terminal sliding control design, effectively achieving synchronization between two hyperchaotic master systems and a hyperchaotic slave system within finite time while mitigating the chattering phenomenon. Practical evaluations through orbital analysis, numerical simulations, and practical implementations further substantiate the efficacy and performance of the proposed system. This study contributes to the advancement of chaotic and hyperchaotic systems, particularly those with dimensions exceeding 5D, offering insights into synchronization techniques and practical applications in engineering and other scientific disciplines.