High-dimensional Chaos Research Articles

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.

We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.

Some Progresses of Chaotic Cipher and Its Applications in Multimedia Secure Communications

This paper reviews the past situation and existing problems in chaotic cipher theories and applications, and reports the recent progress of some issues on theoretical design and hardware implementation of ciphers based on high-dimensional chaos system, including basic theory, design methods, typical applications, and ideas coping with the problems. In the aspect of design of chaotic cipher and its security evaluation, the following progress is summarized: counteracting dynamics degradation of digital chaotic systems with anti-control methods; designing no-degeneration chaotic systems in digital domain; proposing a multi-round chaotic stream cipher with the high-dimensional digital chaotic systems and feedback mechanism of self-loop; evaluating security level of the proposed chaotic ciphers with various methods. In the aspect of application and hardware implementation of multimedia secure communication, the following developments are reported: optimizing a cross platform conducting real-time remote chaotic secure communication, targeting different application businesses of hand-held devices, such as smartphone, computer, ARM, FPGA; and establishing a demonstration platform on chaotic secure communications to verify the effectiveness.

Transition to turbulence from plane Couette flow

Modelling fluid turbulence is perhaps one of the hardest problems in Applied Mathematics. In a recent paper, the author argued that the classical Navier–Stokes equation is not sufficient to describe the transition to turbulence, but that a Reiner–Rivlin type equation is needed instead. This is explored here for the simplest of all viscous fluid flows, the Couette flow, which is a simple shear between two moving plates. It is found that at high wavenumbers, the transition to unstable flow at the critical Reynolds number is characterized by a large number of eigenvalues of the Orr–Sommerfeld equation moving into the unstable zone essentially simultaneously. This would generate high-dimensional chaos almost immediately, and is a suggested mechanism for the transition to turbulence. Stability zones are illustrated for the flow, and a simple asymptotic solution confirms some of the features of these numerical results. doi:10.1017/S1446181115000176

Optical Heterodyne Generation of High-Dimensional and Broadband White Chaos

Physical chaos is a fascinating prospect for high-speed data security by serving as a masking carrier or a key source, but suffers from a colored spectrum that divulges system's intrinsic oscillations and degrades randomness. Here, we demonstrate that physical chaos with a white spectrum can be achieved by the optical heterodyning of two delayed-feedback lasers. A white chaotic spectrum with 1-dB fluctuation in a band of 11 GHz is experimentally obtained. The white chaos also has a perfect delta-like autocorrelation function and a high dimensionality of greater than 10, which makes chaos reconstruction extremely difficult and thus improves security.

TRANSITION TO TURBULENCE FROM PLANE COUETTE FLOW

Modelling fluid turbulence is perhaps one of the hardest problems in Applied Mathematics. In a recent paper, the author argued that the classical Navier–Stokes equation is not sufficient to describe the transition to turbulence, but that a Reiner–Rivlin type equation is needed instead. This is explored here for the simplest of all viscous fluid flows, the Couette flow, which is a simple shear between two moving plates. It is found that at high wavenumbers, the transition to unstable flow at the critical Reynolds number is characterized by a large number of eigenvalues of the Orr–Sommerfeld equation moving into the unstable zone essentially simultaneously. This would generate high-dimensional chaos almost immediately, and is a suggested mechanism for the transition to turbulence. Stability zones are illustrated for the flow, and a simple asymptotic solution confirms some of the features of these numerical results.

Transition from low to high dimensional chaos in a group of pulsations recorded in a broad radiowave interval

We examined the dynamic characteristics of the time series regarding a group of pulsations in broadband spectrum at metric waveband solar radio emission. The data were recorded with the radio polarimeter of the INAF-Trieste Astronomical Observatory at July 17, 2002. The aim is to determine if the underlying process of these pulsations can be describe as a periodic, deterministic chaos or stochastic. The pulsations under inquiry in present paper are rather rare, as we found only one example of similar ones reported in the literature. Unlike most of the previously works where the analyses was done to a broadband pulsating events at one single frequency, we examine the pulsation event as it evolves both in time and in frequency. We found that the dynamics underlying the generation of pulsations can be characterized by a deterministic chaotic process which increases the dimension of chaos with frequency showing a transition from low-dimensional to high-dimensional deterministic chaotic system.

Nonlinear Forecasting of the Generalized Kuramoto–Sivashinsky Equation

The emergence of pattern formation and chaotic dynamics is studied in the one-dimensional (1D) generalized Kuramoto–Sivashinsky (gKS) equation by means of a time-series analysis, in particular, a nonlinear forecasting method which is based on concepts from chaos theory and appropriate statistical methods. We analyze two types of temporal signals, a local one and a global one, finding in both cases that the dynamical state of the gKS solution undergoes a transition from high-dimensional chaos to periodic pulsed oscillations through low-dimensional deterministic chaos while increasing the control parameter of the system. Our results demonstrate that the proposed nonlinear forecasting methodology allows to elucidate the dynamics of the system in terms of its predictability properties.

An image encryption algorithm based on the perturbed high-dimensional chaotic map

Due to the secure problems of low-dimensional chaos in image encryption and cycle degradation caused by computers’ finite precision, a perturbed high-dimensional chaos is designed for image encryption according to Devaney and topological conjugate definition. Perturbation is added to the new high-dimensional chaos to enlarge cycle. The characteristics of the high-dimensional chaotic map are proved by Devaney definition, and the security features are analyzed. Owe to the periodic defects of Cat map, a new separated Cat map is designed for the confusion process. The “confusion–diffusion” structure is used in the image encryption. The security test results demonstrate the proposed methods have high security, and the speed of our algorithm is faster than classical solutions.

A new image encryption algorithm based on the high-dimensional chaotic map

Because precision degradation of low-dimensional chaos has seriously affected the security of encryption, a high-dimensional chaos is designed for image encryption according to Devaney and topological conjugate definition. The characteristics of the high-dimensional chaotic map are proved by Devaney definition and the security features are analysed. Owing to the periodic defects of Cat map, a new separated Cat map is designed for the confusion process. The bilateral-diffusion scheme is used in the novel image encryption. The experiment analysis shows that the algorithm passes the random and pseudo-random sequence of test standard SP800-22 released by the National Institute of Standards and Technology Commission and the DIEHARD pseudo-random sequence test. And the experiment results demonstrate that the encryption scheme has a high security.

Dynamic Nature of Explanatory Variables in Water Demand Forecasting

Researchers have used a wide range of explanatory variables in their water demand forecasting models. In reality, nature of the data used in these modelling techniques is not similar to each other. In fact, not enough attention has been paid to periodic or chaotic nature of the time series deployed in water demand forecasting techniques. The purpose of this study was to quantify chaos in weather information variables used in demand forecasting models. Using correlation dimension method to check the evidence of deterministic chaos, this research proved these explanatory variables can exhibit high dimensional chaos in stochastic systems.

Electrochemical Dissolution in the Presence of Time Delayed Feedback

It is well known that time-delayed systems can exhibit rich dynamical behavior ranging from simple oscillations to high-dimensional chaos.[1, 2] Such systems are studied extensively in optoelectronic[3] and biological[4] systems. In electrochemical systems, time-delayed feedback can be used for the control of dynamics, e.g., for the stabilization of unstable steady states and periodic orbits[5]. In this study, the dynamical features of a prototype, negative differential resistance electrochemical model [6] is studied in the presence of various forms of delayed feedback. Specifically, the double layer potential drop is fed back with delay that is either discrete or in a distributed form using a delay kernel. In the latter case, the system was analyzed in a framework of an extended system of ordinary differential equations using an auxiliary variable. The behavior of the system at different parameter values is explored with a virtual space-time representation [7] technique. The features of various virtual spatial states in the form of rotating waves, cluster formation, and turbulence are discussed. Finally, comments are made on possibility of formation of virtual chimera states[8] in the electrochemical system with external delayed feedback.

High-dimensional chaos from self-sustained collisions of solitons

We experimentally demonstrate chaos generation based on collisions of electrical solitons on a nonlinear transmission line. The nonlinear line creates solitons, and an amplifier connected to it provides gain to these solitons for their self-excitation and self-sustenance. Critically, the amplifier also provides a mechanism to enable and intensify collisions among solitons. These collisional interactions are of intrinsically nonlinear nature, modulating the phase and amplitude of solitons, thus causing chaos. This chaos generated by the exploitation of the nonlinear wave phenomena is inherently high-dimensional, which we also demonstrate.

Bifurcation analysis of a two-DoF mechanical system subject to digital position control. Part II. Effects of asymmetry and transition to chaos

Considering a two DoF system subject to digital position control, of interest for robotic application, we analyze the dynamics of the system at the intersection of two loci of Neimark–Sacker bifurcations, where a double Neimark–Sacker bifurcation is taking place. In the system, the saturation of the control force is the only nonlinear term considered, other than this, the system is piecewise linear. Starting from the analytical investigation already performed in Part I (Habib et al. in Nonlin. Dyn., under review, 2013), in this paper the effects of an asymmetry of the saturation of the control force are investigated, both analytically and numerically. The results show the increasing complexity of the dynamics for a more and more asymmetric system. First, the asymmetry is making the bifurcation transit from supercritical to subcritical, then it generates a stable torus that breaks down into a strange attractor, associated with a chaotic motion. In the last part of the paper, the torus breakdown and the onset of chaos are investigated, furthermore the evolution of complex dynamics through regions of phase locking and higher-dimensional chaos is outlined.

Cryptanalysis of a parallel sub-image encryption method with high-dimensional chaos

Recently, a parallel sub-image encryption method with high-dimensional chaos has been proposed. But there is a fatal flaw in the cryptosystem that the generated keystream remains unchanged when encrypting every image. Based on this point, we could recover the plaintext by applying chosen plaintext attack. Therefore the proposed cryptosystem is not supposed to be used in image transmission system. Experimental results show the feasibility of our attack.

Multiscale analysis of economic time series by scale-dependent Lyapunov exponent

Economic time series usually exhibit complex behavior such as nonlinearity, fractal long-memory, and non-stationarity. Recently, considerable efforts have been made to detect chaos and fractal long-memory in finance. While evidence supporting fractal scaling in finance has been accumulating, it is now generally thought that financial time series may not be modeled by chaos or noisy chaos, since the estimated Lyapunov exponent (LE) is negative. A negative LE amounts to a negative Kolmogorov entropy, and thus implies simple regular dynamics of the economy. This is at odds with the general observation that the economy is highly complicated due to nonlinear and stochastic interactions among component systems and hierarchical regulations in the world economy. To resolve this dilemma, and to provide an effective means of characterizing fractal long-memory properties in non-stationary economic time series, we employ a multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize economic time series. SDLE cannot only unambiguously distinguish low-dimensional chaos from noise, but also detect high-dimensional and intermittent chaos, as well as effectively deal with non-stationarity. With SDLE, we are able to show that the reported negative LE may correspond to large-scale convergence, but not imply the absence of small-scale divergence or noisy chaos in the world economy. Using US foreign exchange rate data as examples, we further show how SDLE can readily characterize fractal, persistent or anti-persistent long-range correlations in economic time series.

Short-term prediction of dynamical behavior of flame front instability induced by radiative heat loss

We apply nonlinear forecasting to the time series of the flame front instability induced by radiative heat loss to test for the short-term predictability and long-term unpredictability characteristic of deterministic chaos in flame front instability. Our results indicate that the flame front instability represents high-dimensional chaos generated via the period-doubling cascade process reported in our previous study [H. Gotoda, K. Michigami, K. Ikeda, and T. Miyano, Combust Theory Modell. 14, 479 (2010)], while its short-term behavior is predictable using a local nonlinear predictor based on the Sugihara-May method [H. Gotoda, H. Nikimoto, T. Miyano, and S. Tachibana, Chaos 20, 013124 (2011); G. Sugihara and R. M. May, Nature 344, 734 (1990)] as well as a generalized radial basis function network as a global nonlinear predictor. The feasibility of a new approach based on short-term prediction is also discussed in this work from the practical viewpoint of combustion systems.

Chaotic ultra-wideband radio generator based on an optoelectronic oscillator with a built-in microwave photonic filter

We induce a microwave photonic bandpass filter into an optoelectronic oscillator to generate a chaotic ultra-wideband signal in both the optical and electrical domain. The theoretical analysis and numerical simulation indicate that this system is capable of generating band-limited high-dimensional chaos. Experimental results coincide well with the theoretical prediction and show that the power spectrum of the generated chaotic signal basically meets the Federal Communications Commission indoor mask. The generated chaotic carrier is further intensity modulated by a 10 MHz square wave, and the waveform of the output ultra-wideband signal is measured for demonstrating the chaotic on-off keying modulation.

Theory of fast nondeterministic physical random-bit generation with chaotic lasers

We theoretically show that completely stochastic fast physical random bit generation at a rate of more than one gigabit per second can be realized by using lasers with optical delayed feedback which creates high-dimensional chaos of laser light outputs. The theory is based on the mixing property of chaos, which transduces microscopic quantum noise of spontaneous emission in lasers into random transitions between discrete macroscopic states.

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Chaos Based Mixed Keystream Generation for Voice Data Encryption

In this paper, a high dimensional chaotic systems based mixed keystream generator is proposed to secure the voice data. As the voice-based communication becomes extensively vital in the application areas of military, voice over IP, voice-conferencing, phone banking, news telecasting etc. It greatly demands to preserve sensitive voice signals from the unauthorized listening and illegal usage over shared/open networks. To address the need, the designed keystream generator employed to work as a symmetric encryption technique to protect voice bitstreams over insecure transmission channel. The generator utilizes the features of high dimensional chaos like Lorenz and Chen systems to generate highly unpredictable and random-like sequences. The encryption keystream is dynamically extracted from the pre-treated chaotic mixed sequences, which are then applied to mask the voice bitstream for integrity protection of voice data. The experimental analyses like auto-correlation, signal distribution, parameterresidual deviation, key space and key-sensitivity demonstrate the effectiveness of the proposed technique for secure voice communication.