Chaos Model Research Articles

An Energy Focusing-Based Scanning and Localization Method for Shallow Underground Explosive Sources

To address the problem of slow speed and low accuracy for recognizing and locating the explosive source in complex shallow underground blind spaces, this paper proposes an energy-focusing-based scanning and localization method. First, the three-dimensional (3D) energy field formed by the source explosion is reconstructed using the energy-focusing properties of the steered response power (SRP) localization model, and the velocity field is calculated based on a multilayered stochastic medium model by considering the random statistical characteristics of the medium. Then, a power function factor is introduced to quantum particle swarm optimization (QPSO) to search for and solve the above energy field and to approach the real location of the energy focus point. Additionally, the initial population is constructed based on the logistic chaos model to realize global traversal. Finally, extensive simulation results based on the real-world dataset show that compared to the baseline algorithm, the focusing accuracy of the energy field of the proposed scheme is improved by 117.20%, the root mean square error (RMSE) is less than 0.0551 m, the triaxial relative error (RE) is within 0.2595%, and the average time cost is reduced by 98.40%. It has strong advantages in global search capability and fast convergence, as well as robustness and generalization.

The Primacy of Doubt: From Quantum Physics to Climate Change, How the Science of Uncertainty Can Help Us Understand Our Chaotic World

The Primacy of Doubt: From Quantum Physics to Climate Change, How the Science of Uncertainty Can Help Us Understand Our Chaotic World

PLC-Based Electric Hybrid Control of Assembly Line Manipulators from Robust Testing to Chaos Modeling

PLC-Based Electric Hybrid Control of Assembly Line Manipulators from Robust Testing to Chaos Modeling

Punctuated chaos and indeterminism in self-gravitating many-body systems

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This paper presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e. punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic.

NOMA security scheme based on constellation camouflage and selective mapping.

This study aims to present a non-orthogonal multiple access (NOMA) security scheme based on constellation camouflage and selective mapping. To improve the security of the system, we use a four-dimensional chaos model to camouflage high-power signals at the transmitter. The constellation diagram of high power is disguised from binary phase-shift keying (BPSK) form to quadrature phase-shift keying (QPSK) form, and after power multiplexing, further camouflaged from 8 points to 16 points. To improve the transmission performance of the designed system as much as possible and not increase the computational complexity, we use the selective mapping method in the process of power multiplexing and use the region decision method for demodulation at the receiving end. The proposed scheme is verified by experiments on a 2-km 7-core optical fiber, and achieves the safety transmission of a power division multiplexing-orthogonal frequency-division multiplexing (PDM-OFDM) signal with a net rate of 97.38 Gb/s without signal damage.The maximum achievable key space of the proposed scheme is 10135. Hence, it is a feasible and secure non-orthogonal multiple access-passive optical network scheme.

A new spatiotemporal chaos model and its application in bit-level image encryption

A new spatiotemporal chaos model and its application in bit-level image encryption

Born approximation study of the strong disorder in magnetized surface states of a topological insulator

In this study we investigate the effect of random point disorder on the surface states of a topological insulator with out-of-plane magnetization. We consider the disorder within a high order Born approximation. The Born series converges to the one branch of the self-consistent Born approximation (SCBA) solution at low disorder. As the disorder strength increases, the Born series converges to another SCBA solution with the finite density of states within the magnetization induced gap. Further increase of the disorder strength leads to a divergence of the Born series, showing the limits of the applicability of the Born approximation. We find that the convergence properties of this Born series are closely related to the properties of the logistic map, which is known as a prototypical model of chaos. We also calculate the longitudinal and Hall conductivities within the Kubo formulas at zero temperature with the vertex corrections for the velocity operator. Vertex corrections are important for describing transport properties in the strong disorder regime. In the case of strong disorder, the longitudinal conductivity is weakly dependent on the disorder strength, while the Hall conductivity decreases with increasing disorder.

Calibration of high-fidelity hydrodynamic models utilizing on-site vessel response measurements

With a focus on weather-restricted marine operations, a comprehensive work on numerical model calibration is conducted to improve the vessel response prediction accuracy in real environmental conditions. This framework involves using existing physics-based models in which the metocean and system parameters are entered as inputs to derive vessel responses. The numerical models based on linear potential flow theory resolve only a part of the physics, i.e., it does not account for viscous damping effects. Moreover, significant uncertainties are associated with the system parameters related to the vessel operational conditions that are not known or measured directly. Therefore, the present paper integrates the vessel model with a derivative-free Lipschitz optimization technique to calibrate the vessel system parameters. For efficient optimization, the most influential variables are identified by conducting a sensitivity study for relevant quantities of interest using a cost-effective Polynomial Chaos model. The sensitivity analysis assumes independent uniform distributions for the system parameters with proper lower and upper limits. Subsequently, estimates of the highly influential variables are optimized using on-site vessel measurements, and in turn, utilized for superior response estimation. The response computations based on the numerical wave spectrum produced superior results in comparison with the results based on the parametric wave spectrum. The calibration algorithm is validated with full-scale measurements using frequency and time domain approaches for simulating vessel motions. It has been quantitatively revealed that the Center of Gravity parameters associated with the vessel operational characteristics and sea state-dependent additional damping coefficients govern the Roll response variation to a great extent. The calibration of these parameters resulted in improved Roll spectra estimations that match the measured spectra closely.

The Genesis of Uncertainty: Structural Analysis of Stochastic Chaos in Finance Markets

The presented article is methodological in nature and is devoted to the analysis of observation series of financial asset quotation changes in capital markets. The most important feature of these processes is their instability, which manifests itself in high sensitivity to seemingly minor disturbing factors. This phenomenon is well-studied in the theory of nonlinear dynamical systems and is described by models of deterministic chaos. However, for the processes considered in the article, the dynamic instability of the immersion environment is exacerbated by stochastic uncertainty caused by random fluctuations in the pricing process. As a result, describing observation series of quotations of financial assets is difficult because it involves stochastic chaos. This article analyzes and classifies chaotic series of observations to help model and forecast related processes.

Buffeting Chaotification Model for Enhancing Chaos and Its Hardware Implementation

Many shortcomings of chaos-based applications stem from the weak dynamic properties of the chaotic maps they use. To alleviate this problem, inspired by the buffeting effect in aeroelasticity, this article proposes the buffeting chaotification model (BCM). Using the especially designed buffeting and modulo operators, the BCM can generate numerous new chaotic maps with strong dynamic properties from existing one-dimensional chaotic maps. The effectiveness of the BCM is mathematically proven according to the Lyapunov exponent, and further numerical experiments confirm the superiority of the chaotic maps generated by the BCM in terms of the dynamic properties. The field-programmable gate array implementation also shows that the BCM owns simplicity in hardware devices. To investigate the practical application, a scheme for constructing the pseudorandom number generator is designed. Performance analyses indicate that our generators have a strong ability to produce high-quality pseudorandom sequences rapidly.

Pointwise Weyl Law for Graphs from Quantized Interval Maps

We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pakoński et al. (J Phys A 34:9303–9317, 2001) as a model for quantum chaos on graphs. Since we allow shrinking spectral windows in the pointwise Weyl law, as a consequence we obtain for these models a strengthening of the quantum ergodic theorem from Berkolaiko et al. (Commun Math Phys 273:137–159, 2007), and show in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. We also examine further the specific case where the interval map is the doubling map.

Quantum Bounds on the Generalized Lyapunov Exponents

We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos.

A Self-Adaptive Image Encryption Scheme Based on Chaos and Gravitation Model

In order to improve the security of image transmission, this paper proposes a self-adaptive image encryption scheme based on chaos and gravitation model. The proposed scheme consists of three stages: key generation stage, double scrambling stage, and double diffusion stage. In the key generation stage, the plaintext-associated key generation mechanism (BPPAKG) based on the bit plane is used to obtain the key associated with the plaintext. Since the key ultimately depends on the plaintext image itself, this scheme has self-adaptability. The first stage of scrambling uses mixed chaotic transform and shift row transform to disturb pixel position, then changes the pixel value of the image based on the concept of gravitation to achieve the first stage of diffusion. To enhance the security performance of the scheme, the traditional encryption structure is extended, and the second stage of scrambling and diffusion is quickly realized by using diagonal scanning transform and XOR operation. Simulation and experimental results show that the scheme has good encryption and decryption performance and strong resistance to attack. In addition, compared with some latest proposed algorithms, it has better security.

Singularly Degenerate Heteroclinic Cycles with Nearby Apple-Shape Attractors

Compared with most known singularly degenerate heteroclinic cycles consisting of two different equilibria of a line or a curve, or two parallel lines of semi-hyperbolic equilibria, little seems to be noticed about the one that connects two perpendicular lines of semi-hyperbolic equilibria, i.e. [Formula: see text] and [Formula: see text], [Formula: see text], which is found in the mathematical chaos model: [Formula: see text], [Formula: see text], [Formula: see text] when [Formula: see text] and [Formula: see text]. Surprisingly, apple-shape attractors are also created nearby that kind of singularly degenerate heteroclinic cycles in the case of small [Formula: see text]. Further, some other rich dynamics are uncovered, i.e. global boundedness, Hopf bifurcation, limit cycles coexisting with one chaotic attractor, etc. We not only prove that the ultimate bound sets and globally exponentially attracting sets perfectly coincide under the same parameters, but also illustrate four limit cycles coexisting with one chaotic attractor, the saddle in the origin, and other two stable nontrivial node-foci, which are also trapped in the obtained globally exponentially attracting set, extending the recently reported results of the Lü-type subsystem. In addition, combining theoretical analysis and numerical simulation, the bidirectional forming mechanism of that kind of singularly degenerate heteroclinic cycles is illustrated, and their collapses may create three-scroll/apple-shape attractors, or limit cycles, etc. Finally, conservative chaotic flows are numerically found in the new system. We expect that the outcome of the study may provide a reference for subsequent research.

A multi-element non-intrusive Polynomial Chaos method using agglomerative clustering based on the derivatives to study irregular and discontinuous Quantities of Interest

A non-intrusive method to get a multi-element Polynomial Chaos model is developed. This method is called ME-ACD, for Multi-Element based on Agglomerative Clustering on Derivatives. It aims at approximating a Quantity of Interest which presents discontinuities or irregularities making it difficult to be accurately approximated by standard Polynomial Chaos models. The method permits to efficiently split the parameter space and to train local polynomial models of lower degrees on every element where the local pieces of the Quantity of Interest are smoother. The algorithm is based on agglomerative clustering of the observations in a well-chosen abstract space taking into account the value of the Quantity of Interest and of its derivatives with respect to the stochastic input parameters. The same observations are used for both partitioning the space and training the local models. Several partitions of the parameter space are tested, and the one leading to local models minimising a cross-validation error is selected. Once the training observations are labelled with a class number indicating the element they are located in, a neural network classifier is trained to determine which local model to use for further evaluations. The method has proven to efficiently split the parameter space for a set of applications of moderate dimension. The piecewise chaos model is compared with the ones of a standard Polynomial Chaos non-intrusive method and of a Gradient Boosted Trees method in terms of accuracy.

High-security SCMA-OFDM multi-core fiber transmission system based on a regular hexagon chaotic codebook.

Sparse code multiple access (SCMA), a new code-domain non-orthogonal technology in the fifth-generation mobile communication (5G), can be modulated by orthogonal frequency division multiplexing (OFDM) to improve the link quality of a single user. In this paper, a high-security SCMA-OFDM multi-core fiber transmission system based on a regular hexagonal chaotic codebook is proposed for next-generation passive optical network (PON). The whole encryption process consists of a regular hexagon chaotic codebook design and frequency domain block scrambling. In designing the regular hexagon chaotic codebook, the optimization of constellation points on orthogonal resources are considered as the starting point. Firstly, the chaos factor generated by the four-dimensional Rossler chaos model is deployed to disturb the mother constellation, and then the corresponding chaotic book is formed by rotating the mother constellation and multiplying the sparse matrix. The designed codebook logically avoids the degradation of transmission performance caused by the rough scrambling of codebook constellation, to find a balance between codebook disturbance and bit error rate (BER). The security and reliability of the transmission system have been verified by performing 42 Gb/s encrypted SCMA-OFDM data transmission experiments in a 2km multi-core fiber. The key space of the encryption scheme can reach 10178, which effectively ensures the security of the transmission system. Furthermore, the performance of the transmission system with a regular hexagon chaotic codebook is improved by 2.5 dB compared with the traditional codebook when the BER is 1 × 10-3. Moreover, the SCMA-OFDM-based transmission architecture and the detection effects of different multi-user detection algorithms in the SCMA-OFDM multi-core fiber transmission system are also studied.

Spectral form factor in a minimal bosonic model of many-body quantum chaos.

We study spectral form factor in periodically kicked bosonic chains. We consider a family of models where a Hamiltonian with the terms diagonal in the Fock space basis, including random chemical potentials and pairwise interactions, is kicked periodically by another Hamiltonian with nearest-neighbor hopping and pairing terms. We show that, for intermediate-range interactions, the random phase approximation can be used to rewrite the spectral form factor in terms of a bistochastic many-body process generated by an effective bosonic Hamiltonian. In the particle-number conserving case, i.e., when pairing terms are absent, the effective Hamiltonian has a non-Abelian SU(1,1) symmetry, resulting in universal quadratic scaling of the Thouless time with the system size, irrespective of the particle number. This is a consequence of degenerate symmetry multiplets of the subleading eigenvalue of the effective Hamiltonian and is broken by the pairing terms. In such a case, we numerically find a nontrivial systematic system-size dependence of the Thouless time, in contrast to a related recent study for kicked fermionic chains.

Ultimate speed limits to the growth of operator complexity

In an isolated system, the time evolution of a given observable in the Heisenberg picture can be efficiently represented in Krylov space. In this representation, an initial operator becomes increasingly complex as time goes by, a feature that can be quantified by the Krylov complexity. We introduce a fundamental and universal limit to the growth of the Krylov complexity by formulating a Robertson uncertainty relation, involving the Krylov complexity operator and the Liouvillian, as generator of time evolution. We further show the conditions for this bound to be saturated and illustrate its validity in paradigmatic models of quantum chaos.

A data-driven non-intrusive polynomial chaos for performance impact of high subsonic compressor cascades with stagger angle and profile errors

The impact of geometric variations due to manufacturing errors on the aerodynamic performance of compressor blades is considerable in engineering practice. Accurate uncertainty quantification (UQ) of aerodynamic performance based on actual statistical information of manufacturing errors is helpful for error detection, aerodynamic shape design, etc. However, the actual manufacturing errors may not be independent of each other, and their marginal distributions are often not standard. To apply UQ to dependent input variables with arbitrary distributions, the present paper first introduced a decorrelation algorithm and proposed a novel data-driven Non-Intrusive Polynomial Chaos (DNIPC) model. The subsequent tests based on high-dimensional function experiments and UQ of the performance for a high subsonic compressor cascade have validated its generality, effectiveness, and efficiency. Then, the simultaneous impact of stagger angle and profile errors on the performance of the compressor cascade was investigated by the proposed DNIPC and Sobol's sensitivity analysis. The distributions of manufacturing error were determined by kernel density estimation based on finite measurement data, then using the parametric B-spline and Bezier curves to build the manufactured geometry of the cascade. Finally, the influence of the manufacturing errors on aerodynamic performance was further discussed through the UQ of the cascade flow field. The results show that the performance impact in the off-design conditions, especially for those below the design incidence, is the most important. It suggested that the uncertainty of profile error should be checked carefully in the process of detecting machining error when the operation condition is at or below the design incidence; in contrast, the influence of stagger angle error calls for more attention when the cascade works at high incidences. The fluctuation of the overall aerodynamic loss is mainly associated with the manufacturing error near the leading edge, for it affects the velocity spike and the development of its downstream boundary layer. Therefore, the leading edge error should be the top focus.

A novel content-selected image encryption algorithm based on the LS chaotic model

Many holistic image encryption schemes have been proposed in recent decades that achieve a certain security level. However, considering the actual application environment, sometimes it is not necessary to encrypt the whole image, but rather a selected specific content of the image. Thus, this paper proposes a chaos-based image encryption algorithm for specific contents with non-regular size. First, we construct a novel LS chaos model that combines the Sine map and Logistic map and then improve its chaotic complexity by perturbing control parameters and introducing delayed states. Second, we obtain the coordinates of each pixel of specific image content using the PSPNet semantic segmentation model trained on the Cityscapes dataset and the Faster-RCNN target detection model trained on the MS-COCO dataset. Finally, the selected pixels are encrypted using a proposed non-regular size encryption algorithm. Several simulation experimental results prove that this new LS chaos model exhibits superior chaotic properties and generates complex pseudo-random sequences. Furthermore, our chaotic encryption algorithm effectively implements image encryption for specific content and shows excellent encryption performance, indicating that our encryption algorithm is highly competitive in terms of security capability.