Complex Chaotic Behavior Research Articles

Analyzing Devaney Chaos of a Sine–Cosine Compound Function System

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

A Novel Look Back N Feature Approach towards Prediction of Crude Oil Price

Prediction of crude oil prices in advance can play a significant role in the global economy. Change in crude oil price affect wide range of application for economic and risk projection. Crude oil price forecasting is a challenging task due to its complex nonlinear and chaotic behavior. During the last decade’s researcher have designed many classification algorithm for crude oil prediction. The major challenge for any unsupervised dataset is to define a class label for every feature of its dataset. This paper, propose a new novel technique, look back N feature (LBNF) algorithm to discover class label. Later the classifier support vector machine (SVM) with k-nearest neighbor (k-NN) has been used to classify the current feature vector to predict the crude indices one day, one weak, one month in advance. We have checked our algorithm with standard recent MCX INR Daily and CFD USD Real Time crude oil dataset. To prove the effectiveness of proposed algorithm we have compared it with recent Grey wave forecasting method and the experimental result is found to be better than this method.Â

Sine-Transform-Based Chaotic System With FPGA Implementation

As chaotic dynamics is widely used in nonlinear control, synchronization communication, and many other applications, designing chaotic maps with complex chaotic behaviors is attractive. This paper proposes a sine-transform-based chaotic system (STBCS) of generating one-dimensional (1-D) chaotic maps. It performs a sine transform to the combination of the outputs of two existing chaotic maps (seed maps). Users have the flexibility to choose any existing 1-D chaotic maps as seed maps in STBCS to generate a large number of new chaotic maps. The complex chaotic behavior of STBCS is verified using the principle of Lypunov exponent. To show the usability and effectiveness of STBCS, we provide three new chaotic maps as examples. Theoretical analysis shows that these chaotic maps have complex dynamics properties and robust chaos. Performance evaluations demonstrate that they have much larger chaotic ranges, better complexity, and unpredictability, compared with chaotic maps generated by other methods and the corresponding seed maps. Moreover, to show the simplicity of STBCS in hardware implementation, we simulate the three new chaotic maps using the field-programmable gate array (FPGA).

Integrated chaotic systems for image encryption

To improve the randomicity behaviors of some existing chaotic maps, this paper proposes two integrated chaotic systems (ICS), which conduct cascade, nonlinear combination, and switch operations to three basic 1D chaotic maps to generate new structures. The developed systems are able to yield several more complicated chaotic maps. Compared with several existing maps, the newly-generated ones have more advanced properties, including wider chaotic ranges, and more complex chaotic behaviors. To demonstrate the effectiveness of ICS, we also design an image encryption scheme based on ICS. Simulation results on different types of images and extensive security analysis demonstrate that the proposed approach has satisfactory properties in image encryption.

One-Dimensional Nonlinear Model for Producing Chaos

Motivated by the concept of circuit design in digital circuit, this paper proposes a one-dimensional (1D) nonlinear model (1D-NLM) for producing 1D discrete-time chaotic maps. Our previous works have designed four nonlinear operations of generating new chaotic maps. However, they focus only on discussing individual nonlinear operations and their properties, but fail to consider their relationship among these operations. The proposed 1D-NLM includes these existing nonlinear operations, develops two new nonlinear operations, discusses their relationship among different nonlinear operations, and investigates the properties of different combinations of these operations. To show the effectiveness of 1D-NLM in generating new chaotic maps, as examples, we provide four new chaotic maps and study their dynamics properties from following three aspects: equilibrium point, stability, and bifurcation diagram. Performance evaluations are provided using the Lyapunov exponent, Shannon entropy, correlation dimension, and initial state sensitivity. The evaluation results show that these new chaotic maps have more complex chaotic behaviors than existing ones. To demonstrate the performance of 1D-NLM in practical applications, we use a pseudo-random number generator (PRNG) to compare new and existing chaotic maps. The randomness test results indicate that new chaotic map generated by 1D-NLM shows better performance than existing ones in designing PRNG.

Secure Image Encryption Based on an Ideal New Nonlinear Discrete Dynamical System

It is well known that encryption algorithms developed based on Logistic map suffer from limited key space due to the narrow regions of system parameters which can be used, potential risk of security in the presence of numerous periodic windows within the key space, and weakness in known-plain-text attack due to the inherent correlation among the chaotic sequence used for encryption. To overcome these existing problems, this paper presents a secure image encryption algorithm based on a new highly nonlinear discrete dynamical system with ideal chaotic characteristics. Transcendental functions are introduced together with modulo operations which physically represent discontinuous time-varying nonlinearities, leading to extremely complex chaotic behavior that is highly sensitive to system parameters and initial conditions, both of which are considered as the key for the cryptosystem. Extensive numerical experiment results have revealed that the proposed image encryption algorithm offers advantages of unlimited key space and high-level security, since those problematic periodic windows are no longer present within the key space, and it is extremely robust against known-plain-text attack, since the chaotic sequence generated bears no correlation whatsoever due to the folding effect of modulo operation. The algorithm makes truly efficient yet highly secure image encryption based on chaotic systems a reality.

Chaotic Oscillation via Edge of Chaos Criteria

In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can be exploited to engineer a phase transition from ordered to chaotic behavior. The frequency of the periodic forcing can be derived from this criteria. In order to generate a periodic or a chaotic oscillation, we have to tune the amplitude of the periodic forcing. For example, we engineer chaotic oscillations in the generalized Duffing oscillator, the FitzHugh–Nagumo model, the Hodgkin–Huxley model, and the Morris–Lecar model. Although forced oscillators can exhibit chaotic oscillations even if the edge of chaos criteria is not satisfied, our computer simulations show that forced oscillators satisfying the edge of chaos criteria can exhibit highly complex chaotic behaviors, such as folding loci, strong spiral dynamics, or tight compressing dynamics. In order to view these behaviors, we used high-dimensional Poincaré maps and coordinate transformations. We also show that interesting nonlinear dynamical systems can be synthesized by applying the edge of chaos criteria. They are globally stable without forcing, that is, all trajectories converge to an asymptotically-stable equilibrium point. However, if we apply a forcing signal, then the dynamical systems can oscillate chaotically. Furthermore, the average power delivered from the forced signal is not dissipated by chaotic oscillations, but on the contrary, energy can be generated via chaotic oscillations, powered by locally-active circuit elements inside the one-port circuit [Formula: see text] connected across a current source.

The Painlevé test for nonlinear system of differential equations with complex chaotic behavior

The Painlevé-analysis was performed for solutions of nonlinear third-order autonomous system of differential equations with quadratic nonlinearities on their right-hand sides. At certain values of two constant parameters incorporated into the system, the latter exhibits complex chaotic behavior. When the parameters attain the values corresponding to complex chaotic behavior, the system was found not to possess the Painlevé property.

Non-periodic and chaotic response of three-multilobe air bearing system

• Effects of rotor mass and bearing number on three-multilobe air bearing are analyzed. • Hybrid DTM&FDM method provides better accuracy than other numerical methods. • Complex periodic, non-periodic and chaotic motion behavior is observed. The dynamic response of a three-multilobe air bearing (TMAB) system is investigated for various values of the rotor mass and bearing number using a hybrid numerical scheme consisting of the differential transformation method (DTM) and the finite difference method (FDM). The validity of the numerical scheme is demonstrated by comparing the results obtained for the rotor center orbit under typical operating conditions with those obtained from the traditional FDM approach and a perturbation method, respectively. The dynamic behavior of the rotor center is then investigated for rotor mass values in the range of 1.0 ≤ m r ≤ 16.0 kg and bearing number values in the range of 1.0 ≤ Λ ≤ 5.0. The phase trajectories, power spectra, bifurcation diagrams, Poincaré maps and maximum Lyapunov exponents show that the TMAB system exhibits a complex dynamic behavior consisting of periodic, quasi-periodic and chaotic motion at certain values of the rotor mass and bearing number. In general, the numerical results obtained in this study provide a useful insight into the dynamic response of TMAB systems. In particular, the results indicate the operating conditions which should be avoided in order to achieve a desirable periodic motion of the system.

A Novel Image Encryption Scheme Using the Composite Discrete Chaotic System

The composite discrete chaotic system (CDCS) is a complex chaotic system that combines two or more discrete chaotic systems. This system holds the chaotic characteristics of different chaotic systems in a random way and has more complex chaotic behaviors. In this paper, we aim to provide a novel image encryption algorithm based on a new two-dimensional (2D) CDCS. The proposed scheme consists of two parts: firstly, we propose a new 2D CDCS and analysis the chaotic behaviors, then, we introduce the bit-level permutation and pixel-level diffusion encryption architecture with the new CDCS to form the full proposed algorithm. Random values and the total information of the plain image are added into the diffusion procedure to enhance the security of the proposed algorithm. Both the theoretical analysis and simulations confirm the security of the proposed algorithm.

Chaos Control of Permanent Magnet Synchronous Motor via Adaptive Sliding Mode Variable Structure Scheme

When system parameters falling into a certain area or under some working conditions, the permanent magnet synchronous motor (PMSM) will appear chaos phenomenon, which threatens the secure and stable operation of drive system. So this paper proposes a robust adaptive sliding mode control strategy aiming at the complex chaotic behavior of PMSM. Using the sliding mode control technique, a time varying surface is constructed. Then an adaptive gain of the robust control law is established considering uncertainties and external interference in PMSM. So the phenomenon of chaos in PMSM system will be eliminated using proposed controller. Finally, stability analysis will be given based on Lyapunov stability theory. Simulation results demonstrate the effectiveness of the proposed adaptive sliding mode control scheme.

Dynamic Parameter-Control Chaotic System.

This paper proposes a general framework of 1-D chaotic maps called the dynamic parameter-control chaotic system (DPCCS). It has a simple but effective structure that uses the outputs of a chaotic map (control map) to dynamically control the parameter of another chaotic map (seed map). Using any existing 1-D chaotic map as the control/seed map (or both), DPCCS is able to produce a huge number of new chaotic maps. Evaluations and comparisons show that chaotic maps generated by DPCCS are very sensitive to their initial states, and have wider chaotic ranges, better unpredictability and more complex chaotic behaviors than their seed maps. Using a chaotic map of DPCCS as an example, we provide a field-programmable gate array design of this chaotic map to show the simplicity of DPCCS in hardware implementation, and introduce a new pseudo-random number generator (PRNG) to investigate the applications of DPCCS. Analysis and testing results demonstrate the excellent randomness of the proposed PRNG.

Rich spectrum of neural field dynamics in the presence of short-term synaptic depression.

In continuous attractor neural networks (CANNs), spatially continuous information such as orientation, head direction, and spatial location is represented by Gaussian-like tuning curves that can be displaced continuously in the space of the preferred stimuli of the neurons. We investigate how short-term synaptic depression (STD) can reshape the intrinsic dynamics of the CANN model and its responses to a single static input. In particular, CANNs with STD can support various complex firing patterns and chaotic behaviors. These chaotic behaviors have the potential to encode various stimuli in the neuronal system.

Chaos control for Willamowski–Rössler model of chemical reactions

Real systems evolving towards complex state encounter chaotic behavior. This behavior is very important in chemical processes or in biological structures because it defines the direction of the evolution of the system. From this point of view, the capability of deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason why theoretical models are useful in these situations. In order to obtain chaos control in chemical reactions, the analysis of the dynamics of Willamowski–Rössler system involving the synchronization of two Minimal Willamowski–Rössler (MWR) systems based on the adaptive feedback method of control is presented in this work. As opposed to previous studies where in order to obtain synchronization 3 controllers were used, implying from a practical point of view the control of the concentrations of three chemical species, in this study we showed that the use of just one is sufficient which in practice is important as controlling the concentration of a single chemical species would be much easier. We also showed that the transient time until synchronization depends on initial conditions of two systems, the strength and number of the controllers and we attempted to identify the best conditions for a practical synchronization.

Standing wave plasmon modes interact in an antenna-coupled nanowire.

In a standing wave optical cavity, the coupling of cavity modes, for example, through a nonlinear medium, results in a rich variety of nonlinear dynamical phenomena, such as frequency pushing and pulling, mode-locking and pulsing, modal instabilities, even complex chaotic behavior. Metallic nanowires of finite length support a hierarchy of longitudinal surface plasmon modes with standing wave properties: the plasmonic analog of a Fabry-Pérot cavity. Here we show that positioning the nanowire within the gap of a plasmonic nanoantenna introduces a passive, hybridization-based coupling of the standing-wave nanowire plasmon modes with the antenna structure, mediating an interaction between the nanowire plasmon modes themselves. Frequency pushing and pulling, and the enhancement and suppression of specific plasmon modes, can be controlled and manipulated by nanoantenna position and shape.

Chaotic Dynamics in Kicked Ratchets

A one-dimensional deterministic continuous dynamical system is studied and shown to exhibit chaotic behavior and complex trans- port properties. Our model is an overdamped rocking ratchet with finite dissipation that is periodically kicked with a delta function driving force, without finite inertia terms or temporal or spatial stochastic forces. This is perhaps the simplest model reported in the literature for a ratchet that exhibits a complex chaotic behavior. We present both numerical and analytical results that predict many key features of the system, such as current reversals, as well as the presence of chaotic behavior and bifurcation. In particular, we show that alternate positive and negative delta functions as the unbiased driving force on a ratchet potential produces both synchronized and chaotic regions.

A Novel Approach for Image Encryption based on Parametric Mixing Chaotic System

image encryption schemes for secure transmission and storage are increasingly needed for a number of applications like medical, military, satellite etc. In this paper, a novel image encryption algorithm based on Logistic and Tinkerbell map is proposed.The proposed method uses two 1-D Logistic maps with different keys and one 2-D Tinkerbell map. The chaotic sequence generated is mixed sequence from the and sequences of Tinkerbell map depending on the chaotic sequences of two logistic maps. The main advantage of such a scheme is complex chaotic behavior of the generated chaotic sequences. The security and performance of the proposed method is analyzed thoroughly by using key-sensitivity, key- space, statistical, entropy, differential and performance analysis. The proposed approach achieves the required level of security with only one round of encryption operation. Hence the proposed method is computationally efficient.

Chaos Control in Cournot-Puu Model

The Cournot-Puu economic model, with a bounded inverse demand function and different constant marginal production costs, exhibits complex bifurcating and chaotic behaviors. In this work, two approaches are developed to control bifurcation and chaos in the model. The first approach is based on the delayed feedback control method and can be viewed as a variant of Chens strategy. The second approach is an adaptive parameter turning algorithm in which the original adaptive method for controlling chaos is implemented. Numerical simulations are conducted to show how bifurcation and chaos can be controlled by the proposed approaches. The possible economic implications of the proposed control strategies are also discussed.

Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent

Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Herein a, b, c, d1 are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.

Existence of heteroclinic orbits in a novel three-order dynamical system

In this paper, we design a novel three-order autonomous system. Numerical simulations reveal the complex chaotic behaviors of the system. By applying the undetermined coefficient method, we find a heteroclinic orbit in the system. As a result, the Ši'lnikov criterion along with some other given conditions guarantees that the system has both Smale horseshoes and chaos of horseshoe type.