Weak Attractor Research Articles

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.

A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit

As a new type of nonlinear electronic component, a memristor can be used in a chaotic system to increase the complexity of the system. In this paper, a flux-controlled memristor is applied to an existing chaotic system, and a novel five-dimensional chaotic system with high complexity and hidden attractors is proposed. Analyzing the nonlinear characteristics of the system, we can find that the system has new chaotic attractors and many novel quasi-periodic limit cycles; the unique attractor structure of the Poincaré map also reflects the complexity and novelty of the hidden attractor for the system; the system has a very high complexity when measured through spectral entropy. In addition, under different initial conditions, the system exhibits the coexistence of chaotic attractors with different topologies, quasi-periodic limit cycles, and chaotic attractors. At the same time, an interesting transient chaos phenomenon, one kind of novel quasi-periodic, and weak chaotic hidden attractors are found. Finally, we realize the memristor model circuit and the proposed chaotic system use off-the-shelf electronic components. The experimental results of the circuit are consistent with the numerical simulation, which shows that the system is physically achievable and provides a new option for the application of memristive chaotic systems.

Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise

This paper is concerned with the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on Rn. We first prove the well-posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors. We finally establish the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains.

Size of the set of attractors for iterated function systems

Abstract We discuss the smallness of the set of attractors for iterated function systems. This paper is an attempt to measure the difference between the family of IFS attractors and a broader family, the set of attractors for the weak iterated function systems. We prove that the IFS attractors form a σ-porous subset of K([0, 1]d), the family of compact subsets of a metric space [0, 1]d. We also show that weak IFS attractors form a first category set.

Dynamics of stochastic reaction–diffusion lattice systems driven by nonlinear noise

This paper is concerned with the dynamics of the stochastic reaction–diffusion lattice systems defined on the entire integer set driven by locally Lipschitz nonlinear noise. We prove the existence and uniqueness of solutions as well as weak pullback random attractors for the mean random dynamical systems generated by the solution operators. The existence of invariant measures in ℓ2 for the stochastic equations is also established. The idea of uniform estimates on the tails of the solutions is employed to show the tightness of a family of probability distributions of the solutions.

Properties of Stationary Statistical Solutions of the Three-Dimensional Navier–Stokes Equations

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier–Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray–Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.

Connectedness of random set attractors

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynamical system we prove connectedness of the pullback attractor on a connected space. Additionally, we provide an example of a weak random set attractor of a random dynamical system with even more restrictive continuity assumptions on an even path-connected space which even attracts all bounded sets and which is not connected. On the way to proving connectedness of a pullback attractor we prove a lemma which may be of independent interest and which holds without the assumption that the state space is connected. It states that even though pullback convergence to the attractor allows for exceptional nullsets which may depend on the compact set, these nullsets can be chosen independently of the compact set (which is clear for \sigma -compact spaces but not at all clear for spaces which are not \sigma -compact).

Random attractor for the 3D viscous primitive equations driven by fractional noises

We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15,16]). The main idea of our method is that we first derive that P-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C([0,T];(H1(℧))3), in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space (H1(℧))3 to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space (H2(℧))3. However, time-uniform a priori estimates in (H2(℧))3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations.

Quantifying wildlife watchers’ preferences to investigate the overlap between recreational and conservation value of natural areas

AbstractNature‐based recreation substantially benefits human wellbeing, for example, by improving physical and mental health. However, recreation can also have severe ecological impacts. The recreational value of landscapes and natural areas is often used to generate support for public spending in conservation. However, we still don't know whether nature‐based recreationists place greater recreational value on natural areas that have high conservation value compared to other green spaces.Here, we determine which attributes of nature‐based tourism provide recreational services. We used pictures of wildlife posted on Flickr to quantify wildlife watching activities in Scotland. We then determined the environmental variables key to attracting wildlife watchers to a destination, such as protected areas (PAs), the perceived naturalness, and the presence of different types of infrastructure.Infrastructure best predicts the intensity of wildlife watching activities in Scotland, while areas of high natural value are rarely used. PAs are weak attractors of wildlife watchers, with PAs designated to protect threatened habitats or species having low recreational value. In accessible and highly visited areas, higher biodiversity increases the intensity of wildlife watching activities.Synthesis and applications. Areas of high natural and conservation value and areas of high recreational value do not tend to overlap. Recreational ecosystem services are mainly provided by the wider countryside and highly transformed landscapes, as opposed to wild ecosystems and protected areas designated to protect environmental features of high conservation value. These results question the synergy between the goals of recreation and those of conservation and the use of recreation as a justification for economic investment in conservation. During wildlife watching activities most people experience an urbanised, highly transformed nature; it is important to determine how this human‐dominated nature can influence support for conservation of wild and remote areas.

Weak Pullback Attractors for Mean Random Dynamical Systems in Bochner Spaces

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.

Minimal random attractors

It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary.

Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models

We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend a relatively long time visiting the attractor’s ruins.

Determining modes for the 3D Navier–Stokes equations

We introduce a determining wavenumber for the forced 3D Navier–Stokes equations (NSE) defined for each individual solution. Even though this wavenumber blows up if the solution blows up, its time average is uniformly bounded for all solutions on the weak global attractor. The bound is compared to Kolmogorov’s dissipation wavenumber and the Grashof constant.

Noise dependent synchronization of a degenerate SDE

We provide an example of an SDE with degenerate additive noise where synchronization depends on the strength of noise and the number of directions in which the noise acts. Here, synchronization means that the weak random attractor consists of a single random point. Indicated by a change of sign of the top Lyapunov exponent, we prove synchronization and no (weak) synchronization respectively.

Fixed points of symmetric complex-valued Hopfield neural networks

Abstract A complex-valued Hopfield neural network (CHNN) is a model of a multistate Hopfield neural network, and has been applied to the storage of multilevel data. Weak noise tolerance, however, is a disadvantage of CHNNs. Symmetric CHNNs (SCHNNs), modified CHNNs, improve the noise tolerance of CHNNs. In the present work, we study the global and local minima of SCHNNs with one training pattern. In CHNNs, the global minima are the training and rotated patterns, and there are no local minima. In SCHNNs, it has been hard to determine all the global and local minima. It is thought that the global minima are the training and reversed patterns, and that there are no local minima in most cases. In the present work, however, we find many local minima, and show that they are very weak attractors, which reduce noise tolerance very little. In addition, we determine all the global minima of SCHNNs.

Ionizing radiation augments glioma tropism of mesenchymal stem cells.

OBJECTIVE Mesenchymal stem cells (MSCs) have been shown to localize to gliomas after intravascular delivery. Because these cells home to areas of tissue injury, the authors hypothesized that the administration of ionizing radiation (IR) to tumor would enhance the tropism of MSCs to gliomas. Additionally, they sought to identify which radiation-induced factors might attract MSCs. METHODS To assess the effect of IR on MSC migration in vitro, transwell assays using conditioned medium (CM) from an irradiated commercially available glioma cell line (U87) and from irradiated patient-derived glioma stem-like cells (GSCs; GSC7-2 and GSC11) were employed. For in vivo testing, green fluorescent protein (GFP)-labeled MSCs were injected into the carotid artery of nude mice harboring orthotopic U87, GSC7-2, or GSC17 xenografts that were treated with either 0 or 10 Gy of IR, and brain sections were quantitatively analyzed by immunofluorescence for GFP-positive cells. These GSCs were used because GSC7-2 is a weak attractor of MSCs at baseline, whereas GSC17 is a strong attractor. To determine the factors implicated in IR-induced tropism, CM from irradiated GSC7-2 and from GSC11 was assayed with a cytokine array and quantitative ELISA. RESULTS Transwell migration assays revealed statistically significant enhanced MSC migration to CM from irradiated U87, GSC7-2, and GSC11 compared with nonirradiated controls and in a dose-dependent manner. After their intravascular delivery into nude mice harboring orthotopic gliomas, MSCs engrafted more successfully in irradiated U87 (p = 0.036), compared with nonirradiated controls. IR also significantly increased the tropism of MSCs to GSC7-2 xenografts (p = 0.043), which are known to attract MSCs only poorly at baseline (weak-attractor GSCs). Ionizing radiation also increased the engraftment of MSCs in strong-attractor GSC17 xenografts, but these increases did not reach statistical significance. The chemokine CCL2 was released by GSC7-2 and GSC11 after irradiation in a dose-dependent manner and mediated in vitro transwell migration of MSCs. Immunohistochemistry revealed increased CCL2 in irradiated GSC7-2 gliomas near the site of MSC engraftment. CONCLUSIONS Administering IR to gliomas enhances MSC localization, particularly in GSCs that attract MSCs poorly at baseline. The chemokine CCL2 appears to play a crucial role in the IR-induced tropism of MSCs to gliomas.

Chaotic dynamics of large-scale structures in a turbulent wake

The dynamics of a 3D bimodal turbulent wake downstream a square-back Ahmed body are experimentally studied in a wind-tunnel through high-frequency wall pressure probes mapping the rear of the model and a horizontal 2D velocity field. The barycenters of the pressure distribution over the rear part of the model and the intensity recirculation are found highly correlated. Both described the most energetic large-scale structures dynamics, confirming the relation between the large-scale recirculation bubble and its wall pressure foot-print. Focusing on the pressure, its barycenter trajectory has a stochastic behavior but its low frequencies dynamics exhibit the same characteristics as a weak strange chaotic attractor system, with two well defined attractors. The low frequencies dynamics associated to the large-scale structures are then analyzed. The largest Lyapunov exponent is first estimated, leading to a low positive value characteristic of strange attractors and weak chaotic systems. Afterwards, analyzing the autocorrelation function of the time-series, we compute the correlation dimension, larger than two. The signal is finally transformed and analyzed as a telegraph signal showing that its dynamics correspond to a quasi-random telegraph signal. This is the first demonstration that the low frequencies dynamics of a turbulent 3D wake are not a purely stochastic process but rather a weak chaotic process exhibiting strange attractors. From the flow-control point of view, it also opens the path to more simple closed-loop flow control strategies aiming at the stabilization of the wake and the control of the dynamics of the wake barycenter.

Synchronization by noise for order-preserving random dynamical systems

We provide sufficient conditions for weak synchronization/stabilization by noise for order-preserving random dynamical systems on Polish spaces. That is, under these conditions we prove the existence of a weak point attractor consisting of a single random point. This generalizes previous results in two directions: First, we do not restrict to Banach spaces, and second, we do not require the partial order to be admissible nor normal. As a second main result and application, we prove weak synchronization by noise for stochastic porous media equations with additive noise.

Weak solutions of the Shigesada–Kawasaki–Teramoto equations and their attractors

We derive the global existence of weak solutions of the Shigesada–Kawasaki–Teramoto systems in any space dimension d≥1 with a rather general condition on the coefficients. The existence is established using finite differences in time with truncations and an argument of Stampacchia’s maximum principle to show the positivity of the solutions. We derive also the existence of a weak global attractor.

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.