Asymptotic Attractor Research Articles

Modified cam clay bounding surface hyper-viscoplastic model

Clays exhibit complex mechanical behaviour with significant viscous, nonlinear, and hysteric characteristics, beyond the prediction capacity of the well-known modified cam clay (MCC) model. This paper extends the MCC model to address these important limitations. The proposed family of models is constructed entirely within the hyperplasticity framework deduced from thermodynamic extremal principles. More specifically, the previously developed MCC hyper-viscoplastic model based on the isotache concept is extended to incorporate multiple internal variables and to capture recent loading history, hysteresis, and smooth response of the material. This is achieved by defining an inelastic free energy and an element that implements a bounding surface within hyperplasticity, resulting in pressure dependency in both reversible and irreversible processes with a unique critical state envelope, and only eight material parameters with a readily measurable viscous parameter. A kinematic hardening in the logistic differential form in stress space is derived that enables the proposed model to function effectively across a wide range of stresses. Based on this kinematic hardening rule, the current stress state acts as an asymptotic attractor for the back/shift stresses whose evolution rates are proportional to their current state.

Statistical physics of learning in high-dimensional chaotic systems

In many complex systems, elementary units live in a chaotic environment and need to adapt their strategies to perform a task by extracting information from the environment and controlling the feedback loop on it. One of the main examples of systems of this kind is provided by recurrent neural networks. In this case, recurrent connections between neurons drive chaotic behavior, and when learning takes place, the response of the system to a perturbation should also take into account its feedback on the dynamics of the network itself. In this work, we consider an abstract model of a high-dimensional chaotic system as a paradigmatic model and study its dynamics. We study the model under two particular settings: Hebbian driving and FORCE training. In the first case, we show that Hebbian driving can be used to tune the level of chaos in the dynamics, and this reproduces some results recently obtained in the study of more biologically realistic models of recurrent neural networks. In the latter case, we show that the dynamical system can be trained to reproduce simple periodic functions. To do this, we consider the FORCE algorithm—originally developed to train recurrent neural networks—and adapt it to our high-dimensional chaotic system. We show that this algorithm drives the dynamics close to an asymptotic attractor the larger the training time. All our results are valid in the thermodynamic limit due to an exact analysis of the dynamics through dynamical mean field theory.

Asymptotic limits and attractors for a laminated beam model

Asymptotic limits and attractors for a laminated beam model

Model system for the transient behavior of double diffusive convection in two miscible layers

We investigate thermal convection in two horizontal layers of miscible fluids with an initial state in which the fluids are at rest and the temperature has a linear continuous vertical distribution, but the concentration takes different constant values in each layer. Because the initial layered state is not a stationary state, most theoretical methods available for immiscible fluids are not applicable to the miscible case. Here we propose a new model of the thermal convection of miscible fluids, which is devised in such a way that the horizontal average of the width of the transitional layer is kept constant. Both the stability of the stationary state in the linear stage and the asymptotic attractors in the nonlinear stage of the model closely describe the transition of the convection pattern in the original system, which is associated with the change of the width of the transitional layer.

Lorenz-63 Model as a Metaphor for Transient Complexity in Climate

Dynamical systems like the one described by the three-variable Lorenz-63 model may serve as metaphors for complex natural systems such as climate systems. When these systems are perturbed by external forcing factors, they tend to relax back to their equilibrium conditions after the forcing has shut off. Here we investigate the behavior of such transients in the Lorenz-63 model by studying its trajectories initialized far away from the asymptotic attractor. Counterintuitively, these transient trajectories exhibit complex routes and, in particular, the sensitivity to initial conditions is akin to that of the asymptotic behavior on the attractor. Thus, similar extreme events may lead to widely different variations before the perturbed system returns back to its statistical equilibrium.

On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

In this paper, we study the nonnegativity and stability properties of the solutions of a newly proposed extended SEIR epidemic model, the so-called SE(Is)(Ih)AR epidemic model which might be of potential interest in the characterization and control of the COVID-19 pandemic evolution. The proposed model incorporates both asymptomatic infectious and hospitalized infectious subpopulations to the standard infectious subpopulation of the classical SEIR model. In parallel, it also incorporates feedback vaccination and antiviral treatment controls. The exposed subpopulation has three different transitions to the three kinds of infectious subpopulations under eventually different proportionality parameters. The existence of a unique disease-free equilibrium point and a unique endemic one is proved together with the calculation of their explicit components. Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. The reproduction numbers in the presence of one or both controls is linked to the control-free reproduction number to emphasize that such a number decreases with the control gains. We also prove that, depending on the value of the basic reproduction number, only one of them is a global asymptotic attractor and that the solution has no limit cycles.

Asymptotic Regularity and Attractors for Slightly Compressible Brinkman–Forchheimer Equations

Slightly compressible Brinkman–Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.

Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.

Dynamics of relaxation and dressing of a quenched Bose polaron

We study the non-equilibrium dynamics of relaxation and dressing of a mobile impurity suddenly immersed--or quenched-- into a zero temperature homogeneous Bose Einstein condensate (BEC) with velocity $v$. A many body generalization of Weisskopf-Wigner theory is implemented to obtain the impurity fidelity, reduced density matrix and entanglement entropy. The dynamics depend crucially on the Mach number $\beta =v/c$, with $c$ the speed of sound of superfluid phonons and features many different time scales. Quantum Zeno behavior at early time is followed by relaxational and dressing dynamics determined by Cerenkov emission of long-wavelength phonons for $\beta >1$ with a decay rate $\Gamma_p \propto (\beta-1)^3$. The polaron dressing dynamics \emph{slows-down} as $\beta \rightarrow 1$ and is characterized by power laws $t^{-\alpha}$ with different exponents for $\beta \lessgtr 1$. The asymptotic entanglement entropy features a sharp discontinuity and the residue features a cusp at $\beta =1$. These non-equilibrium features suggest \emph{universal} dynamical critical phenomena near $\beta \simeq 1$, and are a direct consequence of the linear dispersion relation of long wavelength superfluid phonons. We conjecture on the emergence of an asymptotic dynamical attractor with $\beta \leq 1$.

Convergence to self-similarity for ballistic annihilation dynamics

We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d⩾2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability α∈(0,1) or collide elastically with probability 1−α. Such equation is highly dissipative in the sense that all observables, hence solutions, vanish as time progresses. Following a contribution, by two of the authors, considering well-posedness of the steady self-similar profile in the regime of small annihilation rate α≪1, we prove here that such self-similar profile is the intermediate asymptotic attractor to the annihilation dynamics with explicit universal algebraic rate. This settles the issue about universality of the annihilation rate for this model brought in the applied literature.

Asymptotic Behavior and Attractor of Systems Governing Two-Dimensional Viscoelastic Flows

We establish the existence of a solution to the initial-boundary value problem for the equations of motion of a linear viscoelastic fluid with the adherence condition and describe the behavior of the solution.

Restricted nonlinear model for high- and low-drag events in plane channel flow

A restricted nonlinear (RNL) model, obtained by partitioning the state variables into streamwise-averaged quantities and superimposed perturbations, is used in order to track the exact coherent state in plane channel flow investigated by Toh & Itano (J. Fluid Mech., vol. 481, 2003, pp. 67–76). When restricting nonlinearities to quadratic interaction of the fluctuating part into the streamwise-averaged component, it is shown that the coherent structure and its dynamics closely match results from direct numerical simulation (DNS), even if only a single streamwise Fourier mode is retained. In particular, both solutions exhibit long quiescent phases, spanwise shifts and bursting events. It is also shown that the dynamical trajectory passes close to equilibria that exhibit either low- or high-drag states. When statistics are collected at times where the friction velocity peaks, the mean flow and root-mean-square profiles show the essential features of wall turbulence obtained by DNS for the same friction Reynolds number. For low-drag events, the mean flow profiles are related to a universal asymptotic state called maximum drag reduction (Xi & Graham, Phys. Rev. Lett., vol. 108, 2012, 028301). Hence, the intermittent nature of self-sustaining processes in the buffer layer is contained in the dynamics of the RNL model, organized in two exact coherent states plus an asymptotic turbulent-like attractor. We also address how closely turbulent dynamics approaches these equilibria by exploiting a DNS database associated with a larger domain.

Out-of-equilibrium Gubser flow attractors

We discuss the non-equilibrium attractors of systems undergoing Gubser flow within kinetic theory by means of nonlinear dynamical systems. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories. These attractors are non-planar and the basin of attraction is three dimensional. We compare the asymptotic attractors of each hydrodynamic model with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. Anisotropic hydrodynamics matches, up to high numerical accuracy, the attractor of the exact theory while the other hydrodynamic theories fail to do so. Thus, anisotropic hydrodynamics is an effective theory for far-from-equilibrium fluids, which consists of the dissipative (nonperturbative) contributions at any order in the gradient expansion.

A simple mathematical model to describe antibody-dependent enhancement in heterologous secondary infection in dengue

We develop a mathematical model to describe the role of antibody-dependent enhancement (ADE) in heterologous secondary infections, assuming that antibodies specific to primary dengue virus (DENV) infection are being produced by immunological memory. The model has a virus-free equilibrium (VFE) and a unique virus-presence equilibrium (VPE). VFE is asymptotically stable when VPE is unstable; and unstable, otherwise. Additionally, there is an asymptotic attractor (not a fixed point) due to the fact that the model assumes unbounded increase in memory cells. In the analysis of the model, ADE must be accounted in the initial stage of infection (a window of time of few days), period of time elapsed from the heterologous infection until the immune system mounting an effective response against the secondary infection. We apply the results yielded by model to evaluate ADE phenomonon in heterologous DENV infection. We also associate the possible occurrence of severe dengue with huge viremia mediated by ADE phenomenon.

On the CMC-Einstein-Λ flow

We complement a recent work on the stability of fixed points of the CMC-Einstein-Λ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we consider the notion of the reduced Hamiltonian, originally introduced by Fischer and Moncrief for the standard CMC-Einstein flow. For the analog version of the flow in the presence of a positive cosmological constant we identify the stationary points and relate them to the long-time behavior of the flow on manifolds of different Yamabe types. This entails conjectures on the asymptotic behaviour and potential attractors.

Asymptotic attractors of KdV–KSV equations

In this paper, we apply the method of orthogonal decomposition to prove the existence of asymptotic attractors for the KdV–KSV equations. Moreover, the dimensions estimate of asymptotic attractors is obtained.

Hořava-Lifshitz bouncing Bianchi IX universes: A dynamical system analysis

We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies in Ho\v{r}ava-Lifshitz gravity. The $6$-dim phase space presents two critical points, one asymptotic de Sitter attractor at infinity and a $2$-dim invariant plane. We identified four distinct parameter domains $A$, $B$, $C$ and $D$ for which the pair of critical points engenders distinct features in the dynamics. In the domain $A$ the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces. The center with multiplicity two engenders in its neighborhood the topology of stable and unstable cylinders $R \times S^3$ of orbits. We show that the stable and unstable cylinders coalesce realizing a smooth homoclinic connection to the center manifold, a rare event of regular/non-chaotic dynamics in bouncing Bianchi IX cosmologies. The presence of a saddle of multiplicity two in the domain $B$ engenders a high instability in the dynamics so that the cylinders emerging from the center manifold towards the bounce have four distinct attractors: the center manifold itself, the de Sitter attractor at infinity and two further momentum-dominated attractors with infinite anisotropy. In the domain $C$ we examine the features of invariant manifolds of orbits about a saddle of multiplicity two. The presence of the saddle of multiplicity two engenders bifurcations of the invariant manifold as the energy $E_0$ of the system increases relative to the energy $E_{cr}$ providing structures that were not yet observed in the literature. The domain $D$ is not examined as most of its features are present already in the previous domains.

Partially unstable attractors in networks of forced integrate-and-fire oscillators

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

We consider nonlinear parabolic equations involving fractional diffusion of the form $${\partial_t u + {(-\Delta)}^{s} \Phi(u)= 0,}$$ with $${0 < s < 1}$$ , and solve an open problem concerning the existence of solutions for very singular nonlinearities $${\Phi}$$ in power form, precisely $${\Phi'(u)=c\,u^{-(n+1)}}$$ for some $${0 < n < 1}$$ . We also include the logarithmic diffusion equation $${\partial_t u + {(-\Delta)}^{s} \log(u)= 0}$$ , which appears as the case $${n=0}$$ . We consider the Cauchy problem with nonnegative and integrable data $${u_0(x)}$$ in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The limit solutions we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, which are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as $${t\to\infty}$$ ) of the solutions with general integrable data. A new comparison principle is introduced.

Cosmological constraints on induced gravity dark energy models

We study induced gravity dark energy models coupled with a simple monomial potential ∝ σn and a positive exponent n. These simple potentials lead to viable dark energy models with a weak dependence on the exponent, which characterizes the accelerated expansion in the asymptotic attractor, when ordinary matter becomes negligible. We use recent cosmological data to constrain the coupling γ to the Ricci curvature, under the assumptions that the scalar field starts at rest deep in the radiation era and that the gravitational constant in the Einstein equations is compatible with the one measured in a Cavendish-like experiment. By using Planck 2015 data only, we obtain the 95 % CL bound γ < 0.0017 for n=4, which is further tightened to γ < 0.00075 by adding Baryonic Acoustic Oscillations (BAO) data. This latter bound improves by ∼ 30 % the limit obtained with the Planck 2013 data and the same compilation of BAO data. We discuss the dependence of the γ and ̇ GN/GN (z=0) on n.