High-dimensional Chaotic Systems Research Articles

Precursors-driven machine learning prediction of chaotic extreme pulses in Kerr resonators

Machine learning algorithms have opened a breach in the prediction's fortress of high-dimensional chaotic systems. Their ability to find hidden correlations in data can be exploited to perform model-free forecasting of spatiotemporal chaos and extreme events. However, the extensive feature of these evolutions makes up a critical limitation for full-size forecasting processes. Hence, the main challenge for forecasting relevant events is to establish the set of pertinent information. Here, we identify precursors from the transfer entropy of the system and a deep Long Short-Term Memory network to forecast the complex dynamics of a system evolving in a high-dimensional spatiotemporal chaotic regime. Performances of this triggerable model-free prediction protocol based on the information flowing map are tested from experimental data originating from a passive resonator operating in such a complex nonlinear regime. We have been able to predict the occurrence of extreme events up to 9 round trips after the detection of precursor, i.e., 3 times the horizon provided by Lyapunov exponents, with 92% of true positive predictions leading to 60% of accuracy.We have implemented a process to forecast extreme events in a fully developed turbulent flow. The novelty of our strategy lies in the information's use theory method to detect precursors and use them as the input of a neural network to infer the incoming extreme events. Our process is suitable for all the extended dissipative systems that can only be partially observed or real-world data.

Non-hyperbolicity at large scales of a high-dimensional chaotic system

The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively uniformly hyperbolic, which implies many felicitous statistical properties. We obtain direct and reliable numerical evidence, contrary to the chaotic hypothesis, of the existence of non-hyperbolic large-scale dynamical structures in a mean-field coupled system. To do this, we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev basis transfer operator discretization. This enables us to obtain a high-precision estimate of a homoclinic tangency, implying a failure of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be a priori assumed to hold in all systems, and a better understanding of the domain of its validity is required.

Hybrid mapping algorithm based on 1‐DCM and Lorenz

A hybrid chaotic mapping algorithm(HCM) based on 1-DCM (one-dimensional chaotic map) and Lorenz is proposed to address the security issues of low-dimensional chaotic systems and the complexity of high-dimensional chaotic systems. Firstly, the color images are divided into three high 4-bit planes (BP) and low 4-bit planes. By optimizing the parameters and structure of trigonometric function, the new arcsine-sine (AS) function is designedto generate the 1-DCM and permutation matrix, which is constructed to change the plane position and pixel position of the high 12BP to obtain the scrambling plane, which is encrypted by Lorenz chaotic system to produce the selfencryption matrix, with which the low 12BP is encrypted to generate the low bit encryption planes. Then, the ciphertext image is obtained by combining the high bit scrambling plane and low bit encryption plane. The SHA function is also used as the chaotic system's parameter generator, which improves the correlation between plaintext images and the key and resists selective plaintext attacks. The simulation results show that by modularizing the image's high and low bit planes, the encryption process is simplified and improved. The security and reliability of color image transmission are also improved.

Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method.

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatiotemporal systems including fluid turbulence is supported by nonchaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatiotemporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification, or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent, and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatiotemporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

Image encryption scheme based on discrete cosine Stockwell transform and DNA-level modulus diffusion

A new image encryption scheme is presented by combining six-dimensional non-degenerate discrete hyper-chaotic system, two-dimensional discrete cosine Stockwell transform with DNA-level modulus diffusion. The significant advantages of this scheme are the large key space, strong anti-noise ability and resistance to common attacks. To resist the powerful chosen plaintext attack, the initial conditions of the chaotic systems are generated with the SHA-512 hash function value of the plaintext image and the external key. The transmission burden is reduced by compressing the original image with the two-dimensional discrete cosine Stockwell transform. Then random DNA encoding is performed on the compressed image to obtain the DNA image. To speed up the encryption, the DNA-level modulus diffusion algorithm is designed to scramble and diffuse pixels at the same time. Finally, the final encrypted image is obtained by re-encrypting the diffused DNA image with the bit-level permutation and the improved global dynamic diffusion. The two high-dimensional chaotic systems introduced in the image encryption scheme greatly increases the key space and then the image encryption scheme can resist the brute-force attack. The presented scheme is sensitive to both plaintext images and secret keys. Simulation results show that the proposed image encryption algorithm is feasible, secure and effective.

Secure Communication Scheme for Brain-Computer Interface Systems Based on High-Dimensional Hyperbolic Sine Chaotic System

Brain-Computer Interface (BCI) is a direct communication pathway between the brain and the external environment without using peripheral nerves and muscles. This emerging topic is suffering from serious issues such as malicious tampering and privacy leakage. To address this issue, we propose a novel communication scheme for BCI Systems. In particular, this scheme first utilizes high-dimensional chaotic systems with hyperbolic sine nonlinearity as the random number generator, then decorrelation operation is used to remove the physical characteristics of the output sequences. Finally, each of the sequences is applied in differential chaos shift keying (DCSK). Since each output sequence corresponds to a unique electrode, the communication data of different electrodes will not interfere with each other. Compared with popular multi-user DSCK schemes using Walsh code sequences, this scheme does not require the channel data of all electrodes while decoding. Therefore, this scheme has higher efficiency. Experimental results on communication data indicate that the proposed scheme can provide a high level of security.

Multimedia Image Encryption Analysis Based on High-Dimensional Chaos Algorithm

With the development of network and multimedia technology, multimedia communication has attracted the attention of researchers. Image encryption has become an urgent need for secure multimedia communication. Compared with the traditional encryption system, encryption algorithms based on chaos are easier to implement, which makes them more suitable for large-scale data encryption. The calculation method of image encryption proposed in this paper is a combination of high-dimensional chaotic systems. This algorithm is mainly used for graph mapping and used the Lorenz system to expand and replace them one by one. Studies have shown that this calculation method causes mixed pixel values, good diffusion performance, and strong key performance with strong resistance. The pixel of the encrypted picture is distributed relatively random, and the characteristics of similar loudness are not relevant. It is proved through experiments that the above calculation methods have strong safety performance.

A Bit Shift Image Encryption Algorithm Based on Double Chaotic Systems.

A chaotic system refers to a deterministic system with seemingly random irregular motion, and its behavior is uncertain, unrepeatable, and unpredictable. In recent years, researchers have proposed various image encryption schemes based on a single low-dimensional or high-dimensional chaotic system, but many algorithms have problems such as low security. Therefore, designing a good chaotic system and encryption scheme is very important for encryption algorithms. This paper constructs a new double chaotic system based on tent mapping and logistic mapping. In order to verify the practicability and feasibility of the new chaotic system, a displacement image encryption algorithm based on the new chaotic system was subsequently proposed. This paper proposes a displacement image encryption algorithm based on the new chaotic system. The algorithm uses an improved new nonlinear feedback function to generate two random sequences, one of which is used to generate the index sequence, the other is used to generate the encryption matrix, and the index sequence is used to control the generation of the encryption matrix required for encryption. Then, the encryption matrix and the scrambling matrix are XORed to obtain the first encryption image. Finally, a bit-shift encryption method is adopted to prevent the harm caused by key leakage and to improve the security of the algorithm. Numerical experiments show that the key space of the algorithm is not only large, but also the key sensitivity is relatively high, and it has good resistance to various attacks. The analysis shows that this algorithm has certain competitive advantages compared with other encryption algorithms.

A novel image encryption algorithm based on least squares generative adversarial network random number generator

In cryptosystems, the generation of random keys is crucial. The random number generator is required to have a sufficiently fast generation speed to ensure the size of the keyspace. At the same time, the randomness of the key is an important indicator to ensure the security of the encryption system. The chaotic random number generator has been widely used in cryptosystems due to the uncertainty, non-repeatability, and unpredictability of chaotic systems. However, chaotic systems, especially high-dimensional chaotic systems, have slow calculation speed and long iteration time. This caused a conflict between the number of random keys and the speed of generation. In this paper, we introduce the Least Squares Generative Adversarial Networks(LSGAN)into random number generation. Using LSGAN’s powerful learning ability, a novel learning random number generator is constructed. Six chaotic systems with different structures and different dimensions are used as training sets to realize the rapid and efficient generation of random numbers. Experimental results prove that the encryption key generated by this scheme can pass all randomness tests of the National Institute of Standards and Technology (NIST). Hence, our result shows that LSGAN has the potential to improve the quality of the random number generators. Finally, the results are successfully applied to the image encryption scheme based on selective scrambling and overlay diffusion, and good results are achieved.

A novel color image encryption method based on an evolved dynamic parameter-control chaotic system

The majority of existing image encryption algorithms use single chaotic systems, such as Logistic map or Lorentz system, to be the pseudo-random sequence generator. Actually, neither low nor high dimensional chaotic system can get rid of pseudo-randomness deterioration of chaotic sequence calculated by limited precision computer. Additionally, most of them use only one chaotic pseudo-random sequence throughout the encryption process. These are the more obvious deficiencies. In the current paper, a novel compound chaotic system is applied to color images domain to solve the mentioned problems. Dynamic parameter-control chaotic system can enhance the sequence’s randomness after digitalizing. Corresponding the different sequences generated by the novel chaotic system to each color channel of image is a helpful method to reduce the image’s statistical characteristics in the scrambling process. Finally, the effectiveness and security of the proposed encryption has been illustrated by the experimental, and at the meantime, excellent performance is also demonstrated.

Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database

We study the applicability of tools developed by the computer vision community for features learning and semantic image inpainting to perform data reconstruction of fluid turbulence configurations. The aim is twofold. First, we explore on a quantitative basis, the capability of Convolutional Neural Networks embedded in a Deep Generative Adversarial Model (Deep-GAN) to generate missing data in turbulence, a paradigmatic high dimensional chaotic system. In particular, we investigate their use in reconstructing two-dimensional damaged snapshots extracted from a large database of numerical configurations of 3d turbulence in the presence of rotation, a case with multi-scale random features where both large-scale organised structures and small-scale highly intermittent and non-Gaussian fluctuations are present. Second, following a reverse engineering approach, we aim to rank the input flow properties (features) in terms of their qualitative and quantitative importance to obtain a better set of reconstructed fields. We present two approaches both based on Context Encoders. The first one infers the missing data via a minimization of the L2 pixel-wise reconstruction loss, plus a small adversarial penalisation. The second searches for the closest encoding of the corrupted flow configuration from a previously trained generator. Finally, we present a comparison with a different data assimilation tool, based on Nudging, an equation-informed unbiased protocol, well known in the numerical weather prediction community. The TURB-Rot database, http://smart-turb.roma2.infn.it, of roughly 300K 2d turbulent images is released and details on how to download it are given.

A coupling method of double memristors and analysis of extreme transient behavior

By coupling a variable of the memristor in one memristive chaotic circuit with another memristor, an approach to construct a high-dimensional memristive chaotic system is proposed and a five-dimensional hidden chaotic system with double memristors is obtained. The Lyapunov fractal dimension and dynamics map of the system demonstrate that the system has extreme transient behaviors, such as chaos to period, chaos to quasi-period, quasi-period to another quasi-period, quasi-period to chaos, and chaos to another kind of chaos. The whole transition process experiences as many as 2, 3, 4, or even 5 different status. Besides, an interesting symmetric transient phenomenon is discovered and extreme multistability is also observed when changing the initial conditions. The extreme transient behavior is validated using the spectral entropy algorithm.

Multiple-image encryption based on cascaded gyrator transforms and high-dimensional chaotic system

In order to increase encryption capacity and improve transmission efficiency, this paper investigates an encryption scheme for multiple-image based on cascaded gyrator transforms and high-dimensional chaos. Firstly, each plainimage is converted into a real matrix through modulation, quaternion gyrator transform and sharing. The real matrices are connected in turn by plural coding and gyrator transforms. During the process of cascade, phase truncation and preservation operations are performed. The phase of the preceding transformation is modulated by that of the latter complex matrix to enhance security. Followed by splicing and scrambling, a real-valued ciphertext can be obtained. At the receiver end, all the plaintext images can be restituted by the reverse operation of the encryption process using the authorized keys. The adoption of high-dimensioanl chaos and Kronecker product further guarantees the security of the cryptosystem. The numerical results performed on 300 color images have demonstrated that the average PSNR of correctly decrypted images is up to 303.9939 dB, the NMSE is as low as 2.8024 × 10−28 and SSIM is equal to 1.0000. The analysis of sensitivity, statistical characteristic and robustness proves the feasibility and relibility of the proposed encryption scheme.

An Efficient Spectral Method-based Algorithm for Solving a ‎High-dimensional Chaotic Lorenz System

In this paper, we implement the multidomain spectral relaxation method to numerically study high dimensional chaos by considering the nine-dimensional Lorenz system. Chaotic systems are characterized by rapidly changing solutions, as well as sensitivity to small changes in initial data. Most of the existing numerical methods converge slowly for this kind of problems and this results in inaccurate approximations. Spectral methods are known for their high accuracy. However, they become less accurate for problems characterised by chaotic solutions, even with an increase in the number of grid points. As a result, in this work, we adopt the multidomain approach which assumes that the main interval can be decomposed into a finite number of subdomains and the solution obtained in each of the subdomains. This approach remarkably improves the results as well as the efficiency of the method.

A Fast Encryption Scheme Suitable for Video Surveillance Applications Using SHA-256 Hash Function and 1D Sine–Sine Chaotic Map

This paper proposes an image encryption technique which is fast and secure. The encryption scheme is designed for secure transmission of video surveillance data (keyframes) over insecure network. The image encryption technique employs 1D Sine–Sine system with better chaotic properties than its seed map and faster than higher-dimensional chaotic systems. Further, design of encryption scheme is based on two permutation rounds, which employs pixel swapping operation and diffusion operation which is simple and provides required security against plaintext, differential and various other attacks. Three separate chaotic sequences are generated using 1D Sine–Sine system which enhances the key space of the encryption scheme. Secret keys are updated dynamically with SHA-256 hash value obtained from plain image. Hash values of plain image are efficiently used without loss of any hash value information. This makes the encryption scheme plaintext sensitive and secure against plaintext attacks. Performance and security aspects of encryption scheme is analyzed both quantitatively using predefined security metrics and qualitatively by scrutinizing the internal working of encryption scheme. Computational complexity of encrypting a plain image of size [Formula: see text] is [Formula: see text] and is suitable for encrypting keyframes of video for secure surveillance applications.

Designing spontaneous behavioral switching via chaotic itinerancy.

Chaotic itinerancy is a frequently observed phenomenon in high-dimensional nonlinear dynamical systems and is characterized by itinerant transitions among multiple quasi-attractors. Several studies have pointed out that high-dimensional activity in animal brains can be observed to exhibit chaotic itinerancy, which is considered to play a critical role in the spontaneous behavior generation of animals. Thus, how to design desired chaotic itinerancy is a topic of great interest, particularly for neurorobotics researchers who wish to understand and implement autonomous behavioral controls. However, it is generally difficult to gain control over high-dimensional nonlinear dynamical systems. In this study, we propose a method for implementing chaotic itinerancy reproducibly in a high-dimensional chaotic neural network. We demonstrate that our method enables us to easily design both the trajectories of quasi-attractors and the transition rules among them simply by adjusting the limited number of system parameters and by using the intrinsic high-dimensional chaos.

Systematic identification of causal relations in high-dimensional chaotic systems: application to stratosphere-troposphere coupling

Obtaining reliable causal inference is crucial for understanding the climate system. Convergent Cross Mapping (CCM), a recently developed method to infer causal relationships from time series has been shown to be superior to previous methods which are based on linearity assumptions. However, CCM has so far been only tested on low-dimensional or bivariate models, while real-world systems, like the climate system, are high-dimensional and have many more interacting variables. Here, we demonstrate that standard CCM cannot reliably infer causal relations in high-dimensional chaotic systems. However, by using a hierarchy of conceptual models and observational data we show that time-lagged CCM reliably identifies causal relationships in contrast to standard CCM and Pearson correlation. Furthermore, we systematically demonstrate that time-lagged CCM is able to identify long-distance causal interactions. Moreover, we apply time-lagged CCM to detect causal relations in stratosphere-troposphere coupling, and demonstrate the downward causal chain induced by polar vortex activity.

A novel method of constructing high-dimensional digital chaotic systems on finite-state automata**Project supported by the National Key R&D Program of China (Grant No. 2017YFB0802000) and the Cryptography Theoretical Research of National Cryptography Development Fund, China (Grant No. MMJJ20170109).

When chaotic systems are implemented on finite precision machines, it will lead to the problem of dynamical degradation. Aiming at this problem, most previous related works have been proposed to improve the dynamical degradation of low-dimensional chaotic maps. This paper presents a novel method to construct high-dimensional digital chaotic systems in the domain of finite computing precision. The model is proposed by coupling a high-dimensional digital system with a continuous chaotic system. A rigorous proof is given that the controlled digital system is chaotic in the sense of Devaney’s definition of chaos. Numerical experimental results for different high-dimensional digital systems indicate that the proposed method can overcome the degradation problem and construct high-dimensional digital chaos with complicated dynamical properties. Based on the construction method, a kind of pseudorandom number generator (PRNG) is also proposed as an application.

Hyperchaotic image encryption algorithm based on bit-level permutation and DNA encoding

Many chaos-based image encryption algorithms using low-dimensional chaotic mapping and permutation diffusion structures have been proposed recently. However, low-dimensional chaotic maps are less secure than high-dimensional chaotic systems. Furthermore, the permutation process is independent of the plaintext and diffusion process. As a result, they are not very resistant to chosen plaintext attacks and chosen ciphertext attacks. In this paper, we propose a hyper-chaos-based image encryption algorithm that uses a 6-dimensional hyperchaotic system; the key stream generated by hyperchaotic system is related to the plaintext image. Then, bit-level permutation is employed to strengthen the security of the cryptosystem. Finally, DNA coding and operations are employed to change pixels. Theoretical analysis and numerical simulations demonstrate that the proposed algorithm is safe and reliable for image encryption.

Particle filtering for chaotic dynamical systems using future right-singular vectors

In this paper, we combine tools from the study of chaotic dynamical systems with nonlinear non-Gaussian data assimilation algorithms to produce novel particle filtering algorithms, where the use of dynamical information improves both the accuracy of estimation of the unknown state as well as improves the diversity of the particles that collectively represent the posterior distribution. Unlike past efforts of assimilation in the unstable subspace, we focus not on the deterministic signal and low observation noise case, but instead on the case of moderate signal and observation noise. Filtering algorithms using finite-time Lyapunov vectors, left-singular vectors, and a novel concept of future right-singular vectors, to project observations onto reduced subspaces are developed and tested against two regimes of the single scale Lorenz 1996 model—a weakly chaotic, non-Gaussian regime and a strongly chaotic, near-Gaussian regime. As modeling in the geosciences continue to improve resolution and fidelity of finer physical processes, the models are requiring data assimilation techniques that can handle the fully nonlinear, non-Gaussian case and represent multimodal distributions in high-dimensional spaces. This paper contributes to pushing the boundary of nonlinear data assimilation by aiming to improve particle filtering algorithms for high-dimensional chaotic systems.