Hyperchaotic Behavior Research Articles

An Adaptive Centralized Approach to Control Chaotic and Hyperchaotic Dynamics of Smart Valves Network

Catastrophic chaotic and hyperchaotic dynamical behaviors have been experimentally observed in the so-called “smart valves” network, given certain critical parameters and initial conditions. The centralized network-based control of these coupled systems may effectively mitigate the harmful dynamics of the valve-actuator configuration which can be potentially caused by a remote set and would gradually affect the whole network. In this work, we address the centralized control of two bi-directional solenoid actuated butterfly valves dynamically coupled in series subject to the chaotic and hyperchaotic dynamics. An interconnected adaptive scheme is developed and examined to vanish both the chaotic and hyperchaotic dynamics and return the coupled network to its safe domain of operation.

Dynamic analysis for the hyperchaotic system with nonholonomic constraints

This paper first formulates a new hyperchaotic system for particle motion and analyzes the equilibrium stability of the system and the hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane. We then investigate the influences of two nonlinear nonholonomic constraints on the particle motion. The numerical simulations, including bifurcation diagrams, computation of Lyapunov exponents, Poincare maps, and phase portraits for systems, not only show the hyperchaotic phenomena, but also exhibit the different hyperchaotic behaviors in the constrained parameters space for the various regimes. Additionally, a holonomic constraint is introduced to control the hyperchaotic system.

Hyperchaotic memristive system with hidden attractors and its adaptive control scheme

In this research work a novel 4-D memristive system is presented. The proposed system belongs to the category of dynamical systems with hidden attractors as it displays a line of equilibrium points. Also, it has an hyperchaotic dynamical behavior in a particular range of its parameters space. System’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram, Lyapunov exponents and Poincare map. Next, the case of chaos control of the system with unknown parameters using adaptive control method is investigated. Finally, an electronic circuit realization of the novel hyperchaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.

A novel bit-level image encryption algorithm based on 2D-LICM hyperchaotic map

In this paper, a new two-dimensional Logistic ICMIC cascade map (2D-LICM) is proposed based on cascade modulation couple (CMC) model. Performance evaluations show that it has hyperchaotic behavior, wide chaotic range and large complexity. Based on this map, a novel image encryption algorithm is designed by employing bit-level permutation and diffusion simultaneously. The bit-level permutation is performed by circular shifting, and the bit-level diffusion is carried out by exclusive or (xor) and reverse operations. In addition, the initial values of chaotic system are updated in real time according to the obtained ciphertext and it greatly improves the ability of resisting known plaintext attack and chosen plaintext attack. Simulation results and performance analysis show that this algorithm has good encryption effect and high efficiency. It can resist typical attacks including statistical, brute-force, differential attacks and so forth.

A Technique for Studying a Class of Fractional-Order Nonlinear Dynamical Systems

In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.

Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections

Abstract In this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

Using 0–1 test to diagnose chaos on shape memory alloy dynamical systems

Shape Memory Alloy (SMA) dynamical systems may exhibit a rich response that can include periodic, quasi-periodic, chaotic and hyperchaotic behaviors. In this regard, diagnostic tools are important in order to identify the different types of behaviors. This paper aims to analyze systems with SMA elements through a nonlinear dynamics perspective with a specific focus on the use of 0–1 test to quantify the chaoticity of the dynamical response of SMA oscillators. The investigation includes different constitutive models for the restitution force on both single- and two-degree of freedom oscillators. Results of the 0–1 test are compared with Lyapunov exponents calculated with different algorithms. The analyses show that the 0–1 test can be considered a reliable and computationally efficient alternative as a diagnostic tool of chaotic responses.

SF-SIMM high-dimensional hyperchaotic map and its performance analysis

Derived from Sine map and an iterative chaotic map with infinite collapse (ICMIC), a new high-dimensional hyperchaotic map, sinusoidal feedback Sine ICMIC modulation map (SF-SIMM), is proposed. Two-dimensional (2D) model of SF-SIMM is investigated as an example, and its chaotic performances are evaluated. Results show that it has complicated phase space trajectory, infinite equilibrium points, hyperchaotic behaviors, rather large maximum Lyapunov exponent, three typical bifurcations and multiple coexisting attractors with odd symmetry. Furthermore, it has advantages in complexity, distribution characteristics and zero correlation and can generate two independent pseudo-random sequences simultaneously. Therefore, it has good application prospects in secure communication.

Hyperchaotic Attractor in a Novel Hyperjerk System with Two Nonlinearities

Hyperjerk systems have received considerable interest in the literature because of their simplicity and complex dynamical properties. In this work, we introduce a novel hyperjerk system with an absolute nonlinearity and a quintic term. Interestingly, the hyperjerk system exhibits hyperchaotic behavior. Dynamics and the feasibility of the hyperjerk system are discovered by using numerical analysis and circuit implementation. Moreover, adaptive controllers have been designed for stabilization and synchronization of the new hyperjerk system. The control results have been established by using Lyapunov stability theory, and numerical simulations with MATLAB have been shown to illustrate the validity of the constructed adaptive controllers.

Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors

Although different hyperjerk systems have been discovered, a few hyperjerk systems can exhibit hyperchaotic behavior. In this work, we introduce a new hyperjerk system with hyperchaotic attractors. By investigating dynamics of the system, we have observed the different coexisting attractors such as coexistence of period-2 attractors, or coexistence of period-2 attractor and quasiperiodic attractor. It is worth noting that this striking phenomenon is rarely reported in a hyperjerk system. The proposed system has been realized with electronic components. The agreement between the simulation and experimental results indicates the feasibility of the hyperjerk system. Moreover, chaos control and synchronization of such hyperjerk system have been also reported.

Chaotic Behavior of Random Telegraph Noise in Nanoscale UTBB FD-SOI MOSFETs

Low frequency noise in nanoscale fully depleted ultra-thin body and buried oxide n-MOSFETs is not as random (stochastic) as it appears to be. The fluctuation of the drain current in such nano-devices, under certain bias conditions, exhibits complex random telegraph noise (RTN) that is usually observed in chaotic systems. Indeed, the deterministic chaotic nature of this noise-like variation is confirmed by means of the combined calculation of established nonlinear dynamics metrics. Specifically, the correlation dimension, the phase portrait of the reconstructed chaotic attractor, and the Lyapunov spectrum, conclude for a hyperchaotic behavior of the complex RTN with fractal characteristics.

Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws

The nature is very complex to model with mathematical equations. Some physical problems found in nature could follow the power law; other could follow the Mittag–Leffler law and other the exponential decay law. On the other hand one could observe in nature a physical problem that combines both, it is therefore important to provide a new fractional operator that could possibly be used to model such physical problem. In this paper, we suggest a fractional operator exponential-Mittag–Leffler kernel with two fractional orders. Some very useful properties are obtained. Numerical solutions were obtained for three examples proposed.

Simple hyperchaotic memory system with large topological entropy

The memory elements have been the research hot spot for a long time. However, there is a litter works on the linear memory element. This paper presents a study of a new memory system containing a linear memory element. The study shows that the system not only has two kinds of route to hyperchaos, but also exists many kinds of coexisting attractors in the phase space. Moreover, the system can generate more complex hyperchaotic behaviors. To prove it, we find a new kind of topological horseshoes with two-directional expansions that consist of three disconnected compact sets. This new kind of horseshoes suggests that the topological entropy of the hyperchaotic attractor is larger than other hyperchaotic attractors reported before. For detailed study of the hyperchaotic invariant set, we also demonstrate a method to extract the orbits from the hyperchaotic horseshoes.

An image encryption algorithm based on compound homogeneous hyper-chaotic system

Compound hyper-chaotic system is a chaotic system that combines two or more hyper-chaotic systems. In this paper, we propose a 2-dimensional compound homogeneous hyper-chaotic system (CHHCS) and local binary pattern (LBP)-based image encryption algorithm, which includes the CHHCS-based permutation operation and the LBP-based diffusion operation. Firstly, we employ a new CHHCS and prove the good hyper-chaotic behaviors, and we use CHHCS to permutate the plain image twice to obtain good permutation effect. Then, every permutated pixel is diffused with dynamic LBP operation, which means even the same permutated pixel will be encrypted to different cipher value. Finally, we provide some theoretical analyses and simulations to confirm the security and the validity of the proposed algorithm.

Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

Hyperchaos to Secure Communications According to Synchronization by a High Gain Observer

The purpose of this article is to secure the information message using a new six order continuous hyperchaotic system that we have developed. After studying and verifying the hyperchaotic behavior and stability of this system, a chaotic masking scheme is applied to secure the information between a transmitter and a receiver. The results of the simulations confirm the high performance of the observer designed for this high order system and the proposed method leads to an almost perfect restoration of the original signal.

Computing two dimensional Poincaré maps for hyperchaotic dynamics

Poincaré map (PM) is one of the felicitous discrete approximation of the continuous dynamics. To compute PM, the discrete relation(s) between the successive point of interactions of the trajectories on the suitable Poincaré section (PS) are found out. These discrete relations act as an amanuensis of the nature of the continuous dynamics. In this article, we propose a computational scheme to find a hyperchaotic PM (HPM) from an equivalent three dimensional (3D) subsystem of a 4D (or higher) hyperchaotic model. For the experimental purpose, a standard four dimensional (4D) hyperchaotic Lorenz-Stenflo system (HLSS) and a five dimensional (5D) hyperchaotic laser model (HLM) is considered. Equivalent 3D subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3D subsystems. The quantitative measurement of this comparison is made promising by recurrence quantification analysis (RQA). Various two dimensional (2D) Poincaré mas are computed for several suitable Poincaré sections for both the systems. But, only some of them are hyperchaotic in nature. The hyperchaotic behavior is verified by positive values of both one dimensional (1D) Lyapunov Exponent (LE-I) and 2D Lyapunov Exponent (LE-II). At the end, similarity of the dynamics between the hyperchaotic systems and their 2D hyperchaotic Poincaré maps (HPM) has been established through mean recurrence time (MRT) statistics for both of 4D HLSS and 5D HLM and the best approximated discrete dynamics for both the hyperchaotic systems are found out.

Nonlinear behavior of the tarka flute's distinctive sounds.

The Andean tarka flute generates multiphonic sounds. Using spectral techniques, we verify two distinctive musical behaviors and the nonlinear nature of the tarka. Through nonlinear time series analysis, we determine chaotic and hyperchaotic behavior. Experimentally, we observe that by increasing the blow pressure on different fingerings, peculiar changes from linear to nonlinear patterns are produced, leading ultimately to quenching.

A fast image encryption algorithm based on chaotic map

Derived from Sine map and iterative chaotic map with infinite collapse (ICMIC), a new two-dimensional Sine ICMIC modulation map (2D-SIMM) is proposed based on a close-loop modulation coupling (CMC) model, and its chaotic performance is analyzed by means of phase diagram, Lyapunov exponent spectrum and complexity. It shows that this map has good ergodicity, hyperchaotic behavior, large maximum Lyapunov exponent and high complexity. Based on this map, a fast image encryption algorithm is proposed. In this algorithm, the confusion and diffusion processes are combined for one stage. Chaotic shift transform (CST) is proposed to efficiently change the image pixel positions, and the row and column substitutions are applied to scramble the pixel values simultaneously. The simulation and analysis results show that this algorithm has high security, low time complexity, and the abilities of resisting statistical analysis, differential, brute-force, known-plaintext and chosen-plaintext attacks.

Coexistence and Dynamical Connections between Hyperchaos and Chaos in the 4D Rössler System: A Computer-Assisted Proof

It has recently been reported [P. C. Reich, Neurocomputing, 74 (2011), pp. 3361--3364] that it is quite difficult to distinguish between chaos and hyperchaos in numerical simulations which are frequently “noisy.” For the classical four-dimensional (4D) Rössler model [O. E. Rössler, Phys. Lett. A, 71 (1979), pp. 155--157] we show that the coexistence of two invariant sets with different nature (a global hyperchaotic invariant set and a chaotic attractor) and heteroclinic connections between them give rise to long hyperchaotic transient behavior, and therefore it provides a mechanism for noisy simulations. The same phenomena is expected in other 4D and higher-dimensional systems. The proof combines topological and smooth methods with rigorous numerical computations. The existence of (hyper)chaotic sets is proved by the method of covering relations [P. Zgliczyński and M. Gidea, J. Differential Equations, 202 (2004), pp. 32--58]. We extend this method to the case of a nonincreasing number of unstable directions which is necessary to study hyperchaos to chaos transport. The cone condition [H. Kokubu, D. Wilczak, and P. Zgliczyński, Nonlinearity, 20 (2007), pp. 2147--2174] is used to prove the existence of homoclinic and heteroclinic orbits between some periodic orbits which belong to both hyperchaotic and chaotic invariant sets. In particular, the existence of a countable infinity of heteroclinic orbits linking hyperchaos with chaos justifies the presence of long transient behavior.