Synchronization between variable time delayed systems and cryptography

In this letter we consider a prototype model which is described as an autonomous continuous time delayed differential equation with just one variable. The chaos has been investigated with variable delay time and the synchronization phenomenon is examined both numerically and analytically using the Krasovskii-Lyapunov functions. We have applied adaptive coupling law for synchronization,where the coupling equation also contains delay with modulated time. We also studied the effect of cryptography for this coupled system and the message extraction procedure is illustrated with the help of simulated results.Comment: 9 pages,3 figures. Submitted to EP

Synchronization of chaotic modulated time delay networks in presence of noise

We study the constructive role of noises in a Lorenz system with functional delay. The effect of delay can change the dynamics of the system to a chaotic one from its steady state. Induced synchronization with white and colored (red and green) noises are observed between two identical uncoupled systems and enhancement of synchrony is also observed with unidirectional coupling. We investigate both the phenomena in a globally coupled network in the presence of white and color noises.Comment: 10 pages, 7 figure

Phase synchronization of instrumental music signals

Signal analysis is one of the finest scientific techniques in communication theory. Some quantitative and qualitative measures describe the pattern of a music signal, vary from one to another. Same musical recital, when played by different instrumentalists, generates different types of music patterns. The reason behind various patterns is the psychoacoustic measures - Dynamics, Timber, Tonality and Rhythm, varies in each time. However, the psycho-acoustic study of the music signals does not reveal any idea about the similarity between the signals. For such cases, study of synchronization of long-term nonlinear dynamics may provide effective results. In this context, phase synchronization (PS) is one of the measures to show synchronization between two non-identical signals. In fact, it is very critical to investigate any other kind of synchronization for experimental condition, because those are completely non identical signals. Also, there exists equivalence between the phases and the distances of the diagonal line in Recurrence plot (RP) of the signals, which is quantifiable by the recurrence quantification measure tau-recurrence rate. This paper considers two nonlinear music signals based on same raga played by two eminent sitar instrumentalists as two non-identical sources. The psycho-acoustic study shows how the Dynamics, Timber, Tonality and Rhythm vary for the two music signals. Then, long term analysis in the form of phase space reconstruction is performed, which reveals the chaotic phase spaces for both the signals. From the RP of both the phase spaces, tau-recurrence rate is calculated. Finally by the correlation of normalized tau-recurrence rate of their 3D phase spaces and the PS of the two music signals has been established. The numerical results well support the analysis

Noise induced synchronization of time-delayed semiconductor lasers and authentication based asymmetric encryption.

In this work, we propose to enable security mechanisms on a chaotic communication system based upon common noise induced synchronization between two time-delayed semiconductor laser systems. The cryptosystem subjected to the common additive Gaussian colored noise undergoes a transition to follow identical trajectories. An investigation of the system together with a novel scheme for authentication based message encryption process are presented. The encrypted message is also sent over a public channel, while the key is never transmitted at all. The advantage of the scheme is its security, based on the authentication and asymmetric encryption. Extended statistical tests with the proposed two phase cryptography scheme demonstrate the efficiency of the system being robust and tolerant to different types of statistical attacks

Synchronization of Time Delayed Systems by Common Delay Time Modulations

We investigate the synchronization phenomenon between two identical time delayed systems with the common time delay, modulated by a chaotic or random signal. The phenomenon is verified by the conditional Lyapunov exponent. The relation between the present form of synchronization with generalized one is also discussed

Analysis of Rattleback Chaotic Oscillations

Rattleback is a canoe-shaped object, already known from ancient times, exhibiting a nontrivial rotational behaviour. Although its shape looks symmetric, its kinematic behaviour seems to be asymmetric. When spun in one direction it normally rotates, but when it is spun in the other direction it stops rotating and oscillates until it finally starts rotating in the other direction. It has already been reported that those oscillations demonstrate chaotic characteristics. In this paper, rattleback’s chaotic dynamics are studied by applying Kane’s model for different sets of (experimentally decided) parameters, which correspond to three different experimental prototypes made of wax, gypsum, and lead-solder. The emerging chaotic behaviour in all three cases has been studied and evaluated by the related time-series analysis and the calculation of the strange attractors’ invariant parameters

An exploration of fractal-based prognostic model and comparative analysis for second wave of COVID-19 diffusion

The coronavirus disease 2019 (COVID-19) pandemic has fatalized 216 countries across the world and has claimed the lives of millions of people globally. Researches are being carried out worldwide by scientists to understand the nature of this catastrophic virus and find a potential vaccine for it. The most possible efforts have been taken to present this paper as a form of contribution to the understanding of this lethal virus in the first and second wave. This paper presents a unique technique for the methodical comparison of disastrous virus dissemination in two waves amid five most infested countries and the death rate of the virus in order to attain a clear view on the behaviour of the spread of the disease. For this study, the data set of the number of deaths per day and the number of infected cases per day of the most affected countries, the USA, Brazil, Russia, India, and the UK, have been considered in the first and second waves. The correlation fractal dimension has been estimated for the prescribed data sets of COVID-19, and the rate of death has been compared based on the correlation fractal dimension estimate curve. The statistical tool, analysis of variance, has also been used to support the performance of the proposed method. Further, the prediction of the daily death rate has been demonstrated through the autoregressive moving average model. In addition, this study also emphasis a feasible reconstruction of the death rate based on the fractal interpolation function. Subsequently, the normal probability plot is portrayed for the original data and the predicted data, derived through the fractal interpolation function to estimate the accuracy of the prediction. Finally, this paper neatly summarized with the comparison and prediction of epidemic curve of the first and second waves of COVID-19 pandemic to visualize the transmission rate in the both times

Enhancing chaos in multistability regions of Duffing map for an asymmetric image encryption algorithm

We investigate the dynamics of a two-dimensional chaotic Duffing map which exhibits the occurrence of coexisting chaotic attractors as well as periodic orbits with a typical set of system parameters. Such unusual behaviors in low-dimensional maps is inadmissible especially in the applications of chaos based cryptography. To this end, the Sine-Cosine chaotification technique is used to propose a modified Duffing map in enhancing its chaos complexity in the multistable regions. Based on the enhanced Duffing map, a new asymmetric image encryption algorithm is developed with the principles of confusion and diffusion. While in the former, hyperchaotic sequences are generated for scrambling of plain-image pixels, the latter is accomplished by the elliptic curves, S-box and hyperchaotic sequences. Simulation results and security analysis reveal that the proposed encryption algorithm can effectively encrypt and decrypt various kinds of digital images with a high-level security.Comment: 15 pages, 15 figure

Some time-delay finding measures and attractor reconstruction

Topologically equivalent attractor reconstruction is one of the major issues in nonlinear analysis. This is because of the fact that the underlying dynamical model of some nonlinear phenomena may not be known and thus it is necessary to retrieve the dynamics from the data it generates. One way to achieve this is the reconstruction of the attractor. The basis of such reconstruction is the famous Taken’s embedding theorem, which asserts that an equivalent phase space trajectory,preserving the topological structures of the original phase space trajectory, can be reconstructed by using only one observation of the time series. However, in some cases topologically equivalent attractor reconstructions can also be done by using multiple observations. All these things involve the choice of suitable time-delay(s) and embedding dimension. Various measures are available to find out the suitable time-delay(s). Among them, linear auto-correlation, Average mutual information, higher dimensional mutual information are mostly used measures for the reconstruction of the attractors. Every measures have certain limitations in the sense that they are not always useful in finding suitable time-delay(s). Thus it is necessary to introduce few more nonlinear measures, which may be useful if the aforesaid measures fail to produce suitable time-delay/time-delays. In this chapter, some comparatively new nonlinear measures namely generalized auto-correlation, Cross auto-correlation and a new type of nonlinear auto-correlation of bivariate data for finding suitable time-delay(s) have been discussed. To establish their usefulness, attractors of some known dynamical systems have been reconstructed from their solution components with suitable time-delay(s) obtained by each of the measures. These attractors are then compared with their corresponding original attractor by a shape distortion parameter Sd. This shape distortion parameter actually checks how much distorted the reconstructed attractor is from its corresponding original attractor. The main objective of this chapter is to address the problem of reconstruction of a least distorted topologically equivalent attractor. The reason is that if the reconstructed attractor is least distorted from its original one, the dynamics of the system can be retrieved more accurately from it. This would help in identifying the dynamics of the corresponding system, even when the dynamical model is not known. Out of the three measures discussed in this chapter, the generalized and cross auto-correlation measures produce least distorted topologically equivalent attractor only by consideration of multiple solution components of the dynamical system. On the other hand, by using the measure—new type of nonlinear auto-correlation of bivariate data, one can reconstruct a least distorted topologically attractor from single solution component of the dynamical system. Various numerical results on Lorenz system, Neuro-dynamical system and also on two real life signals are presented to prove the effectiveness of the aforesaid three comparatively new nonlinear time-delay finding measures. Finding of suitable embedding dimension is another important issue for attractor reconstruction. However, this issue has not been highlighted in this chapter because we have restricted this discussion only to three dimensional attractor reconstruction