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In this paper, two important points will be investigated, if the variables have the chaotic behavior by LLE and Henon map and if they have chaotic causality by Hristu-Varsakelis and Kyrtsou causality test. It was determined the chaotic behavior of the Turkish stock market and CDS. We found the evidence of bi-directional causality between CDS and Bist-100.

Quantitative measurements of multi-organ interplay are crucial for the assessment of multivariate physiological dynamics in health and disease. Nevertheless, current quantification of multivariate complexity for nonlinear physiological processes is limited by reliability issues on short-time series, and parameters sensitivity especially in case of a multiscale analysis. To overcome these limitations, we propose a new tool to characterize the complexity of interacting physiological processes that may have different temporal dynamics: the Multichannel Complexity Index (MCI). This metrics relies on a novel method for the reconstruction of the multivariate phase space, where each series is embedded using its proper time delay. MCI accounts for the estimation of phase space distances using fuzzy rules, and may be computed at two different ranges of time-scale values to investigate short- and long-term dynamics. We validated our algorithm using three-channel white gaussian noise and 1/f noise systems, with different levels of coupling. By applying our approach to these data, we demonstrate that the MCI method allows to discern not only the degree of complexity in the system dynamics, but also the across-channel coupling level. Results on synthetic series from the Henon map and Rossler attractor demonstrate that MCI effectively discerns between different dynamical behaviours, outperforming state of the art metrics such as the Refined Composite Multivariate Multiscale Fuzzy Entropy. On publicly-available physiological series, considering heartbeat dynamics and blood pressure variability, results demonstrate a MCI sensitivity to postural changes( p 1 0 − 2 for rest vs. slow-tilt, and p 0 . 05 for rest vs. rapid-tilt/stand-up conditions), as well as a MCI sensitivity to subjects’ age-range (data gathered while watching Fantasia Disney movie, 1940) with p 1 0 − 2 for short scales and p = 0 . 03 for long scales. In conclusion, MCI is a viable tool for an effective multivariate physiological complexity assessment. The Matlab code implementing the proposed MCI algorithm is available online.

As was recently shown by the first author and others in Gaidashev et al. (Renormalization and Siegel disks for complex Henon maps, [12]), golden-mean Siegel disks of sufficiently dissipative complex quadratic Henon maps are bounded by topological circles. In this paper we investigate the geometric properties of such curves, and demonstrate that they cannot be \(C^1\)-smooth.

This paper suggests chaotic confusion-diffusion image encryption based on the Henon map. The proposed chaotic confusion-diffusion image encryption utilizes image confusion and pixel diffusion in two levels. In the first level, the plainimage is scrambled by a modified Henon map for n rounds. In the second level, the scrambled image is diffused using Henon chaotic map. Comparison between the logistic map and modified Henon map is established to investigate the effectiveness of the suggested chaotic confusion-diffusion image encryption scheme. Experimental results showed that the suggested chaotic confusion-diffusion image encryption scheme can successfully encrypt/decrypt images using the same secret keys. Simulation results confirmed that the ciphered images have good entropy information and low correlation between coefficients. Besides the distribution of the gray values in the ciphered image has random-like behavior.

Chaos is widely used in secure communication and cryptography due to its randomness, unpredictability, non-periodicity and high sensitivity to initial values and parameters. However, there are many risks in some existing chaotic image encryption algorithms because they use the the chaotic map without complex dynamic characteristics. To overcome these weaknesses, in this paper introduced an improved Hénon map, and the dynamic analysis results show that the improved Hénon map has more rich chaotic behaviors and better complexity. In addition, we designed a color image encryption scheme using the improved Hénon sequences. For this encryption algorithm, color image is divided into R, G, B primary colors, then R, G, B primary colors are scrambled and diffused by the improved Hénon sequences. The simulation results illustrate that color image encryption algorithm security is advanced by usingimproved Hénon map.

In this paper, a dynamic block image encryption using variable-length key and modified Henon map is proposed. To surmount the defects of narrow chaotic range and frail chaotic behaviors when classical Henon map is implemented in practical situations, we design a simple modified Henon map with wide parameter range and high complexity. The Secure Hash Algorithm (SHA) is employed to obtain a 384-bit hash result of plain image considered as a raw key. A variable-length secret key (at least 128 bits), which is dynamically selective from the raw key, is regarded as a valid key to generate initial points of chaotic map. In confusion, we construct a chaotic but non-repetitive integer sequence to implement a substitution without repetition for disorganizing the high spatial relevance among pixels. This can sufficiently avert the problem of fixed points after a confusion, i.e., for any pixel within the confused image, its position will never overlap with the old one. In diffusion, we apply a multidirectional dynamic block diffusion mechanism to modify pixel values, which may not only spread the influences to whole image but reduce encryption time. The proposed scheme has been tested via various cryptographic analysis, and the security analysis demonstrates it has superior security and good efficiency.

A color image encryption algorithm using the Henon-zigzag map and chaotic restricted Boltzmann machine (CRBM) is proposed in this paper. The proposed pseudo-random number generator, chaotic restricted Boltzmann machine (CRBM), can simultaneously generate three pseudo-random number sequences. The algorithm includes the permutation phase and the diffusion phase. In the Henon-zigzag map-based permutation phase, zigzag map is used to modulate two pseudo-random number sequences generated by Henon map to obtain two new pseudo-random number sequences. The mixing of these two chaotic maps makes the security of the permutation phase significantly improve. Subsequently the two pseudo-random number sequences are used for row permutation and column permutation, respectively. In the diffusion phase, through multiple iterations of CRBM of the 3 × 3 architecture, three pseudo-random number sequences are generated by the state values of three neurons in the visible layer. Then these three pseudo-random number sequences are used for bitxor operation with the R, G and B components of the scrambled image, respectively. A series of numerical experiments and analyses on encrypted images prove that the proposed algorithm is more secure than state-of-the-art algorithms. Furthermore, based on the combined use of blockchain and the proposed algorithm, a novel image encryption/decryption system is proposed. The system has two features: asymmetric encryption/decryption of images and authoritative verification of the integrity of encrypted images. It may provide a better understanding of blockchain in digital image encryption.

In this paper, we propose discrete generalized past entropy based on amplitude difference distribution of horizontal visibility graph as a new complexity measure of nonlinear time series. We use amplitude difference distribution instead of degree distribution to extract information from the network constructed from the horizontal visibility graph, and combine amplitude difference distribution with discrete generalized past entropy to propose the new method. By analyzing the logistic map and Henon map with the proposed method, we find the proposed method not only can assess systems well, but also has higher accuracy and sensitivity than the traditional method in characterizing dynamical systems. Furthermore, we apply the proposed method to the financial data: the six indices from Chinese mainland, Hong Kong and US. The result shows that the US market and the Hong Kong market are more developed than the Chinese mainland market, which is consistent with the reality.

Time irreversibility is one of the most important properties of nonstationary time series. Complex time series often demonstrate even multiscale time irreversibility, such that not only the original but also coarse-grained time series are asymmetric over a wide range of scales. We study the multiscale time irreversibility of time series. In this paper, we develop a method called multiscale multifractal time irreversibility analysis (MMRA), which allows us to extend the description of time irreversibility to include the dependence on the segment size and statistical moments. We test the effectiveness of MMRA in detecting multifractality and time irreversibility of time series generated from delayed Henon map and binomial multifractal model. Then we employ our method to the time irreversibility analysis of stock markets in different regions. We find that the emerging market has higher multifractality degree and time irreversibility compared with developed markets. In this sense, the MMRA method may provide new angles in assessing the evolution stage of stock markets.

We prove John Hubbard’s conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Henon map. In fact, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Henon map is completely determined by the monodromy of the complex Henon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.

Discrete dynamical systems where one or several of their parameters vary randomly every iteration are usually referred to as random maps in the literature. However, very few methodologies have been proposed to control these kinds of systems when chaos is present. Here, we propose an extension of the partial control method, that we call parametric partial control, that can be naturally applied to random maps. We show that using this control method it is possible to avoid escapes from a region of the phase space with a transient chaotic behavior. The main advantage of this method is that it allows to control the system even if the corrections applied to the parameter are smaller than the disturbances affecting it. To illustrate how the method works, we have applied it to three paradigmatic models in nonlinear dynamics, the logistic map, the Henon map and the Duffing oscillator.

This paper studies the hidden dynamics of a class of two-dimensional maps inspired by the Henon map. A special consideration is made to the existence of fixed points and their stabilities in these maps. Our concern focuses on three typical scenarios which may generate hidden dynamics, i.e., no fixed point, single fixed point, and two fixed points. A computer search program is employed to explore the strange hidden attractors in the map. Our findings show that the basins of some hidden attractors are tiny, so the standard computational procedure for localization is unavailable. The schematic exploring method proposed in this paper could be generalized for investigating hidden dynamics of high-dimensional maps.

An important problem in a given dynamical system is to determine the existence of a homoclinic orbit. We improve the results of Qin and Xiao [Nonlinearity, 20 (2007), 2305–2317], who present some sufficient conditions for the existence of a homoclinic/heteroclinic orbit for the generalized H´enon map. Moreover, an algorithm is presented to locate these homoclinic orbits.

This study proposes a chaos-based image encryption scheme using Henon map and Lorenz equation with multiple levels of diffusion. The Henon map is used for confusion and the Lorenz equation for diffusion. Apart from the Lorenz equation, another matrix with the same size as the original image is generated which is a complex function of the original image. This matrix which is configured as a diffusion matrix permits two stages of diffusion. Due to this step, there is a strong sensitivity to input image. This encryption algorithm has high key space, entropy very close to eight (for grey images) and very less correlation among adjacent pixels. The highlight of this method is the ideal number of pixels change rate and unified average changing intensity it offers. These ideal values indicate that the encrypted images produced by this proposed scheme are random-like. Further, a cryptanalysis study has been carried out to prove that the proposed algorithm is resistant to known attacks.

No useful information is leaked out as full encryption is done.A very larger key space of 10195 is achieved.Complex diffusion matrix provides strong sensitivity.A lossless compression ratio of 5:1 is achieved. A chaos based image encryption and lossless compression algorithm using hash table and Chinese Remainder Theorem is proposed. Initially, the Henon map is used to generate the scrambled blocks of the input image. The scrambled block undergoes a fixed number of iterations based on the plain image using Arnold cat map. Since hyper chaos system has complex dynamical characteristics than chaos, the confused image is further permuted using the index sequence generated by the hyper chaos along with hash table structure. The permuted image is divided into blocks and the diffusion is carried out either by using Lorenz equations or by using another complex matrix generated from the plain image appropriately. Along with diffusion, compression is also carried out by Chinese Remainder Theorem for each block. This encryption algorithm has high key space, good NPCR and UACI values and very less correlation among adjacent pixels. Simulation results show the high effectiveness and security features of the proposed algorithm.

This paper presents an efficient block-wise image encryption method based on multiple chaotic maps. The image is divided into four overlapping blocks and each block is permutated with Cat map and its parameters are controlled by Henon map using multiple keys. Due to overlapping division of blocks, it produces effect of double permutation in the middle portion of overlapped image in single permutation itself. For diffusion, the whole image is divided into four non-overlapping blocks and diffused with logistic map. Each block pixel values were completely modified in the diffusion process in order to avoid known-plaintext and chosen-plaintext attacks. For each division of blocks different keys were selected for both permutation and diffusion process in the proposed method. The simulation results of several statistical analysis shows that the proposed cryptosystem is efficient and highly secured

The Stokes geometry for the propagator of the quantum Henon map is studied in the light of recent developments of the exact WKB analysis. As the simplest possible situation the Henon map satisfying the so-called horseshoe condition is closely analyzed, together with listing up local bifurcation patterns of the Stokes geometry. This is exactly in the same spirit as pruning theory for the classical horseshoe system, and the present paper is placed as the first step to establish pruning theory of the Stokes geometry for chaotic systems. Our analysis reveals that the birth and death of the WKB solutions caused by the Stokes phenomenon do not occur in a local but entirely global manner, reflecting topological nature encoded in the Stokes geometry. We derive an explicit general formula to enumerate the number of WKB solutions in the asymptotic region and obtain its growth rate, which is shown to be less than the topological entropy of the corresponding classical dynamics. The relations to the diffraction catastrophe integrals and one-step multiply folding maps are also discussed.

In this paper a method for stabilization of unstable periodic solutions of dynamic systems is given. It is based on the delayed feedback with pulse periodic gain. The obtained algorithm is applicable if the linearized system around the cycle has any number of eigenvalues larger than unity. The method is illustrated with the numerical experiment for Henon map.

A three-dimensional (3D) Hénon map of fractional order is proposed in this paper. The dynamics of the suggested map are numerically illustrated for different fractional orders using phase plots and bifurcation diagrams. Lorenz-like attractors for the considered map are realized. Then, using the linear fractional-order systems stability criterion, a controller is proposed to globally stabilize the fractional-order Hénon map. Furthermore, synchronization control scheme has been designed to exhibit a synchronization behavior between a given 2D fractional-order chaotic map and the 3D fractional-order Hénon map. Numerical simulations are also performed to verify the main results of the study.

The realization of real memristor makes it be a very popular topic in recent years. However, the topic about discrete memristor model is rarely discussed. In this paper, a discrete memristor model is proposed based on the difference theory, and the three fingerprints characteristics are proved for this model according to the definition of the generalized memristor. This discrete model is applied to Henon map, and we designed a new chaotic map called the discrete memristor-based Henon map. Its dynamical behaviors are analyzed by attractor phase diagram, bifurcation diagram, Lyapunov exponent spectrum, and spectral entropy complexity algorithm. Simulation results show the performance of Henon map is improved by applying the discrete memristor.

Nowadays, multimedia information, particularly images are shared via the internet and the security of this shared images is a major consideration due to lack in internet security. In this paper, a novel Conditional Butterfly Network Topology (CBNT) based bit-wise confusion, simple pixel-level based confusion and diffusion, based on simple XOR operations is suggested to securely transfer images over an insecure channel like social networks. First, an adaptive image content based initial vector generation is recommended to realize good plain image sensitivity with the aim of withstanding chosen/known plain text attacks. This will change the keys for every plain image in the encryption process. From these initial vectors, Henon map is iterated to acquire the key values to be utilized over the confusion and diffusion processes. To confuse the bits totally in random manner, conditional structure based Butterfly Network Topology (BNT) is utilized. Pixel-based permutation is performed, to disrupt the neighboring correlation of each pixel. Finally, a simple XOR based diffusion is implemented to diffuse the intensity of each pixel. Different security measures are encountered to investigate the amount of security of the suggested system using simulations. From simulations, Number of Pixel Change Rate (NPCR) and Unified Average Changed Intensity (UACI) values of 99.64% and 33.46% and an entropy value of 7.9995 are achieved. Moreover, the proposed model has very less computation time with high level of security.

Thanks to the complex characteristics of ergodicity, pseudo-randomness and sensitivity in initial conditions, chaotic systems have been widely applied in the field of cryptography. By cascading the Hénon map and the Chebyshev map, a new two-dimensional Hénon–Chebyshev modulation map (2D-HCMM) is proposed in this paper. Several methods of objective assessment, including phase diagrams, bifurcation diagrams, Lyapunov exponents and information entropy, are utilized to analyze the dynamics of the 2D-HCMM. The results show that the 2D-HCMM possesses better ergodicity and unpredictability, with larger chaotic ranges, compared with the original chaotic maps. By using the proposed map and the essential principles of genetic recombination and genetic mutation, a new image encryption scheme is proposed. In this scheme, the bit planes of image are substituted by genetic recombination operation, and the pixel values are scrambled randomly by genetic mutation operation. The simulation results and security analysis demonstrate that the proposed scheme not only can resist various conventional attacks, but also possesses a fast speed, achieving a good trade-off between security and efficiency.

For a Henon map H in C 2 , we characterize the polynomial automorphisms of C 2 which keep any fixed level set of the Green function of H completely invariant. The interior of any non-zero sublevel set of the Green function of a Henon map turns out to be a Short C 2 and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short C 2 's. Further, we prove that if any two level sets of the Green functions of a pair of Henon maps coincide, then they almost commute.

We develop a two-dimensional enhanced hyperchaotic Henon map (2D-EHHM) to overcome the problems of small chaotic range and poor security when the 2-D traditional Henon map is implemented in cryptosystems. The performance evaluations show that the 2D-EHHM has wider chaotic range, higher chaotic complexity, better ergodicity, and hyperchaotic behavior compared with certain existing chaotic maps. Based on the 2D-EHHM, we further design an efficient image encryption algorithm consisting of a multiple block substitution stage (MBSS) and a bidirectional-dynamic diffusion stage (BDDS). In the MBSS, a plain image is divided into several nonoverlapping multiple blocks to carry out permutation operation. In the BDDS, the scrambled image is redivided into nonoverlapping sub-blocks of the same size to be diffused dynamically in the forward and backward directions. Moreover, the SHA 512 function is employed to obtain a 512-bit plain image hash value treated as a raw key. A variable-length secret key (at least 128 bits), which is dynamically selected from the raw key and regarded as a valid key, is utilized to generate the initial values for the chaotic system. Simulation results and security analysis show that the proposed algorithm can resist various cryptanalytic attacks and can be applied to real-time data transmission.

Recent researches of image encryption algorithms have been increasingly based on chaotic systems. This paper, a new image encryption scheme which employs. The idea is to associate the Henon attractor and the logistics map, for the construction of a new secret key cryptosystem. We generate values through of the logistics map that will be added to the pixels of the plaintext image. This result modulo 256 will be permuted to another position of the encrypted image. The calculation of this permutation is deducted from the Henon attractor, which is 2-dimensional, in order to have a significantly increasing the resistance to attacks. The proposed system has the advantage of bigger key space (about 180 bits); high security analysis such as key space analysis, statistical analysis and sensitivity analysis were carried out. The results demonstrate that the proposed system is highly efficient and a robust system.

We have suggested a new method for quantitative determination of masking degree of signals. This method is based on results following from the information and entropy theory. Degree of masking of typical telecommunication signals by white Gaussian noise and well-known chaotic signals such as the logistic map, Henon map, and “accumulation-bursting” map has been determined by numerical experiments. Also we have defined the dependence between masking degree of keying digital signals and bit error rate (BER).

Security in images pose a serious challenge like vulnerability, threats, and distributed denial of service attacks. Chaotic based encryption schemes have been widely used in practice to tackle the issues. At present, many variant types of chaos mapping scheme have been applied with respect to attack mechanisms. Henon and Lorenz are the mapping chaotic scheme has been deployed to evaluate the images free from the loopholes of security. Based on the results, we compare and contrast both scheme to address its significance and merits.

Finding synchronization between the outputs of a dynamic system, which are represented mostly as time series, helps to characterize the system activities during an occurrence. An important issue in analyzing time series is that they may behave chaotically or stochastically. Therefore, applying a reliable synchronization measure which can capture the dynamic features of the system helps to quantify the interdependencies between time series, correctly. In this paper, we employ similarity measures based on visibility graph (VG) algorithms as an alternative and radically different way to measure the synchronization between time series. We assess the performance of VG-based similarity measures on chaotic, noisy and stochastic time series. In our experiments, we use the Rössler system and the noisy Hénon map as representative instances of chaotic systems, and the Kuramoto model for studying detection of synchronization between stochastic time series. Our study suggests that the similarity measure based on the horizontal VG algorithm should be favored to other measures for detecting synchronization between chaotic and stochastic time series.

Annali di matematica pura ed applicata, 185 (Supplement 5), ISSN:0373-3114, ISSN:1618-1891

In the last decades many possible applications of nonlinear dynamics in communication systems and signal processing have been reported. Conversely, techniques usually employed by the signal processing and communication systems communities, as correlation, power spectral density analysis, and linear filters, among others have been used to characterize chaotic dynamical systems. This chapter presents four works that aim to use tools from both fields to generate new and interesting results: (1) a message authentication system based on chaotic fingerprint; (2) a study of the spectral characteristics of the chaotic orbits of the Henon map; (3) an investigation of the chaotic nature of the signals generated by a filtered Henon map, and (4) a communication system that presents equalization and a switching scheme between chaos-based and conventional modulations.

Transient chaos can emerge in a variety of diverse systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. The end of the chaotic episode can either be desired or not, depending on the specific system and application. In both cases, however, a prediction of the end of the chaotic dynamics is required. Despite the general challenges of reliably predicting chaotic dynamics for a long time period, the recent observation of a "terminal transient phase" of chaotic transients provides new insights into the transition from chaos to the subsequent (nonchaotic) regime. In spatially extended systems and also low-dimensional maps it was shown that the structure of the state space changes already a significant amount of time before the actual end of the chaotic dynamics. In this way, the terminal transient phase provides the conceptual foundation for a possible prediction of the upcoming end of the chaotic episode a significant amount of time in advance. In this study, we strengthen the general validity of the terminal transient phase by verifying its existence in another spatially extended model (Gray-Scott model) and the Henon map, where in the latter case the underlying mechanisms can be understood in an intuitive way. Furthermore, we show that the temporal length of the terminal transient phase remains approximately constant, when changing the system size (Gray-Scott) or parameters (Henon map) of the investigated models, although the average lifetime of the observed chaotic transients sensitively depends on these variations. Since the timescale of the terminal transient phase is in this sense relatively robust, this insight might be essential for possible applications, where the ratio between the length of the terminal transient phase and the relevant timescale of the dynamics may probably be crucial when a reasonable prediction (thus a sufficient time before) the end of the chaotic episode is required.

This article was withdrawn and retracted by the Journal of Fundamental and Applied Sciences and has been removed from AJOL at the request of the journal Editor in Chief and the organisers of the conference at which the articles were presented (www.iccmit.net). Please address any queries to editor@jfas.info.

This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.

The investigations of hidden attractors are mainly in continuous-time dynamic systems, and there are a few investigations of hidden attractors in discrete-time dynamic systems. The classical chaotic attractors of the Logistic map, Tent map, Henon map, Arnold’s cat map, and other widely-known chaotic attractors are those excited from unstable fixed points. In this paper, the hidden dynamics of a new two-dimensional map inspired by Arnold’s cat map is investigated, and the existence of fixed points and their stabilities are studied in detail.

In today's scenario, the usage of images has become very popular in many applications like military, medical, social media and industries. Sensitive images are communicated through insecure networks. Therefore some methods are required by which the images can be transmitted securely over the network. The proposed algorithm is a modified image encryption algorithm based on Arnold Cat Map. The scheme involves confusion process in which the position of each pixel of the image is shuffled using Cat Map. The shuffling of pixel position leads to the generation of a permuted image which is secured for transmission. Security analysis for the suggested scheme is performed using correlation coefficient (which is 0.6402 horizontally and 0.6954 vertically) and information entropy (which is 7.9504) analysis.

We study the dynamics of the Henon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Henon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Henon maps over $${\mathbb {Q}}_3$$ , this attractor is infinite and supports an SRB-type measure describing the distribution of all nearby forward orbits. We include some numerical calculations which suggest the existence of such infinite attractors over $${\mathbb {Q}}_5$$ and $${\mathbb {Q}}_7$$ as well. On a different region of the parameter space, we show that the Henon map is topologically conjugate on its filled Julia set to the two-sided shift map on the space of bisequences in two symbols.