Dimension Of Attractor Research Articles

Templex-based dynamical units for a taxonomy of chaos.

Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented BraMAH cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a "minimal" form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units: the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated.

Dimension of Planar Non-conformal Attractors with Triangular Derivative Matrices

We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak Ledrappier-Young formula, we show that the dimensions equal to the root of the subadditive pressure and the Lyapunov dimension, respectively, for almost every choice of parameters. We also exhibit concrete examples satisfying the transversality condition with respect to the translation parameters.

Multi-vortices and lower bounds for the attractor dimension of 2d Navier-Stokes equations

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations, which does not use Kolmogorov flows, is presented. Using this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions.

Exploring nonlinear dynamics in brain functionality through phase portraits and fuzzy recurrence plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Asymptotic analysis of the 2D and 3D convective Brinkman–Forchheimer equations in unbounded domains

In this work, we carry out the asymptotic analysis of two- and three-dimensional convective Brinkman–Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and unbounded Poincaré domains (using asymptotic compactness property). In Poincaré domains, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors with respect to domains for CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains [Formula: see text] of the Poincaré domain [Formula: see text] such that [Formula: see text] as [Formula: see text]. If [Formula: see text] and [Formula: see text] are the global attractors of CBF equations corresponding to [Formula: see text] and [Formula: see text], respectively, then we show that for large enough [Formula: see text], the global attractor [Formula: see text] enters into any neighborhood [Formula: see text] of [Formula: see text]. The presence of the Darcy term in CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss the quasi-stability property of the semigroup associated with CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors.

Exploring Nonlinear Dynamics In Brain Functionality Through Phase Portraits And Fuzzy Recurrence Plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

A robust family of exponential attractors for a linear time discretization of the Cahn-Hilliard equation with a source term

We consider a linear implicit-explicit (IMEX) time discretization of the Cahn-Hilliard equation with a source term, endowed with Dirichlet boundary conditions. For every time step small enough, we build an exponential attractor of the discrete-in-time dynamical system associated to the discretization. We prove that, as the time step tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the continuous-in-time dynamical system associated with the PDE. We also prove that the fractal dimension of the exponential attractor (and consequently, of the global attractor) is bounded by a constant independent of the time step. The results also apply to the classical Cahn-Hilliard equation with Neumann boundary conditions.

Asymptotic analysis of two-dimensional Oldroyd fluids in unbounded domains: Global attractors and their dimension

In this work, we consider the two-dimensional Oldroyd model for the non-Newtonian fluid flows (viscoelastic fluid) in Poincaré domains (bounded or unbounded) and study their asymptotic behavior. We establish the existence of a global attractor in Poincaré domains using asymptotic compactness property. Since the high regularity of solutions is not easy to establish, we prove the asymptotic compactness of the solution operator by applying Kuratowski’s measure of noncompactness, which relies on uniform-tail estimates and the flattening property of the solution. Finally, the estimates for the Hausdorff as well as fractal dimensions of global attractors are also obtained.

Encoding of female mating dynamics by a hypothalamic line attractor

Females exhibit complex, dynamic behaviours during mating with variable sexual receptivity depending on hormonal status1–4. However, how their brains encode the dynamics of mating and receptivity remains largely unknown. The ventromedial hypothalamus, ventrolateral subdivision contains oestrogen receptor type 1-positive neurons that control mating receptivity in female mice5,6. Here, unsupervised dynamical system analysis of calcium imaging data from these neurons during mating uncovered a dimension with slow ramping activity, generating a line attractor in neural state space. Neural perturbations in behaving females demonstrated relaxation of population activity back into the attractor. During mating, population activity integrated male cues to ramp up along this attractor, peaking just before ejaculation. Activity in the attractor dimension was positively correlated with the degree of receptivity. Longitudinal imaging revealed that attractor dynamics appear and disappear across the oestrus cycle and are hormone dependent. These observations suggest that a hypothalamic line attractor encodes a persistent, escalating state of female sexual arousal or drive during mating. They also demonstrate that attractors can be reversibly modulated by hormonal status, on a timescale of days.

Attractors of a thermoelastic Bresse system with mass diffusion

AbstractThis paper concerns the long‐time dynamics of a thermoelastic Bresse system with mass diffusion. We prove the existence of a global attractor by showing that the system is gradient and asymptotically smooth. In addition, the attractor is characterized as an unstable manifold of the set of stationary solutions. The quasi‐stability of the system and the finite fractal dimension of the global attractor are established by a stabilizability inequality. Finally, we prove the upper semicontinuity of the global attractor regarding the parameter in a dense residual set.

Cell-Cell Interactions: How Coupled Boolean Networks Tend to Criticality.

Biological cells are usually operating in conditions characterized by intercellular signaling and interaction, which are supposed to strongly influence individual cell dynamics. In this work, we study the dynamics of interacting random Boolean networks, focusing on attractor properties and response to perturbations. We observe that the properties of isolated critical Boolean networks are substantially maintained also in interaction settings, while interactions bias the dynamics of chaotic and ordered networks toward that of critical cells. The increase in attractors observed in multicellular scenarios, compared to single cells, allows us to hypothesize that biological processes, such as ontogeny and cell differentiation, leverage interactions to modulate individual and collective cell responses.

Multi‐shaft stirred reactors mixing efficiency: Rapid characterization strategy based on chaotic attractors

AbstractThe complex fluid dynamics mechanism of multi‐shaft stirring systems has constrained their application in industrial production. In this work, we have extracted torque signals to correlate power consumption with rotational speeds across single, dual, and triple shaft stirred reactors. Remarkably, the reasons behind the unique linear power consumption‐rotational speed relationship exhibited in triple‐shaft reactor, through the application of Fractional Fourier Transform. Our comparative analysis, through mixing energy, reveals multi‐shaft systems' superiority in energy‐efficient mixing. More importantly, the interaction between impellers and fluid within a reactor is reflected in the evolving of torque chaotic attractors. We have established a novel characterization strategy by fractal dimension of attractors which reflect the reactors' mixing efficiency. By mathematically modeling these attractors, we have demonstrated their potential as a tool for establishing the internal dynamics equations of stirred reactors. This approach offers a new method for evaluating mixing effectiveness and automated control of stirring machinery.

Globally Exponentially Attracting Sets and Heteroclinic Orbits Revealed

Motivated by the open problems on the global dynamics of the generalized four-dimensional Lorenz-like system, this paper deals with the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions. Specifically, not only is a family of mathematical expressions of globally exponentially attracting sets derived, but the existence of a pair of orbits heteroclinic to S0 and S± is also proven with the aid of a Lyapunov function and the definitions of both the α-limit set and ω-limit set. Moreover, numerical examples are used to justify the theoretical analysis. Since the obtained results improve and complement the existing ones, they may provide support in chaos control, chaos synchronization, the Hausdorff and Lyapunov dimensions of strange attractors, etc.

Regularity of the Hausdorff dimension of some hyperbolic systems

We study the regularity of the Hausdorff dimension for certain hyperbolic basic sets, as a function of the dynamical system. The main results ensure that the Hausdorff dimension of certain multidimensional thin hyperbolic attractors and of certain nonconformal iterated function systems varies smoothly and analytically with respect to the diffeomorphism. Similar results hold for average conformal hyperbolic sets and average conformal repellers.

Towards Modelling Mechanical Shaking Using Potential Energy Surfaces: A Toy Model Analysis

In this work, we formalize the effect of mechanical shaking by using various forms of an externally exerted force, which may be constant or may be position-dependent, and we examine the changes in the potential energy surfaces that quantify the chemical reaction. We use a simple toy model to model the potential energy surfaces of a chemical reaction, and we study the effect of a constant or position-dependent externally exerted force for various forms of the force. As we demonstrate, the effect of the force can be quite dramatic on the potential energy surfaces, which acquire new stationary points and new Newton trajectories that are distinct from the original ones that were obtained in the absence of mechanochemical effects. We also introduce a new approach to mechanochemical interactions, using a dynamical systems approach for the Newton trajectories. As we show, the dynamical system attractor properties of the trajectories in the phase space are identical to the stationary points of the potential energy surfaces, but the phase space contains much more information regarding the possible evolution of the chemical reaction—information that is quantified by the existence of unstable or saddle fixed points in the phase space. We also discuss how an experimental method for a suitable symmetric liquid solution substance might formalize the effect of shaking via various forms of external force, even in the form of an extended coordinate-dependent force matrix. This approach may experimentally quantify the Epstein effect of shaking in chemical solutions via mechanochemistry methods.

A Dichotomy for the Dimension of Solenoidal Attractors on High Dimensional Space

A Dichotomy for the Dimension of Solenoidal Attractors on High Dimensional Space

Exploring Iterated Implicit Function Systems: Existence and Properties of Attractors

This paper investigates a type of iterated implicit function systems composed of equations [Formula: see text], where [Formula: see text] is a continuous function, and [Formula: see text] is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation [Formula: see text] containing the implicit function or being [Formula: see text]-contractive about [Formula: see text]. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated.

Multi-vortices and Lower Bounds for the Attractor Dimension of 2D Navier–Stokes Equations

Multi-vortices and Lower Bounds for the Attractor Dimension of 2D Navier–Stokes Equations

Strong solutions and attractor dimension for 2D NSE with dynamic boundary conditions

We consider incompressible Navier–Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.

Fractal dimension of random attractors for nonautonomous stochastic strongly damped wave equations on ℝN

In this article, we investigate the dynamics of a nonautonomous stochastic strongly damped wave equation defined on . We first use the energy equation and tail‐estimates to prove the asymptotic compactness of the solutions and obtain the existence of a unique pullback random attractor for the equation with critical nonlinearity. Then, we give an upper bound of fractal dimension of the random attractor when the nonlinearity is of subcritical growth. The unboundedness of the physical space will impose difficulties on the estimation for the upper bound of fractal dimension of the random attractor.