Chaotic Properties Research Articles

A novel Chua's based 2-D chaotic system and its performance analysis in cryptography.

In this study, the chaotic behavior of a second-order circuit comprising a nonlinear resistor and Chua's diode is investigated. This circuit, which includes a nonlinear capacitor and resistor among its components, is considered one of the simplest nonautonomous circuits. The research explores various oscillator characteristics, emphasizing their chaotic properties through bifurcations, Lyapunov exponents, periodicity, local Lyapunov region, and resonance. The system exhibits both stable equilibrium points and a chaotic attractor. Additionally, the second objective of this study is to develop a novel cryptographic technique by incorporating the designed circuit into the S-box method. The evaluation results suggest that this approach is suitable for secure cryptographic applications, providing insights into constructing a cryptosystem for images and text based on its complex behavior. Real-life data were analyzed using various statistical and performance criteria after applying the proposed methodology. These findings enhance the reliability of the cryptosystems. Moreover, The proposed methods are assessed using a range of statistical and performance metrics after testing the text and images. The cryptographic results are compared with existing techniques, reinforcing both the developed cryptosystem and the performance analysis of the chaotic circuit.

A multi-medical image encryption algorithm based on ROI and DNA coding

Abstract With the rapid development of information technology in the field of electronic medicine, the confidentiality of medical images has received increasing attention. The research on the encryption of multiple medical images holds greater practical significance. In this paper, the encryption algorithm is designed specifically for the region of interest (ROI) in medical images. Different techniques and methods are used to encrypt ROI and region of non-interest (RONI) respectively. By combining improved Zigzag scrambling, DNA coding, and the Fisher-Yates shuffle, we place an emphasis on protecting the ROI, and achieve secure encryption for medical images of any number and size. In addition, a new one-dimensional chaotic system S-LCS with larger parameter space and better chaotic properties is proposed. In this encryption scheme, the information about the ROI serves as the secret key, and the initial values and parameters of the chaotic sequences required for encryption are calculated from this key. This strengthens the relationship between the key and the plaintext, enhancing the security of the key. Through testing and comparative analysis, it has been found that the encryption algorithm has high enough security, can resist various attacks, and has high encryption efficiency in the application scenario of multi-image encryption.

Ergodic and Chaotic Properties of the Heat Equation

AbstractWe consider a semiflow generated by the heat equation on the half-line with zero Neumann boundary condition. If the initial functions are from some weighted space X, then we prove that there exists an invariant mixing measure and X is the topological support of this measure. This result implies chaotic properties of the semiflow.

Grain storage temperature prediction based on chaos and enhanced RBF neural network

Grain storage has very strict temperature requirements. Aiming at the problems of nonlinear characteristics and poor prediction accuracy of temperature parameters in grain storage, a combination of chaos theory and enhanced radial basis neural network is proposed as a temperature prediction model for grain storage (C-ERBF). The model first determines the embedding dimension and time delay of the grain storage temperature sequence using chaos theory. It then calculates the Lyapunov exponent to confirm its chaotic properties and reconstructs the sequence in the phase space to extract the hidden dynamic information and structure behind the sequence. Furthermore, the q-Normalized Least Mean Square Fourth (qXE-NLMF) algorithm is designed to enhance the radial basis function (RBF) neural network model for weight updating, to improve its prediction accuracy, and to accelerate the training speed of the model. As verified by the simulation experiments of Mackey–Glass chaotic time series prediction, the enhanced RBF (ERBF) network has faster convergence speed and lower steady-state error compared to the traditional RBF network. Finally, the optimized dataset from chaos theory is input into the model to achieve accurate predictions of grain storage temperature series. The experimental results show that the proposed C-ERBF model has higher prediction accuracy compared to other time series prediction methods. It can realize the grain pile temperature in advance, and take control measures in advance. This proactive approach significantly reduces the consumption of stored grain and prevents issues before they arise.

Chaotic Properties of Billiards in Circular Polygons

Chaotic Properties of Billiards in Circular Polygons

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

Ergodic and chaotic properties in a Tavis-Cummings dimer: Quantum and classical limit

Ergodic and chaotic properties in a Tavis-Cummings dimer: Quantum and classical limit

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.

Patterns in the Chaos: The Moving Hurst Indicator and Its Role in Indian Market Volatility

Estimating the impact of volatility in financial markets is challenging due to complex dynamics, including random fluctuations involving white noise and trend components involving brown noise. In this study, we explore the potential of leveraging the chaotic properties of time series data for improved accuracy. Specifically, we introduce a novel trading strategy based on a technical indicator, Moving Hurst (MH). MH utilizes the Hurst exponent which characterizes the chaotic properties of time series. We hypothesize and then prove empirically that MH outperforms traditional indicators like Moving Averages (MA) in analyzing Indian equity indices and capturing profitable trading opportunities while mitigating the impact of volatility.

Spread and spectral complexity in quantum spin chains: from integrability to chaos

We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.

Thermodynamics and dynamics of coupled complex SYK models.

It has been known that the large-q complex SYK model falls under the same universality class as that of van der Waals (mean-field) and saturates the Maldacena-Shenker-Stanford bound, both features shared by various black holes. This makes the SYK model a useful tool in probing the fundamental nature of quantum chaos and holographic duality. This work establishes the robustness of this shared universality class and chaotic properties for SYK-like models by extending to a system of coupled large-q complex SYK models of different orders. We provide a detailed derivation of thermodynamic properties, specifically the critical exponents for an observed phase transition, as well as dynamical properties, in particular the Lyapunov exponent, via the out-of-time correlator calculations. Our analysis reveals that, despite the introduction of an additional scaling parameter through interaction strength ratios, the system undergoes a continuous phase transition at low temperatures, similar to that of the single SYK model. The critical exponents align with the Landau-Ginzburg (mean-field) universality class, shared with van der Waals gases and various AdS black holes. Furthermore, we demonstrate that the coupled SYK system remains maximally chaotic in the large-q limit at low temperatures, adhering to the Maldacena-Shenker-Stanford bound, a feature consistent with the single SYK model. These findings establish robustness and open avenues for broader inquiries into the universality and chaos in complex quantum systems. We provide a detailed outlook for future work by considering the ``very" low-temperature regime, where we discuss relations with the Hawking-Page phase transition observed in the holographic dual black holes. We present preliminary calculations and discuss the possible follow-ups that might be be taken to make the connection robust.

Fluctuation-dissipation theorem and entanglement dynamics in one-dimensional low-density Jaynes-Cummings Hubbard model after a quench.

In one-dimensional low-density Jaynes-Cummings Hubbard (JCH) models [Phys. Rev. E 106, 064107 (2022)2470-004510.1103/PhysRevE.106.064107], we proved that the eigenstate thermalization hypothesis (ETH) is valid when the tunneling strength and coupling strength are of the same order. Surprisingly, at the weak tunneling limit, we observed that the entanglement entropy and scaling law of kinetic energy operators also exhibit obvious quantum chaotic properties, this is an unexpected result. To substantiate these findings, we further discuss their nonequilibrium dynamics in this paper. Our analysis reveals that when the model is a weak tunneling limit after the quench and the initial state is an equilibrium state of chaos, the system reaches an equilibrium state. This observation supports the conclusion that the low-density JCH model at the weak tunneling limit is nonintegrable, corroborating our previous results [Phys. Rev. E 106, 064107 (2022)2470-004510.1103/PhysRevE.106.064107]. Additionally, by discussing the validity of the fluctuation-dissipation theorem (FDT) and the evolution behavior of entanglement entropy and fidelity, we numerically demonstrate the differences between the one-dimensional low-density JCH model and general nonintegrable systems. Specifically, in the low-density JCH model, when the Hamiltonian after the quench is integrable, the validity of FDT depends on the thermal behavior of the initial Hamiltonian, and a metastable state is observed during the evolution of entanglement entropy. Our research presents an an intriguing and unique nonintegrable model, enriching the current understanding of nonintegrable systems.

Chaotic nature of the electroencephalogram during shallow and deep anesthesia: From analysis of the Lyapunov exponent

In conscious states, the electrodynamics of the cortex are reported to work near a critical point or phase transition of chaotic dynamics, known as the edge-of-chaos, representing a boundary between stability and chaos. Transitions away from this boundary disrupt cortical information processing and induce a loss of consciousness. The entropy of the electroencephalogram (EEG) is known to decrease as the level of anesthesia deepens. However, whether the chaotic dynamics of electroencephalographic activity shift from this boundary to the side of stability or the side of chaotic enhancement during anesthesia-induced loss of consciousness remains poorly understood. We investigated the chaotic properties of EEGs at two different depths of clinical anesthesia using the maximum Lyapunov exponent, which is mathematically regarded as a formal measure of chaotic nature, using the Rosenstein algorithm. In 14 adult patients, 12 s of electroencephalographic signals were selected during two depths of clinical anesthesia (sevoflurane concentration 2% as relatively deep anesthesia, sevoflurane concentration 0.6% as relatively shallow anesthesia). Lyapunov exponents, correlation dimensions and approximate entropy were calculated from these electroencephalographic signals. As a result, maximum Lyapunov exponent was generally positive during sevoflurane anesthesia, and both maximum Lyapunov exponents and correlation dimensions were significantly greater during deep anesthesia than during shallow anesthesia despite reductions in approximate entropy. The chaotic nature of the EEG might be increased at clinically deeper inhalational anesthesia, despite the decrease in randomness as reflected in the decreased entropy, suggesting a shift to the side of chaotic enhancement under anesthesia.

Audio encryption framework based on chaotic map and DNA encoding

In this work, an audio encryption framework is proposed based on the chaos theory and DNA encoding. Traditional encryption algorithms designed for text data might not be efficient for handling the bulk data of audio files, leading to slow processing times and increased storage requirements. Striking the right balance between robust encryption and efficient processing is crucial. Additionally, some encryption methods can introduce noise or artifacts into the audio signal, impacting its clarity. Maintaining high-fidelity audio quality while ensuring strong encryption remains an ongoing area of research. Despite these challenges, the need for secure audio communication is undeniable. The RCM chaotic map is used to introduce chaotic properties in the encryption framework. A novel pseudorandom bit generator approach is proposed and used for encryption purposes. The proposed work is a symmetric encryption approach and is completely lossless. The proposed approach uses some lightweight computational procedures that make the algorithm simple and computationally less expensive which makes it suitable for real-time applications. Furthermore, the proposed approach shows significant performance and resilience against various cryptographic attacks. A new scrambling algorithm is proposed to add a layer of security. The proposed scramble algorithm uses logistic map beside the RCM map. The proposed approach achieved 98.28 % NSCR value and 33.63 % UACI values on average. The experimental outcomes are encouraging for practical applications of the proposed approach.

Multi-wing chaotic system based on smooth function and its predefined time synchronization

In this paper, the construction of multi-wing chaotic system and predefined time synchronization are investigated. First, a new 4D chaotic system is constructed, and the existence of chaotic attractors is proved by dissipativity, boundedness, and Lyapunov exponents. Then the existence of a saddle focus of index-2 is proved by equilibrium point analysis, and a multi-winged hyperchaotic system is generated by extending the saddle focus with the constructed smooth function, which is found to be more complex and more conducive to synchronization applications by complexity analysis. Then the simulation of the circuit also proves the chaotic properties of the system in a physical sense. Finally, a new robust controller is designed using the fast terminal sliding mode algorithm to achieve predefined time synchronization between heterogeneous chaotic systems, and the numerical simulation results indicate that this synchronization method has a rapid time of convergence and achieves the predefined time effect.

Semiclassical dynamics of a superconducting circuit: chaotic dynamics and fractal attractors

We study here the semiclassical dynamics of a superconducting circuit constituted by two Josephson junctions in series, in the presence of a voltage bias. We derive the equations of motion for the circuit through a Hamiltonian description of the problem, considering the voltage sources as semi-holonomic constraints. We find that the dynamics of the system corresponds to that of a planar rotor with an oscillating pivot. We show that the system exhibits a rich dynamical behaviour with chaotic properties and we present a topological classification of the cyclic solutions, providing insight into the fractal nature of the dynamical attractors.

Unmanned ship image encryption method based on a new four-wing three-dimensional chaotic system and compressed sensing

This paper presents a three-dimensional chaotic map with five equilibrium points, which, unlike the classic Lorenz system, possesses a symmetric four-wing structure. Through detailed dynamical analysis and verification, the excellent chaotic properties of this map have been proven. Based on this system, an improved lightweight unmanned surface vehicle (USV) image information encryption system is designed. Given the constraints of USV equipment, the system first compresses images to enhance their storage capacity and transmission efficiency. Subsequently, an enhanced lightweight Feistel network encryption method is developed, utilizing a diagonal interpolation permutation model to ensure the security of the encryption process and reduce its complexity through a block structure. Simulation experiments and data analysis demonstrate that this algorithm effectively ensures the security of image information within the USV system. The use of compressed sensing and a block encryption structure reduces the complexity of the encryption process, thereby enhancing both the encryption efficiency and the transmission efficiency.

An efficient image encryption model based on 6D hyperchaotic system and symmetric matrix for color and gray images

The security of images is one of the predominant pivotal aspects in the mammoth and still expanding digital domain. Due to chaotic system properties i.e. randomness and unpredictability is very appropriate to encrypt the images. In this research article, we construct an encryption model via 6D hyperchaotic map and a symmetric matrix for both color and grayscale images. We utilize the 6D hyperchaotic map in the confusion stage to change the pixel location and the symmetric matrix is used for changing the pixel value in the diffusion step for each RGB channel extraction from plain or original image. The image encryption model is checked over differential attacks (NPCR and UACI). Histogram analysis, correlation coefficients, and entropy analysis are also performed as statistical attacks. In conclusion, the image pixels are uniformly distributed, and the average entropy value are 7.9992 and 7.9973 for color and grayscale images, subsequently. The average NPCR and UACI for color images are 99.5956 and 33.4061, correspondingly, while the values for grayscale images are 99.5934 and 33.3054, respectively. These values are in the vicinity of optimal ranges. The suggested scheme's great efficiency and the proposed algorithm's resilience to a wide range of cryptanalytic attacks are implied by experimental results, statistical analysis, and differential attacks.

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

New 4D hyperchaotic system’s application in image encryption

In order to protect sensitive information from unauthorized access and illegal copy during network transmission, storage and processing, we propose a new four-dimensional hyperchaotic system (4DHS) and apply it to encryption algorithm. Firstly, the dynamical properties of 4DHS are analyzed according to the structure, and the chaotic properties are verified by dissipation, equilibrium point and lyapunov exponent. Secondly, the chaotic sequence combined with Arnold scrambling method is adopted to scramble the pixel values of the plaintext image, and the scrambled pixel matrix is diffused into the ciphertext image matrix by XOR operation. Finally, we conduct the experiments to validate the effectiveness of the proposed encryption algorithm and achieve satisfactory results. At the same time, we compare the proposed encryption algorithm with other encryption algorithms, and the excellent encryption effect of our encryption algorithm can be proved.