Dimension Of Attractor Research Articles

Global attractors for a partially damped Timoshenko–Ehrenfest system without the hypothesis of equal wave speeds

This paper is concerned with the study of global attractors for a new semilinear Timoshenko–Ehrenfest type system. Firstly we establish the well-posedness of the system using Faedo–Galerkin method. By considering only one damping term acting on the vertical displacement, we prove the existence of a smooth finite dimensional global attractor using the recent quasi-stability theory. Our results holds for any parameters of the system.

Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval

Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.

Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed.

Dynamic of the nonclassical diffusion equation with memory

In this paper, we consider the nonclassical diffusion equation with memory and the nonlinearity of the polynomial growth condition of arbitrary order in the time-dependent space. First, the well-posedness of the solution for the equation is obtained in the time-dependent space mathscr{U}_{t}. Then, we establish the existence and regularity of the time-dependent global attractor. Finally, we also conclude that the fractal dimension of the time-dependent attractor is finite.

On attractors in dynamical systems modeling genetic networks

The system of ordinary differential equations that arises in the theory of genetic networks is studied. Attracting sets of a special kind is the focus of the study. These attractors appear as combinations of attractors of lower dimensions,
 which are stable limit cycles. The properties of attractors are studied. Visualizations and examples are provided.

Multi-scroll chaotic attractors with multi-wing via oscillatory potential wells

The oscillatory potential well can cause the mass points in it to move chaotically, which can be considered as a physical mechanism of chaos generation. Based on this physical mechanism, the oscillatory multi-well potential with multi-concave is used to generate controllable multi-scroll chaotic attractors with multi-wing. These attractors have two kinds of topological units: scroll and wing. The topological structure of the attractor depends on the shape of the potential well, that is, the wells and the concaves correspond to the scrolls and wings of the attractor. By constructing potential wells with different shapes, we can obtain attractors with different topological structure. The number of scrolls and wings of the chaotic attractor can be controlled by the oscillation amplitude of the potential well and damping coefficient, that is, stronger oscillation or smaller damping allows the mass points to enter the higher potential wells or concaves, resulting in attractors with more scrolls or wings. As the number of scrolls or wings increases, the attractors become more complex, as expressed by Poincaré maps and quantified by the Kaplan-Yorke dimensions. Kaplan-Yorke dimensions of attractors can be adjusted continuously from 2 to 3 by the two control parameters. Finally, the results are confirmed by Multisim simulation circuit and actual analog circuit.

Direct Path from Turbulence to Time-Periodic Solutions.

Viscous flows through pipes and channels are steady and ordered until, with increasing velocity, the laminar motion catastrophically breaks down and gives way to turbulence. How this apparently discontinuous change from low- to high-dimensional motion can be rationalized within the framework of the Navier-Stokes equations is not well understood. Exploiting geometrical properties of transitional channel flow we trace turbulence to far lower Reynolds numbers (Re) than previously possible and identify the complete path that reversibly links fully turbulent motion to an invariant solution. This precursor of turbulence destabilizes rapidly with Re, and the accompanying explosive increase in attractor dimension effectively marks the transition between deterministic and de facto stochastic dynamics.

Statistical performance of local attractor dimension estimators in non-Axiom A dynamical systems.

We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system's global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth's climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.

Analysis of separation properties of attractors of the product of fuzzy iterated function systems

The theory of fractals mainly deals with analysis of highly irregular shapes having infinitely complex patterns in all scales. Mathematically, fractal geometry uses iteration of different functions defined on the same space to generate different irregular shapes. Fuzzy fractal is indeed an extensive mathematical combination. The separation properties of finite product of fractals formed by repeated iterations of systems consisting of fuzzy contractions in the standard fuzzy metric space with regard to the standard Hausdorff fuzzy metric is the primary topic of this work. Characterizations for the product of fuzzy iterated function systems to become fuzzy disconnected and fuzzy overlapping are obtained. Some mandatory requirements for the product of fuzzy iterated function systems to be fuzzy incident are also given. Along with that, the separation properties of finite product of fuzzy iterated function systems with condensation are attempted.

Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical equations

Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical equations

Fractal Dimension of Random Attractor for a Stochastic Lattice System with White Noise

Fractal Dimension of Random Attractor for a Stochastic Lattice System with White Noise

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.

NmODE: neural memory ordinary differential equation

Brain neural networks are regarded as dynamical systems in neural science, in which memories are interpreted as attractors of the systems. Mathematically, ordinary differential equations (ODEs) can be utilized to describe dynamical systems. Any ODE that is employed to describe the dynamics of a neural network can be called a neuralODE. Inspired by rethinking the nonlinear representation ability of existing artificial neural networks together with the functions of columns in the neocortex, this paper proposes a theory of memory-based neuralODE, which is composed of two novel artificial neural network models: nmODE and epsilon-net, and two learning algorithms: nmLA and epsilon-LA. The nmODE (neural memory Ordinary Differential Equation) is designed with a special structure that separates learning neurons from memory neurons, making its dynamics clear. Given any external input, the nmODE possesses the global attractor property and is thus embedded with a memory mechanism. The nmODE establishes a nonlinear mapping from the external input to its associated attractor and does not have the problem of learning features homeomorphic to the input data space, as occurs frequently in most existing neuralODEs. The nmLA (neural memory Learning Algorithm) is developed by proposing an interesting three-dimensional inverse ODE (invODE) and has advantages in memory and parameter efficiency. The proposed epsilon-net is a discrete version of the nmODE, which is particularly feasible for digital computing. The proposed epsilon-LA (epsilon learning algorithm) requires no prior knowledge of the number of network layers. Both nmLA and epsilon-LA have no problem with gradient vanishing. Experimental results show that the proposed theory is comparable to state-of-the-art methods.

Ramified continua as global attractors of C 1-smooth self-maps of a cylinder close to skew products

In the paper we consider -smooth self-maps of a cylinder close to -smooth skew products (and satisfying some additional conditions). We study such geometric property of the maps, as existence of -smooth invariant local lamination, and apply this geometric property to the proof of the geometric integrability of maps under consideration. Using obtained results we construct the example of the family of -smooth maps close to skew products so that each map from this family admits the global attractor, which is a one-dimensional ramified continuum with a complicated topological structure. The global attractor of every map from the family under consideration consists of arcs of two types. On the unique circle (which is the arc of first type) the map is mixing; on arcs of second type of different lengths homeomorphic to a closed interval (the family of such arcs has continuum cardinality) the map is not mixing. The topological structure of the global attractor and dynamical properties of trajectories on the attractor lead to the property of dense intermittency (in the complement to the attractor) of attraction sets of different ω-limit sets, the union of which coincides with the global attractor.

On the interaction between liquid slug and vapor bubble in the chaotic operation of pulsating heat pipe

A pulsating heat pipe has shown promising results in heat transfer enhancement for several industrial applications. Despite having multiple advantages in the design and physics-based parameters, its applications are still limited due to the lack of understanding of the dynamics of the chaotic interaction of the liquid slug and vapor bubble. The randomly distributed liquid slug and vapor bubble start oscillating upon applying the heat at the evaporator section. A transition from the self-sustained oscillatory to chaotic operation in the pulsating heat pipe has been observed in various experimental studies. However, there are only a few works explaining the chaotic interaction of liquid slugs and vapor bubbles. We present the analysis of the oscillatory behavior of the position of the liquid slug in the pulsating heat pipe using a nonlinear mathematical model. The identification of various operating regions in the parameter space using the bifurcation analysis reveals the presence of a route to chaos by the period-doubling bifurcation. The fast Fourier transform of the temporal evolutions confirms the presence of the period doubling phenomena. Furthermore, the Lyapunov exponent and the correlation dimension are used to detect chaos and quantify the dimension of the chaotic attractor, respectively. Moreover, the Hurst exponent is used to determine the persistency of the oscillations, which indicates that the oscillations are weakly persistent in the chaotic regime. The transition to the chaotic regime is analyzed by period doubling, and a change in the frequency beyond period-doubling shows a gradual shift to the chaotic operation.

Exploring the limits to our understanding of whether fish feel pain.

Exploring the limits to our understanding of whether fish feel pain.

On a Cahn–Hilliard–Oono model for image segmentation

This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as α → 0. The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor.

Hybrid α-attractors, primordial black holes and gravitational wave backgrounds

We investigate the two-stage inflation regime in the theory of hybrid cosmological α-attractors. The spectrum of inflationary perturbations is compatible with the latest Planck/BICEP/Keck Array results, thanks to the attractor properties of the model. However, at smaller scales, it may have a very high peak of controllable width and position, leading to a copious production of primordial black holes (PBH) and generation of a stochastic background of gravitational waves (SGWB).

Global stability of the Euler- Bernoulli beams excited by multiplicative white noises

This paper considers the global stability of the Euler-Bernoulli beam excited by multiplicative white noise. Based on the theory of global random attractors, the Hausdorff dimensions of the global random attractors for the system is obtained. According to the relationship between Hausdorff dimensions and global Lyapunov exponents, the global stability of the stochastic beam is derived.

Global attractors and synchronization of coupled critical Lamé systems with nonlinear damping

We investigate the global attractors and synchronization phenomenon of a coupled critical Lamé system defined on a smooth bounded domain Ω⊂R3 with nonlinear damping and nonlinear forces of critical growth. The existence of a unique and finitely dimensional global attractor Aϰ is proved in the natural energy space H=[D((−Δe)12)]2×[(L2(Ω))3]2, where ϰ is the coupling parameter and Δe is the Lamé operator. This attractor is further proved to be smooth in the regular space [(H2(Ω))3∩(H01(Ω))3]2×[(H01(Ω))3]2. We also show that the coupled Lamé system can be reduced to a single one when ϰ tends to infinity. Then we can compare dynamics of the coupled and single systems by proving the upper-semicontinuity of their attractors in Hδ=[(H2−δ(Ω))3∩(H01−δ(Ω))3]2×[(H01−δ(Ω))3]2 for any δ∈(0,1) as ϰ→∞. These results are finally used to study the asymptotic and exponential synchronization phenomena for the coupled Lamé system.