Low-dimensional Chaotic Systems Research Articles

A fast encryption algorithm of color image based on four-dimensional chaotic system

As the low complexity of low-dimensional chaotic system and the slow speed of image encryption, this paper proposes a fast encryption algorithm of color image based on four-dimensional chaotic system. Firstly, we propose a new method of designing four-dimensional chaotic system based on the classical equations of three-dimensional chaotic system, to increase the complexity and key space of the encryption algorithm. Secondly, according to the nature of color images’ pixels channel, we design a new pseudo-random sequence generator and reuse the random sequence, to improve the speed of image encryption. Finally, the methods of row-major and column-major are used to diffuse the original image and the cat map with parameter is used to scramble the image pixels, respectively, to achieve the effect of encryption. The results of simulation and security analysis show that the proposed encryption algorithm is of good performance on security, robustness and high encryption speed.

On the Verge of a Grand Solar Minimum: A Second Maunder Minimum?

We analyze the yearly mean sunspot-number data covering the period 1700 to 2012. We show that the yearly sunspot number is a low-dimensional deterministic chaotic system. We perform future predictions trying to forecast the solar activity during the next five years (2013 – 2017). We provide evidence that the yearly sunspot-number data can be used for long-term predictions. To test and prove that our model is able to predict the Maunder Minimum period (1645 – 1715), we perform long-term post-facto predictions comparing them with the observed sunspot-number values. We also perform long-term future predictions trying to forecast the solar activity up to 2102. Our predictions indicate that the present Cycle 24 is expected to be a low-peak cycle. We conclude that the level of solar activity is likely to be reduced significantly during the next 90 years, somewhat resembling the Maunder Minimum period.

An Image Encryption Scheme Based on Hyperchaotic Rabinovich and Exponential Chaos Maps

This paper proposes a new four-dimensional hyperchaotic map based on the Rabinovich system to realize chaotic encryption in higher dimension and improve the security. The chaotic sequences generated by Runge-Kutta method are combined with the chaotic sequences generated by an exponential chaos map to generate key sequences. The key sequences are used for image encryption. The security test results indicate that the new hyperchaotic system has high security and complexity. The comparison between the new hyperchaotic system and the several low-dimensional chaotic systems shows that the proposed system performs more efficiently.

Identification and quantification analysis of nonlinear dynamics properties of combustion instability in a diesel engine.

The cycling combustion instabilities in a diesel engine have been analyzed based on chaos theory. The objective was to investigate the dynamical characteristics of combustion in diesel engine. In this study, experiments were performed under the entire operating range of a diesel engine (the engine speed was changed from 600 to 1400 rpm and the engine load rate was from 0% to 100%), and acquired real-time series of in-cylinder combustion pressure using a piezoelectric transducer installed on the cylinder head. Several methods were applied to identify and quantitatively analyze the combustion process complexity in the diesel engine including delay-coordinate embedding, recurrence plot (RP), Recurrence Quantification Analysis, correlation dimension (CD), and the largest Lyapunov exponent (LLE) estimation. The results show that the combustion process exhibits some determinism. If LLE is positive, then the combustion system has a fractal dimension and CD is no more than 1.6 and within the diesel engine operating range. We have concluded that the combustion system of diesel engine is a low-dimensional chaotic system and the maximum values of CD and LLE occur at the lowest engine speed and load. This means that combustion system is more complex and sensitive to initial conditions and that poor combustion quality leads to the decrease of fuel economy and the increase of exhaust emissions.

Studying geometric structures in meso-scale flows

Geometric shapes of coherent structures such as ramp or cliff like signals, step changes and waves, are commonly observed in meteorological temporal series and dominate the turbulent energy and mass exchange between the atmospheric surface layer and the layers above, and also relate with low-dimensional chaotic systems. In this work a simple linear technique to extract geometrical shapes has been applied at a dataset which was obtained at a location experiencing a number of different mesoscale modes. It was found that the temperature field appears much better organized than the wind field, and that cliff-ramp structures are dominant in the temperature time series. The occurrence of structural shapes was related with the dominant flow patterns and the status of the flow field. Temperature positive cliff-ramps and ramp-cliffs appear mainly during night time and under weak flow field, while temperature step and sine structures do not show a clear preference for the period of day, flow or temperature pattern. Uniformly stable, weak flow conditions dominate across all the wind speed structures. A detailed analysis of the flow field during two case studies revealed that structural shapes might be part of larger flow structures, such as a sea-breeze front or down-slope winds. During stagnant conditions structural shapes that were associated with deceleration of the flow were observed, whilst during ventilation conditions shapes related with the acceleration of the flow.

Regularization of chaos by noise in electrically driven nanowire systems

The electrically driven nanowire systems are of great importance to nanoscience and engineering. Due to strong nonlinearity, chaos can arise, but in many applications it is desirable to suppress chaos. The intrinsically high-dimensional nature of the system prevents application of the conventional method of controlling chaos. Remarkably, we find that the phenomenon of coherence resonance, which has been well documented but for low-dimensional chaotic systems, can occur in the nanowire system that mathematically is described by two coupled nonlinear partial differential equations, subject to periodic driving and noise. Especially, we find that, when the nanowire is in either the weakly chaotic or the extensively chaotic regime, an optimal level of noise can significantly enhance the regularity of the oscillations. This result is robust because it holds regardless of whether noise is white or colored, and of whether the stochastic drivings in the two independent directions transverse to the nanowire are correlated or independent of each other. Noise can thus regularize chaotic oscillations through the mechanism of coherence resonance in the nanowire system. More generally, we posit that noise can provide a practical way to harness chaos in nanoscale systems.

A NEW COLOR IMAGE ENCRYPTION BASED ON HIGH-DIMENSIONAL CHAOTIC SYSTEMS

The high-dimensional chaotic systems (HDCS) have a lot of advantages as more multifarious mechanism, greater the key space, more ruleless for the time series of the system variable than with the low-dimensional chaotic systems (LDCS), etc. Thus, a novel encryption scheme using Lorenz system is suggested. Moreover, we use substitution–diffusion architecture to advance the security of the scheme. The theoretical and experimental results show that the suggested cryptosystem has higher security.

A New Regularity to Generate High-Dimensional Hyperchaotic System

The chaotic system plays an important role in information communication, electrical equipment, computer cryptography and so on. In this paper, four new high-dimensional complex hyperchaotic systems are found. A new overlaying regularity to generate a new high-dimensional complex hyperchaotic system is found by overlaying a series of low-dimensional chaotic system with the Duffing chaotic system. The regularity to generate high-dimensional complex hyperchaotic system is analyzed. The features of chase space maps and Lyapunov exponents’ maps are analyzed. The results of theoretical analysis and experiment show that new systems having strong chaotic features. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.3084 Full Text: PDF

High-Dimensional Chaotic System and Its Circuit Simulation

The chaotic system plays an important role in electronic and electrical equipment, computer cryptography, computer communication and so on. In this paper, we established three new six-dimensional complex chaotic systems and one new ten-dimensional complex chaotic system. The regularity to generate high-dimensional chaotic system is also found by overlaying a series of low-dimensional chaotic systems with Duffing chaotic system. The circiuts of the new high-dimensional complex chaotic system are designed. The simulation experiments of the high-dimensional complex chaotic circuits are tested. The results of theoretical analysis and experiment show that new high-dimensional complex chaotic systems and their circiuts have the hyperchaotic characteristics. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.3076

Streamwise-Localized Solutions at the Onset of Turbulence in Pipe Flow

Although the equations governing fluid flow are well known, there are no analytical expressions that describe the complexity of turbulent motion. A recent proposition is that in analogy to low dimensional chaotic systems, turbulence is organized around unstable solutions of the governing equations which provide the building blocks of the disordered dynamics. We report the discovery of periodic solutions which just like intermittent turbulence are spatially localized and show that turbulent transients arise from one such solution branch.

Encryption algorithm based on double chaos system with mutual feedback

In order to increase the security of the low dimensional chaotic encryption system while not affecting the performance,a composite chaotic encryption system based on Logistic mapping and Tent mapping was presented.Firstly,the secret-key sequences were generated based on two mapping with mutual feedback,which was used to encrypt the plaintext;at the same time,the number of chaotic iterations was changed through cipher-feedback,and the iterative process has certain randomness.The results show,compared with single Logistic chaotic encryption,the algorithm not only has a large key space,high encryption intension and good operation efficiency of low dimensional chaotic encryption system,but also can resist brute-force attack,statistical attack and phase diagrams attack effectively.

Detecting unstable periodic orbits in high-dimensional chaotic systems from time series: Reconstruction meeting with adaptation

Detecting unstable periodic orbits (UPOs) in chaotic systems based solely on time series is a fundamental but extremely challenging problem in nonlinear dynamics. Previous approaches were applicable but mostly for low-dimensional chaotic systems. We develop a framework, integrating approximation theory of neural networks and adaptive synchronization, to address the problem of time-series-based detection of UPOs in high-dimensional chaotic systems. An example of finding UPOs from the classic Mackey-Glass equation is presented.

Predictability and chaotic nature of daily streamflow

The predictability of a chaotic series is limited to a few future time steps due to its sensitivity to initial conditions and the exponential divergence of the trajectories. Over the years, streamflow has been considered as a stochastic system. In this study, the chaotic nature of daily streamflow is investigated using autocorrelation function, Fourier spectrum, correlation dimension method (Grassberger-Procaccia algorithm) and false nearest neighbour method. Embedding dimensions of 6-7 obtained, indicate the possible presence of low-dimensional chaotic behaviour. The predictability of the system is estimated by calculating the system's Lyapunov exponent. A positive maximum Lyapunov exponent of 0.167 indicates that the system is chaotic and unstable with a maximum predictability of only 6 days. These results give a positive indication towards considering streamflow as a low dimensional chaotic system than as a stochastic system. Prediction is done using local polynomial method for a range of embedding dimensions and delay times. The uncertainty in the chaotic streamflow series is reasonably captured through the ensemble approach using local polynomial method.

Predictability and Chaotic Nature of Daily Streamflow

The predictability of a chaotic series is limited to a few future time steps due to its sensitivity to initial conditions and the exponential divergence of the trajectories. Over the years, streamflow has been considered as a stochastic system. In this study, the chaotic nature of daily streamflow is investigated using autocorrelation function, Fourier spectrum, correlation dimension method (Grassberger-Procaccia algorithm) and false nearest neighbour method. Embedding dimensions of 6-7 obtained, indicate the possible presence of low-dimensional chaotic behaviour. The predictability of the system is estimated by calculating the system’s Lyapunov exponent. A positive maximum Lyapunov exponent of 0.167 indicates that the system is chaotic and unstable with a maximum predictability of only 6 days. These results give a positive indication towards considering streamflow as a low dimensional chaotic system than as a stochastic system. Prediction is done using local polynomial method for a range of embedding dimensions and delay times. The uncertainty in the chaotic streamflow series is reasonably captured through the ensemble approach using local polynomial method.

New chaotic image encryption algorithm based on cross-mapping

Chaotic cryptography has been applied to image encryption; however, only the traditional low-dimensional chaotic systems has been widely analyzed or deciphered, which does not show satisfied security and efficiency. To solve this problem, a new algorithm based on cross-chaos map has been created in this article. The image pixels are scrambled under control of high-dimensional chaotic sequence, which is generated by cross chaotic map. The image pixels are substituted by ciphertext feedback algorithm. It can relate encryption required parameters with plaintext and can make a plaintext byte affect more ciphertext bytes. Proved by theoretical analysis and experimental results, the algorithm has higher complex degree and has passed SP800-22 pseudo-random number standard tests, and it has high encryption speed, high security, etc. It can be widely applied in the field of image encryption.

Effect of discrete time observations on synchronization in Chua model and applications to data assimilation

Data assimilation is a tool, which incorporates observations in the model to improve the forecast, and it can be thought of as a synchronization of the model with observations. This paper discusses results of numerical identical twin experiments, with observations acting as master system coupled unidirectionally to the slave system at discrete time instances. We study the effects of varying the coupling constant, the observational frequency, and the observational noise intensity on synchronization and prediction in a low dimensional chaotic system, namely, the Chua circuit model. We observe synchrony in a finite range of coupling constant when coupling the x and y variables of the Chua model, but not when coupling the z variable. This range of coupling constant decreases with increasing levels of noise in the observations. The Chua system does not show synchrony when the time gap between observations is greater than about one-seventh of the Lyapunov time. Finally, we also note that the prediction errors are much larger when noisy observations are used than when using observations without noise.

ERRATUM: “SEARCH FOR CHAOS IN NEUTRON STAR SYSTEMS: IS Cyg X-3 A BLACK HOLE?” (2010, ApJ, 708, 862)

The accretion disk around a compact object is a nonlinear general relativistic system involving magnetohydrodynamics. Naturally the question arises whether such a system is chaotic (deterministic) or stochastic (random) which might be related to the associated transport properties whose origin is still not confirmed. Earlier, the black hole system GRS 1915+105 was shown to be low dimensional chaos in certain temporal classes. However, so far such nonlinear phenomena have not been studied fairly well for neutron stars which are unique for their magnetosphere and kHz quasi-periodic oscillation (QPO). On the other hand, it was argued that the QPO is a result of nonlinear magnetohydrodynamic effects in accretion disks. If a neutron star exhibits chaotic signature, then what is the chaotic/correlation dimension? We analyze RXTE/PCA data of neutron stars Sco X-1 and Cyg X-2, along with the black hole Cyg X-1 and the unknown source Cyg X-3, and show that while Sco X-1 and Cyg X-2 are low dimensional chaotic systems, Cyg X-1 and Cyg X-3 are stochastic sources. Based on our analysis, we argue that Cyg X-3 may be a black hole.

Precipitation, dynamical intermittency, and sporadic randomness

While rainfall intermittency is a dynamical phenomenon, little progress has been made in the literature on the link between rainfall intermittency and atmospheric dynamics. We present the basic dynamical models of intermittency that are phenomenologically most similar to rainfall: Pomeau–Manneville Type-III and On–Off. We then illustrate each type with both a 1-D iterative map and a corresponding stochastic process stressing the appearance of these dynamics in high-dimensional (stochastic) systems as opposed to low-dimensional chaotic systems. We show that the pdf of rainfall intensities, the pdf of “laminar phases” (periods of zero rainfall intensity), and the spectrum of the rainfall series all have power-law behavior that is broadly consistent with intermittency in the classic types. Using a seasonal analysis, we find that summer convective rainfall at daily and sub-daily scales seems consistent with features of Type-III intermittency. The correspondence with Type-III intermittency and a preliminary entropic analysis further suggest that rainfall may be an example of sporadic randomness, blending deterministic and stochastic components.

Stochastic modelling of intermittency

Recently, methods have been developed to model low-dimensional chaotic systems in terms of stochastic differential equations. We tested such methods in an electronic circuit experiment. We aimed to obtain reliable drift and diffusion coefficients even without a pronounced time-scale separation of the chaotic dynamics. By comparing the analytical solutions of the corresponding Fokker-Planck equation with experimental data, we show here that crisis-induced intermittency can be described in terms of a stochastic model which is dominated by state-space-dependent diffusion. Further on, we demonstrate and discuss some limits of these modelling approaches using numerical simulations. This enables us to state a criterion that can be used to decide whether a stochastic model will capture the essential features of a given time series.

Image encryption algorithm based on inter-perturbation of dual chaotic systems

An image encryption algorithm based on inter-perturbations of dual chaotic systems was proposed for the possible degradation of low-dimensional chaotic system and the high computing work of high-dimensional chaotic system. Dual chaotic system was constructed through inter-perturbations of two simple Logistic mappings. And the most prominent feature of inter-perturbations between chaotic systems was that perturbation included constant perturbation and random perturbation simultaneously,which not only ensured the necessary system complexity,but also increased the range of system parameters. Taking dual chaotic inter-perturbed system as sequence key generator,an improved quantization method for converting chaotic sequence to binary sequence was put forward. Random testing and correlation analysis were done on binary sequence. The results show that the binary sequence has good pseudo-randomness and correlation,and appropriate to be encryption key. Simulation results of image encryption applied with the binary sequence also show that the binary sequence can cover up plaintext effectively and safely,and good encryption results are achieved.