Trajectory Attractors Research Articles

Attractors. Then and now

This survey is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, various types of trajectory, pullback and random attractors, exponential attractors, determining functionals and inertial manifolds, as well as the dimension theory for the classes of attractors mentioned above. The theoretical results are illustrated by a number of clarifying examples and counterexamples. Bibliography: 248 titles.

Attractors. Then and now

This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, various types of trajectory, pullback and random attractors, exponential attractors, determining functionals and inertial manifolds, as well as the dimension theory for the classes of attractors mentioned above. The theoretical results are illustrated by a number of clarifying examples and counterexamples. Bibliography: 248 titles.

Semicontinuity of trajectory attractors with respect to exponents for p-Laplacian equation

In this paper, we are concerned with the perturbation of trajectory attractors for dissipative non autonomous p-Laplacian problems with dynamic boundary conditions and no guarantee of the uniqueness of solutions. We prove that under standard conditions, the family of trajectory attractors is upper semicontinuous in the topology of the trajectory space.

Pseudorandom number generator based on novel 2D Hénon-Sine hyperchaotic map with microcontroller implementation.

Recently, chaotic maps have been considered to design pseudorandom number generator (PRNG). However, some chaotic maps present security disadvantages, such as low uniformity and low randomness properties. Nowadays, chaos-based PRNGs are used as the main source for the development of cryptographic algorithms. In this work, to overcome such weaknesses, a novel 2D hyperchaotic map is proposed based on discrete-time feedback by using Hénon map and Sine map. In addition, the dynamics of the hyperchaotic map are enhanced by using the remainder after division function (rem), where better random statistical properties are obtained. A comparison is made between the enhanced Hénon-Sine hyperchaotic map (EHSHM) and the Hénon-Sine hyperchaotic map through Lyapunov exponent analysis, attractor trajectory, histograms and sensitivity at initialization. Then, 8-bit pseudorandom number generator based on the proposed hyperchaotic map (PRNG-EHSHM) is designed and the initial seed of the PRNG is calculated by a secret key of 60 hexadecimal characters. It is implemented in both MATLAB and Arduino Mega microcontroller for experimental results. A complete security analysis is presented from a cryptographic point of view, such as key space, floating frequency, histograms and entropy of the information. Moreover, the randomness is verified with the tests of the National Institute of Standards and Technology (NIST 800-22). Based on the security results obtained, the proposed PRNG-EHSHM can be implemented in embedded cryptographic applications based on chaos.

Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium

This article deals with two-dimensional Navier–Stokes system of equations with rapidly oscillating term in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.

Estimating the time structure of descending activation that generates movements at different speeds.

In targeted movements of the hand, descending activation patterns must not only generate muscle activation but also adjust spinal reflexes from stabilizing the initial to stabilizing the final postural state. We estimate descending activation patterns that change minimally while generating a targeted movement within a given movement time based on a model of the biomechanics, the muscle dynamics, and the stretch reflex. The estimated descending activation patterns predict human movement trajectories quite well. Their temporal structure varies across workspace and with movement speed, from monotonic profiles for slow movements to nonmonotonic profiles for fast movements. Descending activation patterns at different speeds thus do not result from a mere rescaling of invariant templates but reflect varying needs to compensate for interaction torques and muscle dynamics. The virtual attractor trajectories, on which active muscle torques are zero, lie within reachable workspace and are largely invariant when represented in end-effector coordinates. Their temporal structure along movement direction changes from linear ramps to "N-shaped" profiles with increasing movement speed.NEW & NOTEWORTHY The descending activation patterns driving movement must be integrated with spinal reflexes, which would resist movement if left unchanged. We estimate the descending activation patterns at different movement speeds using a model of the stretch reflex and of muscle and limb dynamics. The descending activation patterns we find are temporally structured to compensate for interaction torques as predicted by internal models but also shift the reflex threshold, solving the posture-movement problem.

Trajectory attractors of energy balance climate models with bio-feedback and latent energy flux

Trajectory attractors of energy balance climate models with bio-feedback and latent energy flux

A New Hyperchaotic 4D-FDHNN System with Four Positive Lyapunov Exponents and Its Application in Image Encryption.

In this paper, a hyperchaotic four-dimensional fractional discrete Hopfield neural network system (4D-FDHNN) with four positive Lyapunov exponents is proposed. Firstly, the chaotic dynamics’ characteristics of the system are verified by analyzing and comparing the iterative trajectory diagram, phase diagram, attractor diagram, 0-1 test, sample entropy, and Lyapunov exponent. Furthermore, a novel image encryption scheme is designed to use the chaotic system as a pseudo-random number generator. In the scenario, the confusion phase using the fractal idea proposes a fractal-like model scrambling method, effectively enhancing the complexity and security of the confusion. For the advanced diffusion phase, we proposed a kind of Hilbert dynamic random diffusion method, synchronously changing the size and location of the pixel values, which improves the efficiency of the encryption algorithm. Finally, simulation results and security analysis experiments show that the proposed encryption algorithm has good efficiency and high security, and can resist common types of attacks.

Trajectory Attractors of the 3D Micropolar Equations with Damping Term

Trajectory Attractors of the 3D Micropolar Equations with Damping Term

Design and implementation of a new fractional-order Hopfield neural network system

In this work, a novel chaotic system of fractional-order based on the model of Hopfield Neural Network (HNN) is proposed. The numerical solutions of the 4-neurons-based HNN fractional-order chaotic system are obtained by using the Adomain decomposition method. The dynamical performances of the 4-neurons-based HNN fractional-order chaotic system are explored through attractor trajectories, bifurcation diagrams, Lyapunov exponents, SE complexity and chaotic diagram based on SE complexity. In addition, the 4-neurons-based HNN fractional-order chaotic system is implemented based on the Multisim platform. The experimental results indicate that the 4-neurons-based HNN fractional-order chaotic system has rich dynamic behavior, and the influence of different order on the dynamical properties of the system are particularly great. This research will provide theoretical foundation and experimental basis for the hardware implementation and application of the 4-neurons-based HNN fractional-order chaotic system.

Score Prediction of Sports Events Based on Parallel Self-Organizing Nonlinear Neural Network.

This paper introduces the basic concepts and main characteristics of parallel self-organizing networks and analyzes and predicts parallel self-organizing networks through neural networks and their hybrid models. First, we train and describe the law and development trend of the parallel self-organizing network through historical data of the parallel self-organizing network and then use the discovered law to predict the performance of the new data and compare it with its true value. Second, this paper takes the prediction and application of chaotic parallel self-organizing networks as the main research line and neural networks as the main research method. Based on the summary and analysis of traditional neural networks, it jumps out of inertial thinking and first proposes phase space. Reconstruction parameters and neural network structure parameters are unified and optimized, and then, the idea of dividing the phase space into multiple subspaces is proposed. The multi-neural network method is adopted to track and predict the local trajectory of the chaotic attractor in the subspace with high precision to improve overall forecasting performance. During the experiment, short-term and longer-term prediction experiments were performed on the chaotic parallel self-organizing network. The results show that not only the accuracy of the simulation results is greatly improved but also the prediction performance of the real data observed in reality is also greatly improved. When predicting the parallel self-organizing network, the minimum error of the self-organizing difference model is 0.3691, and the minimum error of the self-organizing autoregressive neural network is 0.008, and neural network minimum error is 0.0081. In the parallel self-organizing network prediction of sports event scores, the errors of the above models are 0.0174, 0.0081, 0.0135, and 0.0381, respectively.

Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines

<p style='text-indent:20px;'>The objective of this paper is to consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. As we know, it is very difficult to obtain the uniqueness of an energy solution for this system even in two dimensions caused by the presence of the strong coupling at the boundary. Thus, we first prove the existence of a trajectory attractor for such system, which is a minimal compact trajectory attracting set for the natural translation semigroup defined on the trajectory space. Furthermore, based on the abstract results (trajectory attractor approach) developed in [<xref ref-type="bibr" rid="b38">38</xref>], we construct trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines.</p>

Pullback attractors for weak solution to modified Kelvin-Voigt model

<p style='text-indent:20px;'>The paper is devoted to the investigation of the qualitative dynamics of weak solutions for the modified Kelvin-Voigt model on the base of the theory of pullback attractors for trajectory spaces. At first, for the studied model, an auxiliary problem is considered, its solvability in the weak sense is proved, and some solution estimates are established. Then, on the base of these estimates, a family of trajectory spaces is determined and the existence of trajectory and minimal pullback attractors for the considered trajectory spaces is proved.</p>

Simulating the interaction of social internet services actors using irregular attractors

The high level of popularity of social Internet services has led to new threats to national security due to information operations in the information space of services. The consequences of manipulating the public opinion of social Internet services users may influence social and political processes in the state, disseminating inter-ethnic ideas and interreligious discord, calls for separatism. As a result of the realization of threats to State information security, the interaction of actors in social Internet services may shift to chaotic dynamics. At the same time, chaotic dynamics can be transformed from social Internet services into real life, which will lead to the fueling of public protests, riots, and other events. The development of effective measures to counter such threats to information security by suppressing chaotic dynamics is based on simulations of virtual communities’ interaction in social Internet services. The paper proposes to formalize the interaction of actors in virtual communities as a system of non-linear differential equations that move to chaotic dynamics, using the Rössler model. Modeling of the interaction of actors in social Internet services under the influence of threats to information security has been carried out, and the conditions for their qualitative change have been defined. Bifurcation diagrams have been constructed to identify critical states of the virtual community of actors in which there are uncertainties regarding the choice of an evolutionary trajectory (attractor). The simulation results made it possible to make recommendations on the system’s operating parameters of differential equations. A reduction of the chaotic dynamics is achieved, and the desired parameters of the state of the information space of service are established.

Strong convergence of attractors of reaction-diffusion system with rapidly oscillating terms in an orthotropic porous medium

A system of reaction-diffusion equations in a perforated domain with rapidly oscillating terms in the equations and in the boundary conditions is considered. It is not assumed that the uniqueness theorem conditions are satisfied for the corresponding initial-boundary value problem. We have proved the strong convergence of the trajectory attractors of this system to the trajectory attractors of the homogenized reaction-diffusion system with a ‘strange term’ (potential).

S-, P- and R-striations as attractors for electron phase trajectories in spatially periodic resonance fields

A new point of view on the appearance of S-, P- and R-striations in a positive column of inert gases is proposed, based on a dynamic analysis of the resonance properties of electron phase trajectories in spatially periodic fields. The positive column may be considered as a resonator containing a set of resonant modes. Like a tuning fork, being disturbed, it responds with one of the modes, in particular with of S-, P-, or R-modes or striations, depending on the discharge conditions. The dynamic approach eliminates the difficulties of the kinetic theory associated with the long length of the solution of Boltzmann equation, which is much greater than the length of the positive column.

Homogenization of Attractors of Reaction–Diffusion System with Rapidly Oscillating Terms in an Orthotropic Porous Medium

In a perforated domain, we consider a reaction-diffusion system with rapidly oscillating terms in the equations and boundary conditions. No Lipschitz condition is imposed, so the uniqueness of a solution to the corresponding initial-boundary value problem is not guaranteed. We prove that the trajectory attractors of the system weakly converge to the trajectory attractors of the homogenized reaction-diffusion systems with a strange term (potential).

Image encryption algorithm based on 2D hyperchaotic map

It is well known that chaos is unpredictability and sensitivity to initial values and parameters. It is a best choice for cryptography application. However, the existing chaotic cryptography systems have many defects because of the chaotic maps that without complexity dynamic performances. To improve these weaknesses, we introduced a 2D hyperchaotic map through two 1D-chaotic maps, linear function and represents multiplier. The dynamic performances of the 2D hyperchaotic map are analyzed by attractor trajectory, 0–1 test, bifurcation diagram, Lyapunov exponents and Permutation entropy. The results illustrate that 2D hyperchaotic map has more complexity dynamic characteristics and randomness, which indicates that 2D hyperchaotic map is more suitable for image encryption algorithm. Therefore, an image encryption algorithm is introduced based on 2D hyperchaotic map. In encryption algorithm, based on 2D hyperchaotic map, the image is scrambled by row shift and column shift, then pixel values are diffused through forward and backward diffusion, finally, the ciphertext image is obtained. The simulation results show that the proposed image encryption algorithm is reliable. Moreover, the security analysis indicate that the introduced algorithm has a better security performances than several image encryption algorithms. This algorithm is not absolutely safe and efficient, and does not limit the improvement and optimization of the algorithm. However, this work provide a theoretical basis in chaotic theory and chaotic image encryption algorithm.

Trajectory and global attractors for a modified Kelvin–Voigt model

Trajectory and global attractors for a modified Kelvin–Voigt model

An adaptive damage detection method based on differential evolutionary algorithm for beam structures

An adaptive singular value decomposition (SVD) method is proposed to detect damage based on the differential evolutionary algorithm for beam structures. B-spline wavelet on the interval finite element method is applied to model the damaged structures. The modal shape data obtained by modal analysis are used to construct attractor trajectory matrix, and SVD-based method is adopted to detect damage locations. In order to accommodate to different damage cases, differential evolutionary algorithm (DE) is used to adaptively optimize the parameters of SVD-based method. Both simulation and experiment researches are set up to verify the proposed method. The results indicate that the presented method is accurately effective to identify the damage for beam structures.