High-dimensional Chaos Research Articles

Templex-based dynamical units for a taxonomy of chaos.

Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented BraMAH cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a "minimal" form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units: the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated.

Measuring the Unpredictability of Disruptive Change: The Comparison of the Inkjet Printer and Digital Photography

Disruptive innovation has become a core topic for technology management and indeed a central aspect of the modern industrial revolution. Although considerable attention has been devoted to its emergent properties and selection outcomes, less focus has been given to understand how disruption emerges through evolving and complex interactions that occur during that process. To address this shortfall, this article draws on chaos theory to examine the nature of disruptive innovation as a nonlinear dynamic system. Disruptive innovation can be considered nonlinear because it is time-dependent and deviation-absorbing, and it revolves around identifiable types of attractors appearing in a deterministic context or trajectory. To illustrate and advance these ideas, we study inkjet printer and digital photography technologies using patent application data collected from the United States Patent and Trademark Office. We employ a nonlinear analysis of the correlation dimension to measure the unpredictability of disruptive change and found that both chosen technologies evolved through a nonlinear dynamic process. Furthermore, in light of chaos theory and by way of comparison, we indicated two types of disruption: high-dimensional and low-dimensional chaos. Implications for innovation theory and business practice are explored herein.

A novel 5D memristor conservative chaotic system with multiple forms of hidden flows

Memristor is one of the basic circuit elements commonly used in circuit model analysis. More complex dynamic characteristics can be observed by coupling memristor into nonlinear circuit. However, there is relatively little attention paid to high-dimensional conservative chaos based on memristors up to now. In this paper, a five-dimensional memristor conservative chaotic system is built after the introduction of the memristor into conservative chaotic system. There is no equilibrium point in this system and the phase trajectory produced by it has hidden properties. Its conservatism is analyzed by bifurcation diagram, Lyapunov exponent spectrum and divergence. The phase trajectory will change with the change of parameters, which Poincaré mapping also verified these dynamic behaviors. In addition, hidden extreme multistability and initial value offset boosting behavior are also found in this system. It is to be noted that this behavior is less in memristor conservative chaotic system without equilibrium points. At the same time, a new transient transition behavior is observed. By introducing spectral entropy algorithm, the complexity of sequences is analyzed and compared with the existing literature. The results show that the system has higher complexity. Finally, the systematic analogous circuit is designed and built whose results are consistent with the MATLAB numerical simulation results, which has laid a solid foundation for the practical application of the system in engineering.

Using covariant Lyapunov vectors to quantify high-dimensional chaos with a conservation law.

We explore the high-dimensional chaos of a one-dimensional lattice of diffusively coupled tent maps using the covariant Lyapunov vectors (CLVs). We investigate the connection between the dynamics of the maps in the physical space and the dynamics of the covariant Lyapunov vectors and covariant Lyapunov exponents that describe the direction and growth (or decay) of small perturbations in the tangent space. We explore the tangent space splitting into physical and transient modes and find that the splitting persists for all of the conditions we explore. In general, the leading CLVs are highly localized in space and the CLVs become less localized with increasing Lyapunov index. We consider the dynamics with a conservation law whose strength is controlled by a parameter that can be continuously varied. Our results indicate that a conservation law delocalizes the spatial variation of the CLVs. We find that when a conservation law is present, the leading CLVs are entangled with fewer of their neighboring CLVs than in the absence of a conservation law.

Synchronization of Boolean networks with chaos‐driving and its application in image cryptosystem

AbstractThis paper proposes a Boolean network model with high dimensional chaos driving and investigates the synchronization of the chaos‐driven Boolean network with a semi‐tensor product. In order to protect the privacy and ensure the security of image transmission, the synchronization results are utilized in the image cryptosystem to achieve compression and encryption. First, the driving chaos system is coupled with multiple local systems and synchronized with the transmitted encrypted signals. Second, the Boolean network is driven and synchronized with derived chaos signals. Finally, images are encrypted and compressed with chaos‐driven Boolean network signals in the transmitter, and then decrypted and recovered with synchronized chaos and Boolean network signals in the recipient. Because of the complexities of high dimensional chaos and Boolean network, the proposed cryptosystem has good security in the secure communication and image process.

Retracted: Multimedia Image Encryption Analysis Based on High-Dimensional Chaos Algorithm

Retracted: Multimedia Image Encryption Analysis Based on High-Dimensional Chaos Algorithm

Maximal Memory Capacity Near the Edge of Chaos in Balanced Cortical E-I Networks.

We examine the efficiency of information processing in a balanced excitatory and inhibitory (E-I) network during the developmental critical period, when network plasticity is heightened. A multimodule network composed of E-I neurons was defined, and its dynamics were examined by regulating the balance between their activities. When adjusting E-I activity, both transitive chaotic synchronization with a high Lyapunov dimension and conventional chaos with a low Lyapunov dimension were found. In between, the edge of high-dimensional chaos was observed. To quantify the efficiency of information processing, we applied a short-term memory task in reservoir computing to the dynamics of our network. We found that memory capacity was maximized when optimal E-I balance was realized, underscoring both its vital role and vulnerability during critical periods of brain development.

Global sensitivity analysis of a coupled multiphysics model to predict surface evolution in fusion plasma–surface interactions

We construct a global sensitivity analysis framework for a coupled multiphysics model used to predict the changes in material properties and surface morphology of helium plasma-facing components in future fusion reactors. The model combines the particle dynamics simulator F-TRIDYN, that predicts the helium implantation profile, with the cluster dynamics simulator Xolotl, that predicts the growth and evolution of subsurface helium gas bubbles. In order to keep the sensitivity analysis tractable, we first construct a sparse, high-dimensional polynomial chaos expansion surrogate model for each output quantity of interest, which allows the efficient extraction of sensitivity information. The sensitivity analysis is performed for two problem settings: one for ITER-like conditions, and one that resembles conditions inside the PISCES-A linear plasma device. We present a systematic comparison of important parameters, for both F-TRIDYN and Xolotl in isolation as well as for the coupled model, and discuss the physical interpretation of these results.

Lyapunov exponents for temporal networks.

By interpreting a temporal network as a trajectory of a latent graph dynamical system, we introduce the concept of dynamical instability of a temporal network and construct a measure to estimate the network maximum Lyapunov exponent (nMLE) of a temporal network trajectory. Extending conventional algorithmic methods from nonlinear time-series analysis to networks, we show how to quantify sensitive dependence on initial conditions and estimate the nMLE directly from a single network trajectory. We validate our method for a range of synthetic generative network models displaying low- and high-dimensional chaos and finally discuss potential applications.

Complex spatiotemporal oscillations emerge from transverse instabilities in large-scale brain networks.

Spatiotemporal oscillations underlie all cognitive brain functions. Large-scale brain models, constrained by neuroimaging data, aim to trace the principles underlying such macroscopic neural activity from the intricate and multi-scale structure of the brain. Despite substantial progress in the field, many aspects about the mechanisms behind the onset of spatiotemporal neural dynamics are still unknown. In this work we establish a simple framework for the emergence of complex brain dynamics, including high-dimensional chaos and travelling waves. The model consists of a complex network of 90 brain regions, whose structural connectivity is obtained from tractography data. The activity of each brain area is governed by a Jansen neural mass model and we normalize the total input received by each node so it amounts the same across all brain areas. This assumption allows for the existence of an homogeneous invariant manifold, i.e., a set of different stationary and oscillatory states in which all nodes behave identically. Stability analysis of these homogeneous solutions unveils a transverse instability of the synchronized state, which gives rise to different types of spatiotemporal dynamics, such as chaotic alpha activity. Additionally, we illustrate the ubiquity of this route towards complex spatiotemporal activity in a network of next generation neural mass models. Altogehter, our results unveil the bifurcation landscape that underlies the emergence of function from structure in the brain.

A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks.

In this paper, a surjective mapping that satisfies the Li-Yorke chaos in the unit area is constructed and a perturbation algorithm (disturbing its parameters and inputs through another high-dimensional chaos) is proposed to enhance the randomness of the constructed chaotic system and expand its key space. An algorithm for the composition of two systems (combining sequence based on quantum random walks with chaotic system's outputs) is designed to improve the distribution of the system outputs and a compound chaotic system is ultimately obtained. The new compound chaotic system is evaluated using some test methods such as time series complexity, autocorrelation and distribution of output frequency. The test results showed that the new system has complex dynamic behavior such as high randomicity, unpredictability and uniform output distribution. Then, a new scheme for generating pseudorandom numbers is presented utilizing the composite chaotic system. The proposed pseudorandom number generator (PRNG) is evaluated using a series test suites such as NIST sp 800-22 soft and other tools or methods. The results of tests are promising, as the proposed PRNG passed all these tests. Thus, the proposed PRNG can be used in the information security field.

Wideband and high-dimensional chaos generation using optically pumped spin-VCSELs.

We propose and numerically demonstrate wideband and high-dimensional chaos signal generation based on optically pumped spin-polarized vertical-cavity surface-emitting lasers (spin-VCSELs). Here, we focus on the chaotic characteristics of spin-VCSELs under two scenarios: one is a spin-VCSEL with optical feedback and the other is optical heterodyning the outputs of two free-running spin-VCSELs. Specifically, we systematically investigate the influence of some key parameters on the chaotic properties, i.e., bandwidth, spectral flatness (SF), time delay signature (TDS), correlation dimension (CD), and permutation entropy (PE), and reveal the route to enhance these properties simultaneously. Our simulation results demonstrate for the first time that spin-VCSELs with simple auxiliary configurations allow for chaos generation with desired properties, including effective bandwidth up to 30 GHz and above, no TDS of greater than 0.2, the flatness of 0.75 and above, and the high complexity/dimensionality over a wide range of parameters under both schemes. Therefore, our study may pave the way for potential applications requiring wideband and high-dimensional chaos.

Crises and chaotic scattering in hydrodynamic pilot-wave experiments.

Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.

Medical image encryption algorithm based on hyper‐chaotic system and DNA coding

With the international development of the medical service informatisation, medical information sharing has become the key standard to measure the degree of medical informatisation. In this process, it is important to ensure the security of patients' medical information such as patients' records, examination reports, images and so on. In this article, an image encryption scheme based on Secure Hash Algorithm 3 (SHA-3), DNA coding and high dimensional chaos system is proposed to promote the security level of medical images in information sharing processes such as over the network. First, SHA-3 algorithm is used to calculate the hash value of the input image, and the result is taken as the initial value of the hyper-chaotic system. Second, the intensity value of the input image is converted into a serial binary digital stream. Third, the pseudo-random sequence generated by a 4-dimensional hyper-chaotic system is used to perturb the bit stream globally so as to achieve the purpose of hiding the effective information of the input image. During the operation of hyper-chaotic sequence and DNA sequence, algebraic and complementary operations are performed on DNA encoding values to enhance encryption performance. Finally, simulations have been applied on an open-source medical image database, and the results demonstrate obvious encryption effectiveness and high security level of the proposed encryption algorithm in the robustness of noise and clipping attacks.

Image Recognition and Encryption Algorithm Based on Artificial Neural Network and Multidimensional Chaotic Sequence.

With the continuous development of Internet technology and technological innovation, image recognition technologies such as face unlocking and face brushing payment have gradually entered daily life. However, it can not be ignored that these technologies not only bring us great convenience but also face great risks. The biological characteristics of a face image are unique, and it will be difficult to modify once it is leaked. If the image information stored in the cloud is leaked because it cannot be properly kept, users have no privacy. The encryption and recognition of face image can effectively solve this problem. Aiming at this, high-dimensional chaos Henon Map and one-dimensional chaos Logistic Map are used to generate a key to complete the encryption of the image in the transformation domain, and the capacity and complexity of the key are further enhanced. Then, combined with BP neural network to achieve face image recognition. Finally, the robustness of the proposed algorithm is verified and analyzed by conventional attacks, geometric attacks, and occlusion attacks.

Chaos along the rc-branch of [formula omitted]-shunted intrinsic Josephson junctions

The appearance of chaos along the “charged” resonance circuit branch of the current-voltage characteristics is examined in a model of intrinsic Josephson junctions shunted by resistive, inductive and capacitive circuit elements. Detailed numerical simulations of the electric charge density, current-voltage characteristics and various chaos indicators, such as Lyapunov exponents and Poincaré sections are performed over a wide range of relevant system parameters. In mapping the parameter space, several distinctly different regions are found: some completely regular, while others are dominated by chaos, where the rc-frequency determines the appearance and properties of those regions. At higher values of the rc-frequency chaos appears at first over relatively narrow regions at the lower-current end of the rc-branch. As the rc-frequency is lowered, the chaotic region at the lowered-current end of the rc-branch may becomes wider, depending sensitively on the combination of system parameters. Examination of the metric entropy and maximal Kaplan–Yorke dimension shows that the dimensions of the chaotic attractors associated with the chaos do not plateau, as in the overdamped case, but surprisingly increase indefinitely with the number of junctions, indicating that the system is capable of truly high-dimensional chaos. The onset of the chaos along the rc-branch is found to occur via a two-frequency quasi-periodic route. Our results provided a guide for specific parameter combinations that could minimize the chaos, thereby making certain applications potentially more viable.

Fast physical random bit generation using a millimeter-wave white noise source.

A broadband millimeter-wave (MMW) white noise signal generated by optical heterodyning of two Fabry-Perot laser diodes (FP-LDs) subject to optical feedback is demonstrated and employed for fast physical random bit generation with a simple least significant bits (LSBs) retaining method. Firstly, under suitable feedback conditions, two external-cavity feedback FP-LDs can be easily driven into chaotic states. In this process, the optical spectra of multi-longitudinal modes are significantly broadened. Then, two spectral broadening multi-longitudinal chaotic signals are mixed and converted into an MMW white noise signal through the heterodyne beating technique combined with a fast photodetector. With such an approach, a high dimensional broadband chaos with perfect characteristics of MMW white noise (3-dB bandwidth beyond 50 GHz without any time-delay signature) is experimentally achieved. Finally, taking the generated MMW white noise as the entropy source, 640 Gb/s physical random bit generation is realized by directly selecting 4-LSBs at 160 GS/s sampling rate after an 8-bit analog-digital-convertor.

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.

Complex network analysis of spatiotemporal dynamics of premixed flame in a Hele-Shaw cell: A transition from chaos to stochastic state.

We study the dynamical state of a noisy nonlinear evolution equation describing flame front dynamics in a Hele-Shaw cell from the viewpoint of complex networks. The high-dimensional chaos of flame front fluctuations at a negative Rayleigh number retains the deterministic nature for sufficiently small additive noise levels. As the strength of the additive noise increases, the flame front fluctuations begin to coexist with stochastic effects, leading to a fully stochastic state. The additive noise significantly promotes the irregular appearance of the merge and divide of small-scale wrinkles of the flame front at a negative Rayleigh number, resulting in the transition of high-dimensional chaos to a fully stochastic state.

Dynamical invariants and inverse period-doubling cascades in multi-delay systems.

We investigate transitions to simple dynamics in first-order nonlinear differential equations with multiple delays. With a proper choice of parameters, a single delay can destabilize a fixed point. In contrast, multiple delays can both destabilize fixed points and promote high-dimensional chaos but also induce stabilization onto simpler dynamics. We show that the dynamics of these systems depend on the precise distribution of the delays. Narrow spacing between individual delays induces chaotic behavior, while a lower density of delays enables stable periodic or fixed point behavior. As the dynamics become simpler, the number of unstable roots of the characteristic equation around the fixed point decreases. In fact, the behavior of these roots exhibits an astonishing parallel with that of the Lyapunov exponents and the Kolmogorov-Sinai entropy for these multi-delay systems. A theoretical analysis shows how these roots move back toward stability as the number of delays increases. Our results are based on numerical determination of the Lyapunov spectrum for these multi-delay systems as well as on permutation entropy computations. Finally, we report how complexity reduction upon adding more delays can occur through an inverse period-doubling sequence.