Search

Well-known optical image encryption technique double random phase encoding scheme has been shown vulnerable to basic attacks. Thereafter triple random phase encoding scheme is proposed to endure basic attacks. Recently, triple random phase encoding scheme is also shown vulnerable by using deep learning based attack. In this paper, an image encryption scheme based on triple random phase encoding with chaotic Henon map in Fourier domain is proposed. Henon map is used to strengthen the image encryption scheme. Henon map has two parameters and two initial conditions which are highly sensitive to its original value. Experiments were carried out on grayscale images to validate the proposed scheme. Statistical attacks such as information entropy, histogram and 3-D plot analysis are successfully endured by the scheme. Performances against noise and occlusion attacks show the robustness of the scheme. The key sensitivity results indicate that the proposed scheme is highly secure.

In this paper, we propose a novel single-channel secure transmission scheme based on the impulsive synchronization of fractional-order discrete chaotic systems. A discrete fractional-order impulsive observer has been developed at the receiver level in order to reconstruct the states of a fractional-order Henon map used at the transmitter level. Robust chaos-based encryption and decryption algorithms have been used to cipher an original color image and retrieve it as soon as the synchronization is established between the transmitter and the receiver.

We introduce and discuss the methodology of the chaotic evolution (CE) algorithms, which is supported by a chaotic system. Because properties of the chaotic systems are the essential factors to influence the performance of the CE algorithms, we design and analyse several CE algorithms using different chaotic systems: logistic map, tent map, Gauss map and Henon map in a well-designed chaotic evolution framework. We propose a new CE algorithm using the combination of these chaotic systems to accelerate the convergence speed and improve convergence performance and analyse its effectiveness by comparing these new algorithms with our previous proposed CE algorithm with the logistic map. From the evaluation, our proposed CE helps the search to avoid the local optimum.

A novel quantum image compression and encryption algorithm with Daubechies $$ {D^{(4)}} $$ quantum wavelet transform (DQWT) and 3D hyper-chaotic Henon map is presented. The quantum image is firstly scrambled by the iterative generalized Arnold transforms to eliminate its block effect. Then, the produced quantum image is compressed with DQWT and measurement matrix, which could be implemented with Hadamard gate. Subsequently, a quantum key image is constructed by a hyper-chaotic Henon sequence generated by 3D hyper-chaotic Henon map under the control of three initial values and two parameters. The quantum key image is XORed with the produced quantum compression image. The key space is relatively large enough since there are three initial values and two parameters involved. Numerical simulations demonstrate that the proposed quantum image compression and encryption algorithm is feasible, secure and efficient.

Chimera state refers to coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of Chimera, on one hand is essential due to its applicability in various areas including neuroscience, and on other hand is challenging due to its widely varied appearance in different systems and the peculiar nature of its profile. Therefore, a simple yet universal method for its identification remains an open problem. Here, we present a very distinctive approach using machine learning techniques to characterize different dynamical phases and identify the chimera state from given spatial profiles generated using various different models. The experimental results show that the performance of the classification algorithms varies for different dynamical models. The machine learning algorithms, namely random forest, oblique random forest based on tikhonov, parallel-axis split and null space regularization achieved more than $96\% $ accuracy for the Kuramoto model. For the logistic-maps, random forest and tikhonov regularization based oblique random forest showed more than $90\%$ accuracy, and for the H\'enon-Map model, random forest, null-space and axis-parallel split regularization based oblique random forest achieved more than $80\%$ accuracy. The oblique random forest with null space regularization achieved consistent performance (more than $83\%$ accuracy) across different dynamical models while the auto-encoder based random vector functional link neural network showed relatively lower performance. This work provides a direction for employing machine learning techniques to identify dynamical patterns arising in coupled non-linear units on large-scale, and for characterizing complex spatio-temporal patterns in real-world systems for various applications., Comment: 20 Pages, 4 Figures; Comments welcome

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 30, 061104 (2020) and may be found at https://doi.org/10.1063/5.0008902. We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.

We consider the transmission dynamics of COVID-19 which is characterized by two distinct features. One is the existence of asymptomatic carriers which is the hidden variable in the problem. The other is the issue of latency which means that among the symptomatic carriers there could be a fraction whose symptoms develop after a couple of days. We write down a Henon-like map to take these effects into account and find that the map has an unusual fixed point which corresponds to a very high ratio of asymptomatic to symptomatic infected persons. This fixed point has its own stability zone and influences the dynamics overall. Although a rather elementary model, we show that it is not totally devoid of reality.

Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspective of information flow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information flow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information flow. Therefore, contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write G e o C y &rarr, x . This avoids some of the boundedness issues that we show exist for the transfer entropy, T y &rarr, x . We will highlight our discussions with data developed from synthetic models of successively more complex nature: these include the H&eacute, non map example, and finally a real physiological example relating breathing and heart rate function.

Machine Learning (ML) inspired algorithms provide a flexible set of tools for analyzing and forecasting chaotic dynamical systems. We here analyze the performance of one algorithm for the prediction of extreme events in the two-dimensional H\'enon map at the classical parameters. The task is to determine whether a trajectory will exceed a threshold after a set number of time steps into the future. This task has a geometric interpretation within the dynamics of the H\'enon map, which we use to gauge the performance of the neural networks that are used in this work. We analyze the dependence of the success rate of the ML models on the prediction time $T$ , the number of training samples $N_T$ and the size of the network $N_p$. We observe that in order to maintain a certain accuracy, $N_T \propto exp(2 h T)$ and $N_p \propto exp(hT)$, where $h$ is the topological entropy. Similar relations between the intrinsic chaotic properties of the dynamics and ML parameters might be observable in other systems as well., Comment: 9 pages, 12 figures

We describe a renormalization method in maps of the plane \begin{document}$ (x, y) $\end{document} , with constant Jacobian \begin{document}$ b $\end{document} and a second parameter \begin{document}$ a $\end{document} acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. \begin{document}$ |b| = 1-\varepsilon $\end{document} ), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the \begin{document}$ (x, y, a) $\end{document} space, in sequences of diverging period, that we call "branches". We define a renormalization approach, by "hopping" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter \begin{document}$ a $\end{document} (see [ 3 ]) and in other ranges of the period for the dynamical plane \begin{document}$ (x, y) $\end{document} . For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane \begin{document}$ (x, y) $\end{document} . We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map. The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [ 7 ] for highly dissipative systems.

In recent years, the environments like military, government, medical field, cloud computing, and social networks deal with a large number of confidential images transmitted over the Internet. Therefore, it is very important to protect the image from unauthorized access during the transmission in an open network. Encryption is the most convenient technique to guarantee the security of images over public networks. To maximize the network utilization, the compression technique is used to reduce the size of the image by the channel provider who is having plenty of computational resources. This paper proposes an image Encryption-Then-Compression (ETC) system. Hybrid Chaotic method is used to encrypt the image, where Arnold map is used for confusion and Henon map is used for diffusion. Asymmetric Numerical Method (ANM) is used to compress the encrypted image. The experiment result shows that the proposed system is better in terms of compression performance, security, and computation time.

Recently, the properties and applications of Julia sets from more general complex maps were investigated. Saitoh et al. [210] discussed a complex system extended from Logistic map, and got the condition under which the Julia set is Cantor set. Fornaess [211] studied the complex Henon map and pointed out that the complex Henon map’s maximum entropy set equals to its Julia set. By extending the complex map from integral expression into rational expression, Blanchard et al. [212, 213] investigated the fractal behavior of the following system

In this paper, we propose a multidimensional scaling (MDS) method based on complexity-invariant distance (CID) and generalized complexity-invariant distance (GCID) to analyze and classify complex time series like traffic signals and financial stock indexes. Three types of simulation time series from the $$\upalpha $$ -map model, the 2D Henon map model and the Lozi map model as well as two real-world time series are used to illustrate the practicability of the proposed MDS method. Results from two traditional MDS and the MDS based on the mutual information are compared with the MDS based on CID and GCID, which demonstrate the proposed method is more effective and reasonable.

Chaotic return maps are widely used to model various dynamical systems such as charged particle movement, laser beam dynamic, celestial body orbiting and many others. Return maps are commonly obtained by discretization of continuous equations using the Euler–Cromer operator with the only motivation that it is the simplest symplectic operator. Recent progress in geometric integration raised considerable interest to symmetric operators due to their ability to preserve some geometric properties of continuous flows this way yielding better agreement between discrete and continuous dynamical systems. Here we compare symmetric and asymmetric discretization approaches applied to several examples of Hamiltonian systems. In particular, we suggest symmetric modifications of Chirikov and Henon maps and show explicitly that the implied symmetric integration procedure yields reflectional symmetry in the phase space. For verification, we show that a smooth even perturbation function used instead of a discontinuous delta pulse provides asymptotically similar results. Numerical experiments using several statistical methods show that symmetric and asymmetric maps, while yielding similar asymptotic behavior, often exhibit considerably different statistical properties for intermediate regimes providing smoother transitions that are more reminiscent to those observed in various natural phenomena. We believe that the proposed approach may be useful for modeling empirical systems by preserving their keynote physical properties.

It was recently shown in Gaidashev and Yampolsky [Golden mean Siegel disk universality and renormalization. Preprint, 2016, arXiv:1604.00717] that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalizations in the Hénon family considered in de Carvalho et al [Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121 (5/6) (2006), 611–669]. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Hénon map is non-smoothly rigid.

In this paper, the classical problem of uncovering the maximal invariant set of a (discrete-time) dynamical system is illuminated from a novel perspective, which in particular leads to a novel sample-based computational procedure to compute the invariant set. The mathematical description of these new insights can be formulated in strikingly basic set-theoretic terms, and more importantly, be efficiently realized computationally in terms of different sample-based implementations. We illustrate the simplicity and efficiency of the computational method on three examples with a maximal invariant set that is unstable in both time directions, the classical Henon map, a three-dimensional analogue of the Henon map, and a Van der Pol oscillator.

The present work develops validated numerical methods for analyzing continuous branches of connecting orbits—as well as their bifurcations—in one parameter families of discrete time dynamical systems. We use the method of projected boundaries to reduce the connecting orbit problem to a finite dimensional zero finding problem for a high dimensional map. We refer to this map as the connecting orbit operator. We study one parameter branches of zeros for the connecting orbit operator using existing methods of computer assisted proof know as the radii-polynomial approach and validated continuation. The local stable/unstable manifolds of the fixed points are analyzed using validated numerical methods based on the parameterization method. The validated branches of zeros found using our argument correspond to continuous families of transverse connecting orbits for the original dynamical system. Validation of a saddle node bifurcations for the connecting orbit operator correspond to a proof of existence for a tangency. We illustrate the implementation of our method for the Henon map.

We study a continuous Hénon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations. Ministerio de Economía y Competitividad Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía)

In this paper, a new two dimensional modified Henon map (2D-MHM) which is derived from Henon map is proposed. Its chaotic performance is analyzed through bifurcation diagram, Lyapunov exponent spectrum and Lyapunov dimension. The map has broad chaotic regime over an extensive range of system parameters, maximum Lyapunov exponent and better chaotic performance when compared to existing chaotic maps. Further, a novel image cryptosystem is proposed based on 2D-MHM and sine map. The algorithm employs confusion and diffusion operations in consecutive manner which is different from traditional chaos based cryptosystems. Hybrid chaotic shift transform (HCST) is introduced to perform confusion operation which is controlled by 2D-MHM. The principle of diffusion is achieved by using chaotic matrix generated from sine map and exclusive or (XOR) operation. Extensive simulation results and performance analysis demonstrate that the proposed image cryptosystem is able to resist various cryptanalytic attacks. Furthermore, the comparison results reveal that the algorithm outperforms traditional and existing encryption schemes. The proposed algorithm is also applicable for speech signals and data encryption of other multimedia.

Delay-coordinate maps have been widely used recently to study nonlinear dynamical systems, where there is only access to the time series of one of their variables. Here, we show how the partial control method can be applied in this kind of framework in order to prevent undesirable situations for the system or even to reduce the variability of the observable time series associated with it. The main advantage of this control method is that it allows to control delay-coordinate maps even if the control applied is smaller than the external disturbances present in the system. To illustrate how it works, we have applied it to three well-known models in the field of nonlinear dynamics with different delays such as the two-dimensional cubic map, the standard map and the three-dimensional hyperchaotic Henon map. For the first time we show here how hyperchaotic systems can be partially controlled.

The convergence and divergence of oscillatory behavior of different brain regions are very important for the procedure of information processing. Measurements of coupling or correlation are very useful to study the difference of brain activities. In this study, EEG signals were collected from ten subjects under two conditions, i.e. eyes closed state and idle with eyes open. We propose a nonlinear algorithm, symbolic joint entropy, to compare the coupling strength among the frontal, temporal, parietal and occipital lobes and between two different states. Instead of decomposing the EEG into different frequency bands (theta, alpha, beta, gamma etc.), the novel algorithm is to investigate the coupling from the entire spectrum of brain wave activities above 4Hz. The coupling coefficients in two states with different time delay steps are compared and the group statistics are presented as well. We find that the coupling coefficient of eyes open state with delay consistently lower than that of eyes close state across the group except for one subject, whereas the results without delay are not consistent. The differences between two brain states with non-zero delay can reveal the intrinsic inter-region coupling better. We also use the well-known Henon map data to validate the algorithm proposed in this paper. The result shows that the method is robust and has a great potential for other physiologic time series.

The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to detect and stabilize all the UPOs up to a maximum length of the period. The role played by each involved parameter is investigated and additional results to those reported in the literature are presented.

In this work, we propose a method of control and anti-control of chaos based on the moving largest Lyapunov exponent using reinforcement learning. In this method, we design a reward function for the reinforcement learning according to the moving largest Lyapunov exponent, which is similar to the moving average but computes the corresponding largest Lyapunov exponent using a recently updated time series with a fixed, short length. We adopt the density peaks-based clustering algorithm to determine a linear region of the average divergence index so that we can obtain the largest Lyapunov exponent of the small data set by fitting the slope of the linear region. We show that the proposed method is fast and easy to implement through controlling and anti-controlling typical systems such as the Henon map and Lorenz system.

We report the discovery of wide nested sequences of period-adding complexification routes spawning the whole control parameter space of the Henon map, a discrete-time proxy of single-mode loss-modulated CO 2 lasers. For a realistic map, the new adding routes found reproduce analogous phenomena previously observed in continuous-time differential equations describing, for instance, oscillatory phases in an enzyme reaction model. In contrast to differential equations, the map is analytically tractable to a large extent, and provides a prototypic framework to investigate intricate features of generic dissipative flows.

A double color image encryption method based on DNA (deoxyribonucleic acid) computation and chaos is proposed. Differently from the conventional algorithms, double color images are encrypted at the same time so that we can save information of each other, which makes the encryption more safe and reliable. In addition, a new chaotic fractional order (FO) discrete improved Henon map (FODIHM) is proposed as a pseudo-random number generator. To ensure the plain-image sensitivity of the encryption algorithm, the initial value of FODIHM is calculated from the hash value of the color image (SHA-256) and from the three additional keys entered by the user. Furthermore, a Rubik’s cube transform scrambles the pixels in each color component of the two images. Then, each pixel in each color component of the two images is diffused by means of different DNA coding rules. Finally, the CAT transform, based on FO discrete Logistic map and the classic XOR, is used to further improve the security performance. The key space size of the proposed algorithm is of order 1 0 135 , which is about 30 orders of magnitude higher than those available in the literature. The information entropies are 7.9974 and 7.9973, which are very close to the ideal entropy value of 8. The values of the unified average changing intensity (NPCI) are 99.630 and 99.623, while the number of pixels change rate (UACR) are 33.473 and 33.553, which are also close to the ideal NPCR and UACR value of 99.6094 and 33.4635, respectively. The numerical results and security analysis prove that the algorithm has good resistance to several classic attacks.

This paper proposes a maiden application of a two dimensional Sine Logistic map based chaotic sine cosine algorithm (2D-SLCSCA) optimized classical PID controller for load frequency control (LFC) of an islanded microgrid (MG). In comparison to random variables and 1D chaotic sequences, the 2D chaotic sequences are more ergodic and possess a wider chaotic range, thereby enhancing the global convergence speed and search capability of an algorithm. Initially, the proposed 2D-SLCSCA is tested on eight classical benchmark test functions and its performance is compared with 1D Logistic map based chaotic SCA (1D-LCSCA), 1D Sine map based chaotic SCA (1D-SCSCA), and the SCA incorporating random variables. Test results reveal that the proposed algorithm exhibits better convergence characteristics, statistics, and execution time. Finally, the proposed 2D-SLCSCA is implemented for the LFC analysis of the islanded MG. To establish the competence of the proposed algorithm in this regard, its performance is compared with 2D Henon map based chaotic SCA, 2D Lozi map based chaotic SCA, improved salp swarm algorithm (ISSA), SCA, grey wolf optimizer (GWO), and particle swarm optimization (PSO) algorithm considering diverse load disturbance patterns in the MG. Simulation results affirm that the proposed control scheme augments the frequency response of the MG exhibiting a maximum percentage improvement of 78.89%, 78.86%, and 96.51% in peak overshoot ( O S p e a k ), peak undershoot ( U S p e a k ) and objective function ( O F I T S E ) value, respectively as compared to the other algorithms. Furthermore, sensitivity of the proposed 2D-SLCSCA is validated considering ± 30% variation in the MG parameters.