Lorenz System Research Articles

Mixformer: An improved self-attention architecture applied to multivariate chaotic time series prediction

Multivariate chaotic time series prediction has been a challenging problem, especially the coupling relationship between multiple variables need to be carefully considered. We propose a self-attention architecture with an information interaction module, called mixformer, applied to the multivariate chaotic time series prediction task. First, Based on rethinking the multivariate data structure, we compensate the deficiency of phase space reconstruction with a group of cross-convolution operator that can automatically update parameters, and propose an explicitly designed feature reconstruction module. Then, to address the problem of interactive fusion of series information and channel information, we propose an information interaction module that enables the feature communication by expanding and contracting dimensions. By breaking the communication barrier between series information and channel information, the feature representation capability is enhanced. Finally, we construct mixformer that combines locally sparse features and global context features. And notably, mixformer integrates the information interaction module and the feature reconstruction module to form a continuous solution. By comparing the results with existing models using multiple simulated data (Lorenz, Chen, and Rossler systems) and real-world application (power consumption), it is verified that our proposed model achieves surprising performance in practice.

Based on the Improved PSO-TPA-LSTM Model Chaotic Time Series Prediction

In order to enhance the prediction accuracy and computational efficiency of chaotic sequence data, issues such as gradient explosion and the long computation time of traditional methods need to be addressed. In this paper, an improved Particle Swarm Optimization (PSO) algorithm and Long Short-Term Memory (LSTM) neural network are proposed for chaotic prediction. The temporal pattern attention mechanism (TPA) is introduced to extract the weights and key information of each input feature, ensuring the temporal nature of chaotic historical data. Additionally, the PSO algorithm is employed to optimize the hyperparameters (learning rate, number of iterations) of the LSTM network, resulting in an optimal model for chaotic data prediction. Finally, the validation is conducted using chaotic data generated from three different initial values of the Lorenz system. The root mean square error (RMSE) is reduced by 0.421, the mean absolute error (MAE) is reduced by 0.354, and the coefficient of determination (R2) is improved by 0.4. The proposed network demonstrates good adaptability to complex chaotic data, surpassing the accuracy of the LSTM and PSO-LSTM models, thereby achieving higher prediction accuracy.

A general theory to estimate Information transfer in nonlinear systems

A general theory for computing information transfers in nonlinear stochastic systems driven by deterministic forcings and additive and/or multiplicative noises, is presented, satisfying quite general boundary conditions in the state-space: either closed, periodic or satisfying a vanishing probability density function. It extends the Liang-Kleeman (LK) framework of causality inference to nonlinear cases based on information transfer across system variables, which is presented in detail in (Liang, 2016. Information flow and causality as rigorous notions ab initio. Phys. Rev. E, 94: 052201. DOI: 10.1103/PhysRevE.94.052201). We present an effective method of computing formulas of the rates of Shannon entropy transfer (RETs) between selected causal and consequential variables, the ‘Causal Sensitivity Method’ (CSM), relying on the estimation from data of conditional expectations of the system forcings and their derivatives. Those expectations are approximated by nonlinear differentiable regressions, leading to a much easier and more robust way of computing RETs than the ‘brute-force’ approach which calls for the computation of numerical integrals over the state-space and the knowledge of the multivariate probability density function of the system. The CSM is furthermore fully adapted to the case where no model equations are available, starting with a nonlinear model fitting from data of the consequential variables, with the subsequent application of CSM to the fitted model. RETs are decomposed into deterministic and stochastic components, being compensated by the self generation of entropy in ergodic conditions. Moreover, RETs are decomposed into sums of single one-to-one RETs plus synergetic terms (of pure nonlinear nature) accounting for the joint causal effect of groups of variables. State-dependent (or specific) RET formulas are also introduced, puting in evidence where in state-space the entropy transfers and local synergies are more relevant. A comparison of the RETs estimations is performed between: 1) the ‘brute-force’, expensive (taken as benchmark), probability-density-based approach (AN), 2) the CSM-based approach with and/or without model fitting, and 3) the multivariate linear (ML) approach, in the context of two different models: (i) a model derived from a potential function and (ii) the classical chaotic Lorenz system, both forced by additive and/or multiplicative noises. The analysis demonstrates that the CSM estimations are robust, cheaper, and less data-demanding than the AN-reference values in the different experiments, providing evidence of the possibilities and generalizations offered by the method (e.g. causality diagnostics between subspaces) and opening new perspectives on real-world applications.

Secure blind watermarking using Fractional-Order Lorenz system in the frequency domain

This paper investigates two different blind watermarking systems in the frequency domain with the development of a Pseudo Random Number Generator (PRNG), based on a fractional-order chaotic system, for watermark encryption. The methodology is based on converting the cover image to the YCbCr color domain and applying two different techniques of frequency transforms, Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT), to the Y channel. Then, the encrypted watermark is embedded in the middle-frequency band and HH band coefficients for the DCT and DWT, respectively. For more security and long encryption key size, the fractional-order Lorenz system is used to double the encryption key size and make it secure against brute-force attacks. The proposed algorithms successfully detect the hidden watermark by using the statistical properties of the embedding media, where the PRNG is examined using statistical tests and the watermarking systems are evaluated using standard imperceptibility and robustness measures. Common attacks such as noise-adding attacks, image enhancement attacks and geometric transformation attacks are discussed. Results of the PRNG demonstrate sensitivity to the system parameters, and results of the watermarking systems show good imperceptibility while keeping the robustness measures in a good range.

A simple chaotic system using signum function

Starting from Lorenz System, a simpler system using signum function was obtained, which shows chaotic behavior. The conditions for stability were explained. Lyapunov and bifurcation analyses were carried out to exhibit the chaotic behavior. The system’s suitability for circuit realization was shown with an electronic circuit implementation which is based mainly on opamps, whose simulation results were in harmony with Matlab results. Finally, as an application, a random number generator (RNG) scheme was designed, in which binary data were obtained from x,y,z variables. In order to reveal that the RNG scheme is functional, NIST-Test was applied to the obtained binary data.

The Two- to Four-Day Predictability of Midlatitude Cyclones: Don’t Sweat the Small Stuff

Abstract We use convection-permitting idealized simulations of moist midlatitude cyclones to compare the growth of synoptic-scale perturbations derived from an adjoint model with the growth of equal-energy-norm, monochromatic-wavelength perturbations at the smallest resolved scale. For initial magnitudes comparable to those of initial-condition uncertainties in present-day data assimilation systems, the adjoint perturbations produce a “forecast bust,” significantly changing the intensity and location of the cyclone and its accompanying precipitation. In contrast, the small-scale-wave perturbations project strongly onto the moist convection, but the upscale growth from the random displacement of individual convective cells does not significantly alter the cyclone’s development nor its accompanying precipitation through 2–4-day lead times. Instead, the differences in convection generated at early times become negligible because the development of subsequent convection is driven by the mostly unchanged synoptic-scale flow. Reducing the perturbation magnitudes by factors of 10 and 100 demonstrates that nonlinear dynamics play an important role in the displacement of the cyclone by the full-magnitude adjoint perturbations, and that the upscale growth of small-magnitude, small-scale perturbations is too slow to significantly change the cyclone. These results suggest that a sensitive dependence on the synoptic-scale initial conditions, analogous to that of the Lorenz (1963) system, may be more relevant to 2–4-day midlatitude-cyclone forecast busts than the upscale error growth in the Lorenz (1969) model.

2D compressed sensing of encrypted images based on complex‐valued measurement matrix

AbstractWhen using untrusted third parties to compress and transmit images in real‐life scenarios, it is vital to encrypt them before compression. In order to better address the issues of low security in the original image and poor reconstruction quality of the encrypted image during compressed sensing, this paper proposes a 2D compressed sensing scheme for encrypted images based on complex‐valued measurement matrix (2DCS‐CVM). Firstly, the SHA‐256 algorithm generates keys for the hyperchaotic Lorenz system, and then the chaotic sequences are used to create encrypted images with increased security through subtractive diffusion and global permutation. Secondly, the complex‐valued Vandermonde measurement matrix is utilized for 2D compressed sensing on the encrypted image, and the two‐dimensional projected gradient with embedding decryption algorithm is used to generate recovered images with improved reconstruction performance. Finally, the measurement matrix's computational complexity and transmission bandwidth are reduced through structural sparsification with sparse random matrices. Simulation results demonstrate that this scheme offers an optimal balance between storage, computational complexity, hardware implementation, and reconstruction performance while providing excellent security and robustness.

Data-driven dynamics reconstruction using RBF network * *This work was supported by NSFC (No. 32171094) and National Key R&D Program of China (2019YFA0709503).

Constructing the governing dynamical equations of complex systems from observational data is of great interest for both theory and applications. However, it is a difficult inverse problem to explicitly construct the dynamical equations for many real complex systems based on observational data. Here, we propose to implicitly represent the dynamical equations of a complex system using a radial basis function (RBF) network trained on the observed data of the system. We show that the RBF network trained on trajectory data of the classical Lorenz and Chen system can faithfully reproduce the orbits, fixed points, and local bifurcations of the original dynamical equations. We also apply this method to electrocardiogram (ECG) data and show that the fixed points of the RBF network trained using ECG can discriminate healthy people from patients with heart disease, indicating that the method can be applied to real complex systems.

Selective medical image encryption based on 3D Lorenz and Logistic system

Selective medical image encryption based on 3D Lorenz and Logistic system

ANEW ARCHITECTURE OF KEY GENERATION USING DWT FOR IMAGE ENCRYPTION WITH THREE LEVELS ARNOLD TRANSFORM PERMUTATION

The security of image transmission is an important issue in digital communication. As well as it is necessary to preserve and protect important information for several applications like military, medical and other services related to high confidentiality over the Internet or other unprotected networks. In this paper a proposed encryption scheme was introduced that using Lorenz system with a circular convolution and discreet cosine transform (DCT). The diffusion process was achieved using three levels of Arnold transform permutation: - namely, block level, inside each block and pixel level. The image was divided into blocks of sizes 8x8 pixels and shuffle by applying Fisher-Yates permutation image pixels. The DCT was applied to each block and multiplied by the H kernel matrix to achieve the circular convolution. Next the logistic map is used for diffusion to get the cipher image. A new key generation method is applied in order to produce the key values for generating the chaos numbers sequence. Finally applying the discreet wavelet transform (DWT) to the image will produced four quarters (LL, LH, HL and HH). The cosine of each pixel in the LL and HH quarters were computed. The sine of each pixel in the LH and HL quarters were also computed in order to generate the keys for the chaos sequence. By using this way of keys generation in every image, the differential attack possibility will be negligible

LMIs-based exponential stabilization for interval delay systems via congruence transformation: Application in chaotic Lorenz system

This paper investigates the stochastic Takagi-Sugeno (T-S) fuzzy uncertainty-dependent state feedback sampled-data control, asymptotical stability and exponential stability analysis for a class of stochastic T-S fuzzy system with measurable uncertainties, mixed interval time varying delays and Gaussian white noise. Firstly, the T-S fuzzy model and stochastic Bernoulli theory are employed to approximate the system plant. Secondly, the stochastic T-S fuzzy uncertainty-dependent state feedback sampled-data controller is designed via uncertainty-dependent parameters and the hold times series of sampled-data. Thirdly, the Bernoulli probability distribution is employed to describe the random occurrence characteristic of multipath packet dropouts. Three classes of delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) are derived for the closed-loop system. The multiple integral fuzzy-basis-dependent Lyapunov functional and augmented state variable are employed to derive the delay-dependent stability conditions, then the closed-loop system is asymptotically stable and exponentially stable. The congruence transformation method and prescribed H-infinity performance functional are employed to derive the less conservative delay-dependent stability conditions, and the prescribed H-infinity performance is guaranteed. The relaxation matrix and congruence transformation method are employed to design the linear transformation equalities and congruence transformation matrices, then the controller gain matrix is determined and computation complexity of solving LMIs is reduced. Finally, the proposed method is applied in the chaotic Lorenz system to show the effectiveness and advantage of proposed method.

An Improved Approach for Implementing Dynamic Mode Decomposition with Control

Dynamic Mode Decomposition with Control is a powerful technique for analyzing and modeling complex dynamical systems under the influence of external control inputs. In this paper, we propose a novel approach to implement this technique that offers computational advantages over the existing method. The proposed scheme uses singular value decomposition of a lower order matrix and requires fewer matrix multiplications when determining corresponding approximation matrices. Moreover, the matrix of dynamic modes also has a simpler structure than the corresponding matrix in the standard approach. To demonstrate the efficacy of the proposed implementation, we applied it to a diverse set of numerical examples. The algorithm’s flexibility is demonstrated in tests: accurate modeling of ecological systems like Lotka-Volterra, successful control of chaotic behavior in the Lorenz system and efficient handling of large-scale stable linear systems. This showcased its versatility and efficacy across different dynamical systems.

Thermodynamic efficiency of atmospheric motion governed by the Lorenz system

Thermodynamic efficiency of atmospheric motion governed by the Lorenz system

Generalized limited penetrable vector visibility graph

The paper introduced a new way for converting the multivariate time series to complex network combining the vector visibility graph and limited penetrable theory, and the concept is discussed on the generalized limited penetrable range, termed as generalized limited penetrable vector visibility graph. The proposed method is verified by the Lorenz system. We use the four kinds of network structure characteristic to evaluate the noise resistance of the method quantitatively, and provide the strategy with guidance for choosing the optimal limited penetrable range. The results show that the limited penetrable vector visibility graph inherits the noise resistance of limited penetrable visibility graph in the multivariate time series analysis. The generalized limited penetrable range is discussed to clarify the effect on the network structure.

Chaotic Behavior of Lorenz-Based Chemical System under the Influence of Fractals

This research examines a chaotic chemical reaction system based on the variation of the Lorenz system. This study demonstrates that although the first phase portraits of the chemical models under consideration and the Lorenz models are comparable, they do not fully follow all the features of the Lorenz system. Questions about the existence of fractals in systems based on chemical reactions are addressed in the current work. Moreover, we have worked on the hidden information inside in each wings of a chaotic system generated through fractal process, for the first time, with the aid of basin for fractals. Additionally, we looked closely at the dynamics of the model across the basin, which revealed additional details regarding the existence of hidden and cyclic attractors inside each wing. We also produced multi-wings for system (1) in the current study, demonstrating in a general manner that the number of cyclic attractors increase in a direct relation to the number of wings. Moreover, Julia approach is used to accomplish the work of multi-wings, whereas for searching cyclic attractors inside each extra wing, we have used fifteen million initial conditions and compiled them as a basin set. The data generated in this work is also provided within this paper for the ease of readers.

Random templex encodes topological tipping points in noise-driven chaotic dynamics.

Random attractors are the time-evolving pullback attractors of deterministically chaotic and stochastically perturbed dynamical systems. These attractors have a structure that changes in time and that has been characterized recently using Branched Manifold Analysis through Homologies cell complexes and their homology groups. This description has been further improved for their deterministic counterparts by endowing the cell complex with a directed graph (digraph), which encodes the order in which the cells in the complex are visited by the flow in phase space. A templex is a mathematical object formed by a cell complex and a digraph; it provides a finer description of deterministically chaotic attractors and permits their accurate classification. In a deterministic framework, the digraph of the templex connects cells within a single complex for all time. Here, we introduce the stochastic version of a templex. In such a random templex, there is one complex per snapshot of the random attractor and the digraph connects the generators or "holes" of successive cell complexes. Tipping points appear in a random templex as drastic changes of its holes in time, through their birth, splitting, merging, or death. This paper introduces random templexes and computes them for the noise-driven Lorenz system's random attractor.

Reservoir computing with error correction: Long-term behaviors of stochastic dynamical systems

The prediction of stochastic dynamical systems and the capture of dynamical behaviors are profound problems. In this article, we propose a data-driven framework combining Reservoir Computing and Normalizing Flow to study this issue, which mimics error modeling to improve traditional Reservoir Computing performance and integrates the virtues of both approaches. With few assumptions about the underlying stochastic dynamical systems, this model-free method successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors. We verify the effectiveness of the proposed framework in several experiments, including the stochastic Van der Pal oscillator, El Niño-Southern Oscillation simplified model, and stochastic Lorenz system. These experiments consist of Markov/non-Markov and stationary/non-stationary stochastic processes, which are defined by linear/nonlinear stochastic differential equations or stochastic delay differential equations. Additionally, we explore the noise-induced tipping phenomenon, relaxation oscillation, stochastic mixed-mode oscillation, and replication of the strange attractor.

Chaotic behavior learning via information tracking

Chaotic time series are often encountered in real-world applications, where modeling and understanding such time series present a significant challenge. Existing simulation-based methods for characterizing chaotic behavior may be sensitive to the respective model settings, while data-driven methods cannot adapt well to irregularities and aperiodicity in chaotic time series during the prediction stage, leading to nonlinearly aggregated forecasting errors after a finite number of time steps. In our preliminary work (Ren et al., 2023), we introduced a data-driven method that incorporates the idea of information tracking to capture and adapt to the chaotic changes over time, and tested its effectiveness on a real-world task—decadal temperature prediction. While the practical usefulness of the method has been initially validated on one domain-specific application scenario, a set of open questions is yet to be answered: Is the concept of information tracking generalizable to model and solve a broad spectrum of learning tasks that involve chaotic behavior? If yes, what are the underlying principles and learning behaviors that make the general method work well? Moreover, how can we assess the efficacy of the method in different scenarios? This study constitutes a significant step forward by systematically investigating and addressing the questions mentioned above. Specifically, we first provide a rigorous definition of the formalism and computational mechanism of the general method, with all necessary variables, symbols, and formulations. More importantly, we theoretically characterize and understand the generalizability of the method by mathematically uncovering the fundamental principles and behavior of information tracking-based chaotic behavioral learning, as prescribed by two chaotic behavioral indexes, namely, the Lyapunov and the Hurst exponents. Further, we comprehensively demonstrate the generalizability and robustness of the method by empirically examining a variety of chaotic behavioral learning problems generated by three representative chaotic systems: The logistic map, the double pendulum, and the Lorenz system.

A Robust Image Encryption Scheme Based on Block Compressive Sensing and Wavelet Transform

In this paper, a modified robust image encryption scheme is developed by combining block compressive sensing (BCS) and Wavelet Transform. It was achieved with a balanced performance of security, compression, robustness and running efficiency. First, the plain image is divided equally and sparsely represented in discrete wavelet transform (DWT) domain, and the coefficient vectors are confused using the coefficient random permutation strategy and encrypted into a secret image by compressive sensing. In pursuit of superior security, the hyper-chaotic Lorenz system is utilized to generate the updated secret code streams for encryption and embedding with assistance from the counter mode. This scheme is suitable for processing the medium and large images in parallel. Additionally, it exhibits superior robustness and efficiency compared with existing related schemes. Simulation results and comprehensive performance analyses are presented to demonstrate the effectiveness, secrecy and robustness of the proposed scheme. The compressive encryption model using BCS with Walsh transform as sensing matrix and WAM chaos system, the scrambling technique and diffusion succeeded in enhancement of secure performance.

Physics-Informed Long-Short-Term Memory Neural Network for Parameters Estimation of Nonlinear Systems

This paper proposes a physics-informed long-short-term memory neural network optimisation method to estimate all parameters of an arbitrary nonlinear system initialised using any initial conditions. Although the technique is applicable for parameters' estimation of any nonlinear system, in this paper, two well-known chaotic systems are considered with different initial conditions. The three-dimensional Lorenz chaotic system and the four-dimensional Rössler hyperchaotic system are considered. The Lorenz system exhibits stable, chaotic and periodic behaviours for some sets of parameters. Thus, different sets of parameters of the Lorenz system are individually estimated, leading to stable, periodic and chaotic behaviours. In 10000 epochs, the stable and periodic parameters' estimation accuracy is more than 99%; in 15000 epochs, chaotic parameters' estimation accuracy is more than 99%. Upto same epochs, physics-informed neural network is employed for parameter estimation of stable, periodic and chaotic behaviours of the Lorenz system. The comparison results of estimated parameters are shown with dataset obtained using ode45. Next, Circuit realisation of the Lorenz system is done and the dataset is obtained using multisim circuit simulation for stable, chaotic and periodic behaviour. The proposed physics-informed long-short-term memory neural network is used to obtain parameters from the practical data. Then, the estimation of the Lorenz system with a continuum variation of parameters' sets from stable to chaotic to stable behaviours is successfully achieved using the proposed physics-informed long-short-term memory neural network. Finally, the hyperchaotic Rössler system parameters are successfully estimated using the proposed technique with more than 99% accuracy in 20000 epochs.