Chaotic Behavior Of System Research Articles

Random property enhancement of a 1D chaotic PRNG with finite precision implementation

The random-like behavior of chaotic systems make them perfect candidates for the core of numerous pseudo-random number generators. The coupling of two or more chaotic maps has been proven to increase the random property robustness of the generated sequences. However, these properties are also strongly influenced by the datatype implementation of these generators owing to the effect of the sensitivity to initial conditions. Hence, several factors must be considered concerning the hardware implementation of these coupled chaotic map-based generators. This paper first examines the effect of fixed-point precision implementation on the periodicity of a single skew tent map and the extension of this effect on the statistical properties of the generated sequence from coupled skew tent maps. Then, the effect on cross-coupled skew tent maps is discussed. Further, a modification to the cross-coupled scheme is presented; this achieves a reduction in the fixed-point fraction length required for generating a sequence suitable for cryptographic applications while expanding the dependency on the control parameters of the maps. The generated sequences from the proposed modification met all the applied statistical and correlation test requirements, demonstrating that they possess acceptable random properties and are suitable for cryptographic applications.

Dynamic analysis and synchronization of conformable fractional-order chaotic systems

In this paper, chaotic systems based on the conformable fractional-order derivative are investigated. First, some comments on the discrete Conformable Adomian decomposition method (discrete CADM) proposed recently are made. We demonstrate that the discrete CADM is not globally accurate for conformable fractional-order differential equations and some conclusions about conformable fractional-order chaotic systems in present papers based on the discrete CADM are inappropriate. Second, we restudy the conformable fractional-order simplified Lorenz system. And then, the conformable fractional-order Liu system is introduced and studied. The basic dynamical behaviors of the systems are analyzed in detail, such as dissipation, the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits and time series of states. Some interesting dynamic behaviors different from the Caputo or Riemann-Liouville fractional-order chaotic system are discovered. It is found that the system order of the commensurate case for chaos is $q \in (0,1]$ . However, for the incommensurate case, dissipation and stability of equilibrium points depend on system order qi and time t which may cause chaotic transients or period-doubling oscillation. In addition, the dynamic behaviors of conformable fractional-order chaotic systems will be static as $t \rightarrow + \infty$ which is named “dynamic death”. Finally, synchronization between conformable fractional-order simplified Lorenz system and conformable fractional-order Liu system is realized.

Measure of chaos and adaptive synchronization of chaotic satellite systems

In this paper, we analyze the chaotic behaviour of satellite system through the dissipative, equilibrium points, bifurcation diagrams, Poincare section maps, Lyapunov exponents and Kaplan–Yorke dimension. We observe the qualitative behaviour of satellite systems through these tools to justify the chaos in the system. We obtain the equilibrium points of chaotic satellite system. At each equilibrium point we yield the eigenvalue of Jacobian matrix of satellite system and verify the unstable regions. We calculate Kaplan–Yorke dimension, $$D_{KY}= 2.1905$$ . Adaptive synchronization for two identical satellite systems is presented. The qualitative and simulated results are provided for verification of systems.

Control of Pyragas Applied to a Coupled System with Unstable Periodic Orbits.

We apply a Pyragas-type control in order to synchronize the solutions of a glycolytic model that exhibits an aperiodic behavior. This delay control is used to stabilize the orbits of ordinary differential nonlinear equations systems. Inspired by several works that studied the chaotic behavior of diverse systems for the enzymatic reactions in the presence of feedbacks, the control to two of these models is analyzed.

Uncertainty and disturbance estimator based robust synchronization for a class of uncertain fractional chaotic system via fractional order sliding mode control

This paper deals with a finite time robust synchronization problem of a class of uncertain fractional chaotic/hyper-chaotic systems with a novel fractional sliding mode control technique. Firstly, a fractional order sliding surface is proposed to mimic the behavior of master chaotic system. Then, a fractional order sliding mode control (FOSMC) methodology is derived analytically for convergence of all the synchronizing errors to zero in finite time. Finally, the derived control strategy is augmented with an auxiliary control based on uncertainty and disturbance estimator (UDE) for ensuring the robustness of the closed loop system dynamics in the presence of system uncertainties. Further, the uncertainties with unknown bounds are tackled for depicting the practical scenario and these results are also applicable to the N-dimensional uncertain chaotic as well as hyper-chaotic systems. Moreover, Mittag-Leffler and fractional order Lyapunov results are utilized to prove the stability and finite time convergence. Also, the proposed method delivers chatter-free control signal which is a major issue in sliding mode. MATLAB simulations are carried out to verify the efficacy and robustness of the derived results by considering two examples from literature.

Analysis of the Time Series Generated by a New High-Dimensional Discrete Chaotic System

The chaotic behavior of low-dimensional digital chaotic systems is seriously degraded, and the output sequence has a short period. In this study, a digital sequence generator based on a high-dimensional chaotic system is proposed to ensure performance and security. The proposed generator has low resource consumption, and the digital pseudo-random output sequence has a large period. To avoid the nonchaotic state, the multistability in the high-dimensional discrete chaotic system is analyzed. The statistical performance of the output sequence of the proposed digital high-dimensional chaotic system is evaluated, and the results demonstrate that it is a suitable candidate for a long-period pseudo-random sequence generator.

Efficient encryption technique for H.264/AVC videos based on CABAC and logistic map

Nowadays, the use of real time video communication is growing with a rapid pace. For example, Search and Rescue (SAR) applications like earthquake rescue, avalanche victims, wildfire monitoring in highway and surveillance are some of the applications of real time video communication. Most of these video communications require secure transmission of signals. For real time video communication, the processing time is considered as the most important metric to be optimized. Thus, finding a simple and time efficient encryption technique for securing the transmitted data becomes mandatory. The motive behind this work is to design a system that can process the video signal in real time providing a secure communication. In this paper, we present an efficient encryption technique which has low computation complexity. The proposed technique is based on Context-Adaptive Binary Arithmetic Coding (CABAC) where the bin-string of Intra-Prediction Mode, Motion Vector Difference (MVD) and residue coefficients are encrypted with the random behavior of chaotic systems. The random bit streams used for encryption are generated with chaotic systems using Logistic map. The experimental results show that the proposed technique is very effective for real-time applications and robust against different types of attacks.

Bifurcation and chaos analysis of a gear pair system with multi-clearance

In order to investigate the characteristics of bifurcation and chaos for a spur gear pair system, a three-degree-of-freedom nonlinear dynamic model with multi-clearance is established, in which time-varying meshing stiffness, static transmission error, gear backlash and bearing clearance are comprehensively taken into account. Through introducing a relative generalized coordinate, the dimensionless dynamic equations of motion of system are derived and then solved by using Runge-Kutta numerical integration method. And the bifurcation and chaos features of gear pair are systematically analyzed and discussed from bifurcation diagrams with meshing frequency, gear backlash, bearing clearance and damping ratio as control parameters under different loaded conditions. Meantime, with the help of Poincaré map and phase diagram, the motion forms of system are accurately identified. The analysis results reveal that as meshing frequency increases, the system shows various types of motion states which contain periodic motion, quasi-periodic motion and chaotic motion. Similarly, with the increasing of gear backlash, the system undergoes complex motion forms under lightly loaded condition, whereas it is only in period-one motion state under heavily loaded condition. Furthermore, the system motion state is gradually switched from chaos to periodic or quasi-periodic motion under lightly loaded condition when bearing clearance changes. However, under heavily loaded condition, the bearing clearance has a weak effect on dynamic behavior of the gear system. Apparently, the system tends to be more stable under heavily loaded condition than that under lightly loaded condition. In addition, the growing damping ratio can effectively suppress the chaotic behavior and control nonlinear vibration of gear system. The research results provide useful guidance for dynamic design and vibration control for gear set.

Stability of dynamical behavior of a new hyper chaotic system in certain range and its hybrid projective synchronization behavior

This article gives an insight for important tools to study the stability of dynamical behavior of newly constructed hyper chaotic systems within some certain parameter range. Further, hybrid projective synchronization (HPS) of two identical new hyper chaotic systems is defined and scheme of HPS is developed by using tracking control method. A new hyper chaotic system has been constructed and then response system. Numerical simulations verify the effectiveness of this scheme, which has been performed by mathematica and MATLAB.

ОБ ОДНОМ ПОДХОДЕ К ПОСТРОЕНИЮ ХАОТИЧЕСКИХ СИСТЕМ-ХАМЕЛЕОНОВ

Now it is well known that dynamical systems can be categorized into systems with selfexcited attractors and systems with hidden attractors. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. Hidden attractors play the important role in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing. In addition, complex behaviors of chaotic systems have been applied in various areas from image watermarking, audio encryption scheme, asymmetric color pathological image encryption, chaotic masking communication to random number generator. Recently so-called chameleons systems have been found out by researchers. These systems were so are named for the reason, that they shows self-excited or hidden oscillations depending on the value of parameters entering into them. In the present work the simple algorithm of synthesizing of oneparametrical chameleons systems is offered. Evolution Lyapunov exponents and Kaplan-Yorke dimension of such systems at change of parameter is traced.

On some dynamical chameleon systems

It is now well known that dynamical systems can be categorized into systems with self-excited attractors and systems with hidden attractors. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. Hidden attractors play the important role in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing. In addition, complex behaviors of chaotic systems have been applied in various areas from image watermarking, audio encryption scheme, asymmetric color pathological image encryption, chaotic masking communication to random number generator. Recently, researchers have discovered the so-called “chameleon systems”. These systems were so named because they demonstrate self-excited or hidden oscillations depending on the value of parameters. The present paper offers a simple algorithm of synthesizing one-parameter chameleon systems. The authors trace the evolution of Lyapunov exponents and the Kaplan-Yorke dimension of such systems which occur when parameters change.

Out-of-time-order fluctuation-dissipation theorem

We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order n-partite OTOCs as well as in the form of generalized covariance.

Expectations, Price Fluctuations and Lorenz Attractor

This paper describes expectations and Buy-Sell transactions of selected Stokes between economic agents and Exchange on economic space as ground for modeling trading volume and price fluctuations. We study simple model of mutual relations between transactions and expectations and derive economic equations that describe disturbances of price, trading volume and expectations. We obtain simple harmonic oscillations for price fluctuations. We show that our model economic equations can take form of Lorenz attractor. Our approximation of transactions and expectations and economic equations on disturbances of price, trading volume and expectations allows apply dynamical systems methods for modeling chaotic behavior of economic and financial systems.

Multistability in Horizontal Platform System with and without Time Delays

Chaotic behavior and bifurcation analysis of horizontal platform systems (HPS) have been investigated widely by many researchers. However, the multistable features of such systems have not been investigated, and hence we identified the multistable parameter and investigated the coexisting attractors of the HPS. To understand the effects of time delays on the nonautonomous and autonomous HPS, we introduced a constant time delay in the state feedback variable. Investigation of the bifurcation of the time delayed HPS with time delay and parameters reveals that the system behavior differs between the autonomous and nonautonomous HPS. To investigate the multistability existence in time delayed HPS, we plot the bifurcation of the autonomous HPS and show the multistability and coexisting attractors.

Chaotic versus stochastic behavior in active-dissipative nonlinear systems

We study the dynamical state of the one-dimensional noisy generalized Kuramoto-Sivashinsky (gKS) equation by making use of time-series techniques based on symbolic dynamics and complex networks. We focus on analyzing temporal signals of global measure in the spatiotemporal patterns as the dispersion parameter of the gKS equation and the strength of the noise are varied, observing that a rich variety of different regimes, from high-dimensional chaos to pure stochastic behavior, emerge. Permutation entropy, permutation spectrum, and network entropy allow us to fully classify the dynamical state exposed to additive noise.

A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

Nonlinear chaotic systems yield many interesting features related to different physical phenomena and practical applications. These systems are very sensitive to initial conditions at each time-iteration level in a numerical algorithm. In this article, we study the behavior of some nonlinear chaotic systems by a new numerical approach based on the concept of Galerkin–Petrov time-discretization formulation. Computational algorithms are derived to calculate dynamical behavior of nonlinear chaotic systems. Dynamical systems representing weather prediction model and finance model are chosen as test cases for simulation using the derived algorithms. The obtained results are compared with classical RK-4 and RK-5 methods, and an excellent agreement is achieved. The accuracy and convergence of the method are shown by comparing numerically computed results with the exact solution for two test problems derived from another nonlinear dynamical system in two-dimensional space. It is shown that the derived numerical algorithms have a great potential in dealing with the solution of nonlinear chaotic systems and thus can be utilized to delineate different features and characteristics of their solutions.

Chaos, complexity, and random matrices

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an mathcal{O}(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

The design and implementation of hybrid RSA algorithm using a novel chaos based RNG

The number of attempts to break modern encryption methods increases day by day and therefore the need for the design of systems to ensure higher security has occurred. This study aims, an encryption algorithm that combines the strong of asymmetric encryption algorithm and the rich dynamic behaviors of chaotic systems is developed. In this study, firstly a new chaotic system design with high dynamic features is performed and then circuit realization and analyses are made. A chaos based RNG (random number generator) is designed with the help of the new developed chaotic system, NIST and FIPS tests are run. Chaos based hybrid RSA (CRSA) encryption algorithm design in which RNG and RSA algorithms are used together is performed. Text and image encryption is carried out with the algorithm and security analyses of these applications are made. Security analyses results are compared with classical RSA algorithm. It is observed that the developed CRSA algorithm provided better results than RSA algorithm in security tests.

Turing bifurcation analysis for a predator-prey reaction-diffusion system

In this article a physical model of predator-prey reaction-diffusion is considered. The Routh-Hurwitz stability criterion is used to analyze the stability of the dynamical system. Turing instability conditions are derived to investigate the Turing bifurcation. Stability conditions are used to get simulation results for the bifurcation diagram with the variation in diffusion. The effects of parameters, time delay and predator rate, are investigated on the nonlinear behavior of the system. Moreover, the chaotic behavior of the system is discussed through phase portraits and time history. The sensitive dependence of the system on the initial condition is presented graphically through time history maps.

Modified function projective feedback control for time-delay chaotic Liu system synchronization and its application to secure image transmission

This paper is devoted to synchronize time-delay Liu chaotic system in order to develop a secure image transmission system. An adaptive modified function projective synchronization (MFPS) method is proposed to control the behavior of the uncertain Liu chaotic system in the presence of time-delay state variables and uncertainty in system parameters. The stability and convergence of the present method are analyzed by means of Lyapunov stability theorem. Numerical simulations were used to validate the effectiveness of the proposed MFPS method. Furthermore, a novel image cryptosystem is generated for secure image transmission. Image data is embedded within the response signals and is transmitted to the receiver. The image data can be extracted at the receiver by using of the drive signals. The proposed image cryptosystem was applied to a plain image, and also, performance of the image cryptosystem was evaluated by some mathematical tools and the results were satisfactory, in both cases.