Rossler Chaotic System Research Articles

Deep learning based classification of time series of Chen and Rössler chaotic systems over their graphic images

In this study, the graphic images of time series of different chaotic systems are classified with deep learning methods for the first time in the literature. For the classification, a datasetcontains images of time series of Chen and Rossler chaotic systems for different parameter values, initial conditions, step size and time length are generated. Then, high accuracy classifications are performed with transfer learning methods. The used transfer learning methods are SqueezeNet, VGG-19, AlexNet, ResNet50, ResNet-101, DenseNet-201, ShuffleNet, and GoogLeNet. According to the problem, classifications accuracy is varying between 89% and 99.7% in this study. Thus, this study shows that identifying a chaotic system from its graphic image of time series is possible.

Image encryption based on computer generated hologram and Rossler chaotic system

Image encryption based on computer generated hologram and Rossler chaotic system

Synchronization of Rossler chaotic systems via hybrid adaptive backstepping/sliding mode control

In this paper, a hybrid control approach combining adaptive backstepping and fuzzy sliding mode control is proposed for the synchronization of two Rossler chaotic systems. The proposed approach provides fast, high accuracy chaos synchronization in presence of unknown system parameters and external disturbances. The adaptive backstepping control law is developed to stabilize the synchronization error dynamics arising from the mismatch between the drive and response systems. Both unknown system parameters and unknown external disturbance are estimated by the adaptive backstepping control approach in a way that the overall system with the controller becomes stable. As a result, the requirement to have a priori knowledge about the unknown parameters, e.g., disturbance bounds, is nullified. The fuzzy sliding mode controller is designed to accompany the adaptive backstepping to confine the synchronization error and enhance the synchronization speed between the two chaotic systems. Performance of the proposed approach is investigated for a numerical example and compared against commonly used chaos synchronization techniques including optimal control, active control, and adaptive backstepping control method. Simulation results indicate the superior performance of the hybrid control method over state-of-the-art for the synchronization of Rossler chaotic systems.

An Efficient Method of Image Encryption Using Rossler Chaotic System

In this study, a new approach of image encryption has been proposed. This method is depends on the symmetric encryption algorithm RC4 and Rossler chaotic system. Firstly, the encryption key is employed to ciphering a plain image using RC4 and obtains a ciphered-image. Then, the same key is used to generate the initial conditions of the Rossler system. The system parameters and the initial conditions are used as the inputs for Rossler chaotic system to generate the 2-dimensional array of random values. The resulted array is XORed with the ciphered-image to obtain the final encrypted-image. Based on the experimental results, the proposed method has achieved high security and less computation time. Also, the proposed method can be resisted attacks like (statistical, brute-force, and differential).

A 2D logistic map and Lorenz-Rossler chaotic system based RGB image encryption approach

This paper presents a novel color image encryption approach. The proposed approach utilizes the basic concepts of DNA cryptography along with Lorenz and Rossler chaotic system and 2D logistic map. The proposed approach encrypts RGB images using DNA cryptography techniques. In diffusion phase, at pixel level Lorenz and Rossler chaotic system is used to encrypt the three channels of test images. Afterwards, at bit level 2D logistic map is used for performing bitwise chaotic ponytail process on these diffused Red, Green, and Blue channels in confusion phase. Simulation of the proposed approach on test images reveals that the color images have been encrypted very efficiently.

A modified chaotic oscillator with megastability and variable boosting and its synchronisation using contraction theory-based control which is better than backstepping and nonlinear active control

This paper reports a modified 2D periodically forced oscillator. Numerical simulation results using the phase portrait and Lyapunov exponents’ spectrum say that the proposed oscillator depicts megastability. Offset boosting in the new system is shown using the variable boostable phenomenon. Very limited research is available on such type of oscillator. Further, in this paper, a controller based on contraction theory is designed for the synchronisation between the two identical modified chaotic oscillators. The performances of the designed controller are compared with two widely used and well-known controllers for the chaotic systems. These controllers are (i) nonlinear active controller (NAC) and (ii) backstepping controller (BSC). It is found that the contraction theory-based controller performs better in terms of the less synchronisation time, negligible steady-state error and low control energy. Further, synchronisation between two identical Rossler chaotic systems is also presented to validate the effectiveness of the contraction theory. The simulation results validate the objectives of this paper.

New technique to quantify chaotic dynamics based on differences between semi-implicit integration schemes

Many novel chaotic systems have recently been identified and numerically studied. Parametric chaotic sets are a valuable tool for determining and classifying oscillation regimes observed in nonlinear systems. Thus, efficient algorithms for the construction of parametric chaotic sets are of interest. This paper discusses the performance of algorithms used for plotting parametric chaotic sets, considering the chaotic Rossler, Newton-Leipnik and Marioka-Shimizu systems as examples. In this study, we compared four different approaches: calculation of largest Lyapunov exponents, statistical analysis of bifurcation diagrams, recurrence plots estimation and introduced the new analysis method based on differences between a couple of numerical models obtained by semi-implicit methods. The proposed technique allows one to distinguish the chaotic and periodic motion in nonlinear systems and does not require any additional procedures such as solutions normalization or the choice of initial divergence value which is certainly its advantage. We evaluated the performance of the algorithms with the two-stage approach. At the first stage, the required simulation time was estimated using the perceptual hash calculation. At the second stage, we examined the performance of the algorithms for plotting parametric chaotic sets with various resolutions. We explicitly demonstrated that the proposed algorithm has the best performance among all considered methods. Its implementation in the simulation and analysis software can speed up the calculations when obtaining high-resolution multi-parametric chaotic sets for complex nonlinear systems.

Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

An Encryption Scheme Based on Discrete Quantum Map and Continuous Chaotic System

Chaotic encryption is a growing field for competently shielding visual data. As well as quantum encryption and substitution also play a significant role in applied fields owing to its prospective use in secure communication. This article offers new method for the encryption by utilizing quantum chaotic maps and continuous chaotic dynamical systems. Since chaotic maps are itself suitable for quality encryption coupled with quantum logistic encryption, substitution box stipulates highly secure encryption program. Quantum logistic map is combined with Liu and Rossler chaotic systems with the intention of improvement in efficiency of data randomness. Designed scheme ensures the security of data with the minimum time of encryption. Outcomes acquired from performance evaluation signifies that projected chaotic encryption program reveals high complexity and security.

Stabilization of Chaotic Systems With T–S Fuzzy Model and Nonuniform Sampling: A Switched Fuzzy Control Approach

This paper studies the problems of stability and stabilization of a class of chaotic systems (CSs) with Takagi–Sugeno fuzzy model and nonuniform sampling. It designs a fuzzy sampled-data protocol based on a switched idea to tackle the stability issue of such systems. Novel criteria on stabilization of CSs are established by employing a fuzzy membership function (FMF)-dependent Lyapunov functional and the information of the time derivative of FMFs, which significantly utilize the available characteristics of the actual sampling pattern and FMFs simultaneously. Unlike the existing works, a larger sampling interval is obtained by this new approach. A simulation example on chaotic Rossler's system is employed to demonstrate the superiority and reduced conservatism of the proposed method.

A Chaos-Based Image Encryption Technique Utilizing Hilbert Curves and H-Fractals

Image encryption is the most direct and effective technical means for protecting the security of image information. Based on the space filling property of the Hilbert curve and the infinite property of the H-geometric fractal, a new image encryption technique is proposed, which combines the pseudo-randomness of a hyperchaotic system and the sensitivity to initial values. First, the hash value of a plaintext image is calculated using the secure hash algorithm 3 (SHA-3) as the initial value of the piece-wise linear chaotic map (PWLCM) and Rossler chaotic systems, which associates the key with the plaintext. In addition, the chaotic sequences that are generated by the chaotic systems are used to scramble the global pixel positions and the pixel values of the images, thereby disturbing the distribution of the pixel positions and the pixel values. Second, the Hilbert curve and H-fractal are alternately used to scramble the local pixel positions and diffuse the pixel values twice. Finally, the ciphertext feedback is used to further enhance the confusion and diffusion characteristics of the algorithm in order to achieve higher security. The experimental results and security analysis show that the encryption technique has enough key space to resist exhaustive attacks and can effectively resist statistical attacks, differential attacks, noise attacks, and cropping attacks. It can be used for military, judicial, and other privacy-related digital images secure storage and network security transmissions.

A RGB image encryption technique using Lorenz and Rossler chaotic system on DNA sequences

In this paper, a robust color image encryption system using Lorenz-Rossler chaotic map is proposed. The proposed encryption system uses hybrid of two chaotic systems namely Lorenz and Rossler to generate the random sequence. These generated sequences are used for encryption of red, green and blue channels of color image. Rules of DNA cryptosystem are used to encode the plain image in proposed approach. Cross channel operation is proposed to increase randomness in plain image. The proposed encryption approach is tested over different well-known images that are taken from USC-SIPI image dataset. Its performance is compared with recently developed eight image encryption techniques. The experimental results reveal that the proposed approach performs better than the existing techniques in terms of correlation coefficient. The security analyses such as statistical analysis and key sensitivity analysis are performed to validate the security of proposed encryption approach. The key space of proposed approach is large enough to resist against brute force attacks.

Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria

This paper reports a new 4-D dissipative chaotic system. When compared with the available 4-D chaotic/hyperchaotic systems, the unique properties of the proposed system are (i) the system exhibits coexistence of different kind of asymmetric hidden attractors, (ii) the system has a curve of equilibria and (ii) the system is simple. This system does not satisfy Shilnikov criterion for the existence of chaos. The system has a total of eight terms including only one nonlinear term. The coexistence of rich chaotic dynamics in the system is investigated through Lyapunov spectrum, bifurcation diagram, Poincaré map, frequency spectrum and attractors plot. The coexistence of the asymmetric hidden dynamics is shown using chaotic attractors, 3- torus and 2-torus plots. The system is developed from the type-IV 3-D Rossler chaotic system.

Nonlinear characterization of a Rossler system under periodic closed-loop control via time-frequency and bispectral analysis

This study has two primary objectives; they are to investigate the nonlinear interactions (or quadratic phase-coupling) in a chaotic Rossler system under periodic closed-loop control via wavelet bispectral analysis; and to further identify the component mechanisms of synchronization. It is observed that a fixed-gain, fixed-frequency controller produces quadratic phase-coupling and decoupling along lines of constant frequency and that are perpendicular to the diagonal of the bicoherence matrix. Further, it was also observed that for synchronization to occur, both frequency entrainment and quadratic phase-coupling must be present. It was found that forcing the Rossler system with a constant frequency did not reduce the amplitude of the resulting period-1 orbit at sufficiently high gains. For the controller with a fixed gain and time-varying error signal, it was found that the time varying forcing frequency (adjusted by an extremum seeking feedback loop) linearizes the Rossler system and in doing so, suppresses the phase coherence completely. The time-varying forcing frequency removes the conditions for frequency entrainment by providing broadband attenuation; the result is suppression without synchronization.

Cryptanalysis of symmetric key image encryption using chaotic Rossler system

The paper cryptanalyses the symmetric key image encryption scheme proposed by Mrinal et al. [1]. The encryption scheme proposed by Mrinal et al. uses Rossler chaotic system for generating three chaotic sequences which are used for creating the cipher image. A secret key of 128 bits is used to generate the initial conditions for the Rossler chaotic system. The equations used for generating the initial conditions in Mrinal et al. encryption scheme have got a loophole. The loophole is utilized to carry out the cryptanalysis on the cipher image generated using Mrinal et al. encryption scheme. Simulation results show that the encryption scheme can be easily cryptanalysed to revealed the exact plain image. An improved version is proposed with statistical and security analyses to ascertain the practicability of the encryption scheme.

Adaptive coupled synchronization among three coupled chaos systems and its application to secure communications

In this paper, the adaptive synchronization method of three coupled identical chaos systems is proposed. New generic criteria of global chaos synchronization and strict theoretical proofs are given. The unidirectional (bidirectional) coupled system will synchronize by adding three adaptive feedback gain equations. As an example, the three Rossler chaotic systems with unidirectional coupling are given to realize the adaptive coupled synchronization; at the same time, the three Lorenz chaotic systems donate the bidirectional coupling adaptive synchronization. This paper also extends the above adaptive synchronization method to secure chaos communications through the method of chaos masking. Based on the three coupled identical Lorenz chaotic systems, a novel method of chaos encryption is proposed, and the basic idea is that after encrypting two times in two transmitters and one receiver, information signal can be recovered exactly at the receiver end. In consideration of interfering random white noise, the results of encoding and decoding are simulated and analyzed for the multistage chaos synchronized system. By analyzing the characteristics of noise, this paper presents a de-noising method using wavelet transform. At last, a secure communications method for high amplitude information signal transmission based on chaotic masking is studied. With the MATLAB, the effects of signal de-noising could be demonstrated.

Adaptive Robust Stabilization of Rossler System With Time-Varying Mismatched Parameters Via Scalar Input

This paper presents an adaptive robust controller for a class of uncertain chaotic Rossler system with time-varying mismatched parameters. The proposed controller is designed based on Lyapunov stability theory, and it is shown that using this controller all signals of the closed-loop system are uniformly ultimately bounded (UUB). In addition, the proposed scheme is such that it does not require a priori information about the bound of uncertainties. Furthermore, since all the signals are UUB, the control signal is smooth and feasible to implement. Simulation results on a third-order Rossler system with time-varying parameters confirm the effectiveness of the proposed controller.

Lorenz and Rossler Chaotic System for Speech Signal Encryption

this paper an algorithm for speech encryption based on three dimension chaotic maps is proposed. The proposed algorithm consists of three main units: generation of keys, samples substitution and samples permutation process. In order to maximize the benefits of the substitution process, it is performed in two stages with cipher feedback, for the system. Moreover bit-level permutation for sample is introduced as substitution mechanism in the permutation stage. The Lorenz and Rossler chaotic system are employed as generation of keystream used for substitution and permutation process respectively. From the experimental results, it is concluded that the proposed algorithm has the advantages of very low residual intelligibility, key sensitivity and high quality recovered signal, and moreover the proposed algorithm can resist known- plaintext attacks and supports large key space make brute-force attacks infeasible.

Anticipating chaotic synchronization via act-and-wait coupling

Act-and-wait concept is utilized to design a coupling law for anticipating synchronization of chaotic systems. The time-delay coupling term in this algorithm is periodically switched on and off such that the phase space of the whole system remains finite-dimensional despite the presence of the delay. Consequently, the stability of the anticipated synchronization manifold is easily achieved, since it is definite by a finite number of Lyapunov exponents. We show that the stable synchronization regime with considerably large anticipation time can be attained even for single-input single-output systems. The results are demonstrated with the Rossler, Chua and Lorenz chaotic systems.

Nonlinear optimal stabilization of unstable periodic orbits

The framework for the optimal stabilization of unstable periodic orbits in nonlinear systems is proposed. First, the stable manifold method for approximately solving Hamilton-Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Then, this method is applied for optimal stabilization problem of error dynamics around an unstable periodic orbit. The effectiveness of the proposed method is evaluated via numerical simulations for the Rossler chaotic system.