Stochastic Chaotic Systems Research Articles

Fractal–fractional order stochastic chaotic model: A synchronization study

This manuscript provides the existence and uniqueness of solution and synchronization of fractal dimensional fractional order stochastic chaotic systems in Rn space. Based on fractal–fractional analysis, Rn stochastic settings, fixed point rout, stability theory and suitable feedback controller, a valuable adequate useful sufficient conditions are determined for existence and uniqueness of solution and synchronization study of proposed model. We have addressed the nonlinear terms of the systems under consideration in this study by using local assumptions. The theory’s findings are then validated by a numerical simulation.

Numerical treatments for some stochastic–deterministic chaotic systems

We aim in this paper to propose classes of nonlinear ordinary differential equations based on the concept of piecewise differentiation. These classes are known as piecewise deterministic–stochastic differential equations or vice versa. Two deterministic–stochastic chaotic systems will be presented in this paper. These models are the hyperchaotic finance model and Bloch system with time delay, where the hybrid fractional derivatives are used. Existence, uniqueness and stability analysis are presented. In order to study the behavior of these models, three different cases for each model are presented. Moreover, two numerical methods are introduced to study these models. These methods are the nonstandard Milstein method to solve the stochastic models and Grünwald–Letnikov nonstandard finite difference to solving the hybrid fractional order deterministic models. Comparative study and numerical simulation are carried out.

SYNCHRONIZATION OF THE RÖSSLER-LORENZ SYSTEMS WITH FRACTIONAL BROWNIAN MOTION

<abstract> The current paper is devoted to the dynamics of stochastic chaotic systems with fractional Brownian motion with $H\in(\frac{1}{2}, 1)$. The existence and uniqueness of the so-called stochastic Rössler-Lorenz system driven by fractional Brownian motion is established. Moreover, the stochastic synchronization of stochastic Rössler-Lorenz system is proved, and some numerical simulations are provided to verify the theoretical results. </abstract>

Regularity of fractional stochastic convolution and its application to fractional stochastic chaotic systems

The current paper is devoted to the regularity of Caputo-type fractional Onstein–Ulenbeck processes with fractional Brownian motion, and it is applied to establish the existence and uniqueness of the mild solution for Caputo-type fractional Rössler equation and fractional Chen equation with fractional Brownian motion. Some numerical simulations and examples are provided to support the theoretical results.

Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems

This paper proposes a new pathwise sensitivity estimator for chaotic SDEs. By introducing a spring term between the original and perturbated SDEs, we derive a new estimator by importance sampling. The variance of the new estimator increases only linearly in time $T,$ compared with the exponential increase of the standard pathwise estimator. We compare our estimator with the Malliavin estimator and extend both of them to the Multilevel Monte Carlo method, which further improves the computational efficiency. Finally, we also consider using this estimator for the SDE with small volatility to approximate the sensitivities of the invariant measure of chaotic ODEs. Furthermore, Richardson-Romberg extrapolation on the volatility parameter gives a more accurate and efficient estimator. Numerical experiments support our analysis.

An On-Line Tracker for a Stochastic Chaotic System Using Observer/Kalman Filter Identification Combined with Digital Redesign Method

This is the first paper to present such a digital redesign method for the (conventional) OKID system and apply this novel technique for nonlinear system identification. First, the Observer/Kalman filter Identification (OKID) method is used to obtain the lower-order state-space model for a stochastic chaos system. Then, a digital redesign approach with the high-gain property is applied to improve and replace the observer identified by OKID. Therefore, the proposed OKID combined with an observer-based digital redesign novel tracker not only suppresses the uncertainties and the nonlinear perturbations, but also improves more accurate observation parameters of OKID for complex Multi-Input Multi-Output systems. In this research, Chen’s stochastic chaotic system is used as an illustrative example to demonstrate the effectiveness and excellence of the proposed methodology.

Hybrid control of stochastic chaotic system based on memristive Lorenz system with discrete and distributed time‐varying delays

This study investigates the problem of hybrid control of stochastic chaotic system based on memristive Lorenz system with discrete and distributed time-varying delays. First of all, the stochastic switched chaotic system is proposed based on Lorenz system modelled by piecewise linear memristor. Then, a novel hybrid controller is designed and the criteria are established to guarantee that the trivial solution of the corresponding stochastic system is exponentially stable in mean square. Furthermore, a novel technique is proposed to handle the problem of the difference, which cannot be depicted directly by stochastic Newton–Leibniz formula, between two state values of different instants between which there may exist impulse instants. Finally, an example is given to illustrate the efficiency and feasibility of the presented method.

Complexity and synchronization in stochastic chaotic systems

We investigate the complexity of a hyperchaotic dynamical system perturbed by noise and various nonlinear speech and music signals. The complexity is measured by the weighted recurrence entropy of the hyperchaotic and stochastic systems. The synchronization phenomenon between two stochastic systems with complex coupling is also investigated. These criteria are tested on chaotic and perturbed systems by mean conditional recurrence and normalized synchronization error. Numerical results including surface plots, normalized synchronization errors, complexity variations etc show the effectiveness of the proposed analysis.

Robust Adaptive Exponential Synchronization of Two Different Stochastic Perturbed Chaotic Systems with Structural Perturbations

The robust adaptive exponential synchronization problem of stochastic chaotic systems with structural perturbations is investigated in mean square. The stochastic disturbances are assumed to be Brownian motions that act on the slave system and the norm-bounded uncertainties exist in all parameters after decoupling. The stochastic disturbances could reflect more realistic dynamical behaviors of the coupled chaotic system presented within a noisy environment. By using a combination of the Lyapunov functional method, the robust analysis tool, the stochastic analysis techniques, and adaptive control laws, we derive several sufficient conditions that ensure the coupled chaotic systems to be robustly exponentially synchronized in the mean square for all admissible parameter uncertainties. This approach cannot only make the outputs of both master and slave systems reach H∞ synchronization with the passage of time between both systems but also attenuate the effects of the perturbation on the overall error system to a prescribed level. The main results are shown to be general enough to cover many existing ones reported in the literature.

Adaptive Modified Function Projective Lag Synchronization for Two Different Chaotic Systems with Stochastic Unknown Parameters

In this paper, we investigate the modified function projective lag synchronization for two different stochastic chaotic systems using adaptive control method. We design an adaptive controller to make the mean square of synchronization error convergence to an arbitrarily small bound around zero depending on the controller feedback gain according to the Lyapunov stability theory. One example is presented to demonstrate the effectiveness of the proposed controller.

Lyapunov-Based Controller for a Class of Stochastic Chaotic Systems

This study presents a general control law based on Lyapunov’s direct method for a group of well-known stochastic chaotic systems. Since real chaotic systems have undesired random-like behaviors which have also been deteriorated by environmental noise, chaotic systems are modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of Wiener process which eventually generates an Ito differential equation. Proposed controller not only can asymptotically stabilize these systems in mean-square sense against their undesired intrinsic properties, but also exhibits good transient response. Simulation results highlight effectiveness and feasibility of proposed controller in outperforming stochastic chaotic systems.

Adaptive Synchronization for Two Different Stochastic Chaotic Systems with Unknown Parameters via a Sliding Mode Controller

This paper investigates the problem of synchronization for two different stochastic chaotic systems with unknown parameters and uncertain terms. The main work of this paper consists of the following aspects. Firstly, based on the Lyapunov theory in stochastic differential equations and the theory of sliding mode control, we propose a simple sliding surface and discuss the occurrence of the sliding motion. Secondly, we design an adaptive sliding mode controller to realize the asymptotical synchronization in mean squares. Thirdly, we design an adaptive sliding mode controller to realize the almost surely synchronization. Finally, the designed adaptive sliding mode controllers are used to achieve synchronization between two pairs of different stochastic chaos systems (Lorenz-Chen and Chen-Lu) in the presence of the uncertainties and unknown parameters. Numerical simulations are given to demonstrate the robustness and efficiency of the proposed robust adaptive sliding mode controller.

Generalized Synchronization of Stochastic Discrete Chaotic System with Poisson Distribution Coefficient

This paper addresses the generalized synchronization of stochastic discrete chaotic systems with Poisson distribution coefficient. Firstly, based on the orthogonal polynomial approximation theory of discrete random function in Hilbert spaces, the discrete chaotic system with random parameter is transformed into its equivalent deterministic system. Secondly, a general method for the generalized synchronization of discrete chaotic system with random parameter is presented by Lyapunov stability theory and contraction theorem. Finally, two synchronization examples numerically illustrated that the proposed control scheme is effective for any stochastic discrete system.

Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential Equations

Dynamical aspects of initial/boundary value problems for ordinary differential equations have become a rapidly growing area of research in the theory of differential equations and dynamical systems and have gathered substantial research interests during the last decades. The attractiveness of this field not only is derived from theoretical interests but also is motivated by the insights that such dynamical aspects could reveal in several phenomena observed in applied sciences. The current special issue places its emphasis on the study of the dynamical aspects of initial/boundary value problems for ordinary differential equations. Call for papers has been carefully prepared by the guest editors and posted on the journals web page, which has attracted many researchers to submit their contribution on wide topics such as oscillation theory, delay differential equation, impulsive differential equation, multipoint boundary value problems, stochastic mutualism system, chaotic system, homoclinic solutions, Hamiltonian systems, stability and bifurcation, exponential extinction, singular elliptic problem, nonuniform exponential contraction and dichotomy. All manuscripts submitted to this special issue went through a thorough peer-refereeing process. Based on the reviewers’ reports, we collect twenty-five original research articles by more than fifty active international researchers in differential equations and from different countries such as Korea, China, Malaysia, Singapore, Czech Republic, Turkey, Slovenia, India, and USA. Besides, one survey on recent results for the existence of singular periodic problems is also contained. It is certainly impossible to provide in this short editorial note a more comprehensive description for all articles in this special issue. However, the team of the guest editors believes that the results included reflect some recent trends in research and outline new ideas for future studies of dynamical aspects of initial/boundary value problems for ordinary differential equations.

Adaptive Q–S synchronization between coupled chaotic systems with stochastic perturbation and delay

This work investigates the adaptive Q–S synchronization of coupled chaotic (or hyper-chaotic) systems with stochastic perturbation, delay and unknown parameters. The sufficient conditions for achieving Q–S synchronization of two stochastic chaotic systems are derived based on the invariance principle of stochastic differential equation. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the stochastic Q–S synchronization of non-identical chaotic (or hyper-chaotic) systems is to be obtained. Finally, two illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.

Hidden imperfect synchronization of wall turbulence

Instantaneous amplitude and phase concept emerging from analytical signal formulation is applied to the wavelet coefficients of streamwise velocity fluctuations in the buffer layer of a near wall turbulent flow. Experiments and direct numerical simulations show both the existence of long periods of inert zones wherein the local phase is constant. These regions are separated by random phase jumps. The local amplitude is globally highly intermittent, but not in the phase locked regions wherein it varies smoothly. These behaviors are reminiscent of phase synchronization phenomena observed in stochastic chaotic systems. The lengths of the constant phase inert (laminar) zones reveal a type I intermittency behavior, in concordance with saddle-node bifurcation, and the periodic orbits of saddle nature recently identified in Couette turbulence. The imperfect synchronization is related to the footprint of coherent Reynolds shear stress producing eddies convecting in the low buffer.

Four twins for a paradox: On “sensitive” twins and the biological counterpart of the “twin paradox”

Monozygotic twin (MZT) epigenetic development, i.e., aging, diverges largely in time despite the initially very small genetic differences between MZTs. This fact is interpreted as a "sensitivity to initial conditions" phenomenon, a common property of either deterministic or stochastic chaotic systems. Some of the biotheoretical implications stemming from this empirical observation are briefly discussed here, while an actual measure of MZT epigenetic time divergence is given through an estimate of the (Stochastic) Lyapunov exponents (LEs) (i.e., the rate of exponential time divergence). These results suggest a reconsideration of the Langevin-Einstein thought experiment known as the "twin paradox." At least four twins are necessary in order to take into account the inertially independent divergent aging described here. Alternatively, LE estimates, like those given here, should be used. Finally suggested in the actual special-relativity experiments is the replacement of clocks with some nonlinear (chaotic) forced oscillator.

Study on phase synchronization of stochastic chaotic system

This paper detects and characterizes the diverse roles played by bounded noise in chaotic phase synchronization (CPS) of weakly coupled nonlinear stochastic systems. Analysis of a paradigmatic model of two bidirectional coupled three-level food chains is carried out by various statistical measures such as Shannon entropy and mutual information. The results indicate that inside the synchronous regime, CPS is considerably reduced under the influence of bounded noise; near the onset of phase synchronization, temporal phase locking is diversely changed with the increase of noise, i.e., either weak or strong noise also degrades the degree of CPS, while intermediate noise enhances CPS remarkably, and an optimal noise intensity is detected that maximizes the enhancement.

Lower-order state-space self-tuning control for a stochastic chaotic hybrid system

In this paper, subject to acceptable closed-loop performance, an effective lower-order tuner for a stochastic chaotic hybrid system is designed using the observer/Kalman filter identification (OKID) method, in which the system state in a general coordinate form is transformed to one in an observer form. The OKID method is a time-domain technique that identifies a discrete input–output map by using known input–output sampled data in the general coordinate form, through an extension of the eigensystem realization algorithm. Moreover, it provides a lower-order realization of the tracker, with computationally effective initialization, for on-line “auto-regressive moving average process with exogenous model” -based identification and a lower-order state-space self-tuning control technique. Finally, the chaotic Chen's system is used as an illustrative example to demonstrate the effectiveness of the proposed methodology.

Equivalent linear observer-based tracker for stochastic chaotic system with delays and disturbances

Equivalent linear observer-based tracker for stochastic chaotic system with delays and disturbances