Asymptotic Dynamics Research Articles

Time-dependent global attractors for strongly damped wave equations with time-dependent memory kernels

In this paper, we consider the longtime behavior for the strongly damped wave equation with time-dependent memory kernels on a bounded domain . The main novelty of our result is that the memory kernel depends on time, allowing to describe the dynamics of aging meterials. We first investigate the existence and uniqueness of weak solutions and then, we obtain the existence of the time-dependent global attractors in . We also prove the regularity of the time-dependent global attractor , i.e, is bounded in , with a bound independent of t. Finally, when approaches a multiple of the Dirac mass at zero as , we prove that the asymptotic dynamics of our problem is close to the one of its formal limit describing viscoelastic solids of Kelvin-Voigt type.

Cellular automata and bootstrap percolation

We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics by proving two results that do not hold in the subclass of monotone automata. First, it is undecidable whether the automaton almost surely fills the space when initialized on a Bernoulli random configuration with density p, for some/all 0<p<1. Second, there exists an automaton whose space-filling property depends on p in a non-monotone way.

Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation

We prove, in the framework of measure solutions, that the equal mito-sis equation present persistent asymptotic oscillations. To do so we adopt a duality approach, which is also well suited for proving the well-posedness when the division rate is unbounded. The main difficulty for characterizing the asymptotic behavior is to define the projection onto the subspace of periodic (rescaled) solutions. We achieve this by using the generalized relative entropy structure of the dual problem.

Asymptotic dynamics for fractionally damped swelling porous elastic soils with memory

Asymptotic dynamics for fractionally damped swelling porous elastic soils with memory

Asymptotic and transient dynamics of SEIR epidemic models on weighted networks

We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission.

Dynamics of the General Bianchi IX Model near a Cosmological Singularity

Half a century ago, Belinsky and Khalatnikov proposed a generic solution of the Einstein equations near their cosmological singularity, basing on a generalization of the homogeneous model of Bianchi type IX. The consideration of the evolution of the most general non-diagonal case of this model is significantly simplified, if it is assumed that, when approaching the singularity t = 0, it reduces to the so-called asymptotic dynamics, at which inequality Γ1 ≫ Γ2 ≫ Γ3 holds. It has been suggested that this inequality continues to be true from the moment of its first fulfilment up to the singularity of space-time. We analyze this assumption and show that it is incorrect in the general case. However, it is shown that in any case there exists a time t0, after which this assumption becomes true. The value of t0 is the smaller, the less is the degree of non-diagonality of the model. Some details of the behavior of the non-diagonal homogeneous model of Bianchi type IX are considered at the stage of asymptotic dynamics of approaching the singularity.

Hyperbolic Inflation in the Jordan Frame

We consider a multi-scalar field model in the Jordan frame, which can be seen as a two-scalar field model where the Brans–Dicke field interacts in the kinetic part with the second scalar field. This theory under a conformal transformation reduces to the hyperbolic inflation. We show that scaling solutions and the de Sitter universe are provided by the theory. In the study of asymptotic dynamics, we determine an attractor where all the fluid sources contribute in the cosmological fluid. This attractor is always a spiral, and it can be seen as the analogue of the hyperbolic inflation in the Jordan frame.

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

We study the effects of IID random perturbations of amplitude $\epsilon > 0$ on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a \in [0,1]\}$ of smooth multimodal maps which predominantly expanding, i.e., $|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0 \}$. We obtain, for any $\epsilon > 0$, a \emph{checkable, finite-time} criterion on the parameter $a$ for random perturbations of the map $f_a$ to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the situation for the deterministic dynamics of $f_a$, the chaotic regimes of which are determined by typically uncheckable, infinite-time conditions. Moreover, our finite-time criterion depends on only $k \sim \log (\epsilon^{-1})$ iterates of the deterministic dynamics of $f_a$, which grows quite slowly as $\epsilon \to 0$.

Linking bifurcation analysis of Holling–Tanner model with generalist predator to a changing environment

AbstractBifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation.

The nonclassical diffusion equations with time-dependent memory kernels and a new class of nonlinearities

AbstractIn this study, we consider the nonclassical diffusion equations with time-dependent memory kernels \begin{equation*} u_{t} - \Delta u_t - \Delta u - \int_0^\infty k^{\prime}_{t}(s) \Delta u(t-s) ds + f( u) = g \end{equation*} on a bounded domain $\Omega \subset \mathbb{R}^N,\, N\geq 3$ . Firstly, we study the existence and uniqueness of weak solutions and then, we investigate the existence of the time-dependent global attractors $\mathcal{A}=\{A_t\}_{t\in\mathbb{R}}$ in $H_0^1(\Omega)\times L^2_{\mu_t}(\mathbb{R}^+,H_0^1(\Omega))$ . Finally, we prove that the asymptotic dynamics of our problem, when $k_t$ approaches a multiple $m\delta_0$ of the Dirac mass at zero as $t\to \infty$ , is close to the one of its formal limit \begin{equation*}u_{t} - \Delta u_{t} - (1+m)\Delta u + f( u) = g. \end{equation*} The main novelty of our results is that no restriction on the upper growth of the nonlinearity is imposed and the memory kernel $k_t(\!\cdot\!)$ depends on time, which allows for instance to describe the dynamics of aging materials.

The importance of ecological dynamics in evolutionary processes: A host-bacteriophage model revisited

A recent study of adaptive dynamics of lysis propensity in temperate phages suggested that full lysogeny emerges as the outcome of bacteriophage evolution in a simple host-phage system. The conclusion is based on the premise that mutant strains necessarily appear in equilibrium host-phage environments. Revisiting the model, we show that the ecological system exhibits richer asymptotic dynamics and that, in a certain parameter regime, evolution may in fact drive lysis propensity towards an evolutionary singularity in which a non-zero proportion of phages initiate infection in a lytic cycle. These singularities act as points of evolutionary diversification, leading to periodic coexistence of two distinct phage strains on the evolutionary time-scale. One of the two strains in the dimorphic evolutionary singularity is fully lysogenic (in the sense that cell infection always leads to lysogeny), while the other is partially lytic. Our study thus highlights the importance of ecological interactions as a driver of evolution.

Mathematical comparison and empirical review of the Monod and Droop forms for resource-based population dynamics

Almost all biological models use either the Droop or Monod form to describe the resource-based growth of a living organism. Empirical evidence overwhelmingly suggests the Droop form describes data more accurately than the Monod form, however, the Monod form is more popular due to its simplicity. Focusing on phytoplankton, we illustrate the underlying logics behind these two forms via conceptual comparison, experimental data validation, transient, and asymptotic dynamics. The conceptual illustration provides the primary difference in their mechanisms via a paradox. Data validation is tested via field and laboratory experiments. The Droop and Monod forms have consistent asymptotic dynamics in the closed nutrient case, whereas the transient dynamics are significantly different when the nutrient uptake rate is small. In addition, we decipher Michael Droop’s private last statements on unifying the Droop and Monod forms as well as simplifying the Droop form. This article aims to guide future model development with any resource-based growth.

Relaxation at different length scales in models of many-body localization

We study dynamical correlation functions in the random-field Heisenberg chain, which probes the relaxation times at different length scales. Firstly, we show that the relaxation time associated with the dynamical imbalance (examining the relaxation at the smallest length scale) decreases with disorder much faster than the one determined by the dc conductivity (probing the global response of the system). We argue that the observed dependence of relaxation on the length scale originates from local nonresonant regions. The latter have particularly long relaxation times or remain frozen, allowing for nonzero dc transport via higher-order processes. Based on the numerical evidence, we introduce a toy model that suggests that the nonresonant regions asymptotic dynamics are essential for the proper understanding of the disordered chains with many-body interactions.

Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators

In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operatorsut+A(t)u=f(t,u), t>τ;u(τ)=u0, in a Banach space X. Under suitable conditions, we obtain regularity results for ut(t,x) with respect to its spatial variable x and estimates for ut in stronger spaces (Xα). We then apply those results to a nonautonomous reaction-diffusion equationut−div(a(t,x)∇u)+u=f(t,u) with Neumann boundary condition and time-dependent diffusion. From the regularity of ut we derive the existence of classical solutions and from the estimates for ut we prove that the variation of the solution u is bounded in the long-time dynamics. We also prove the existence of pullback attractor, as well as the existence of a compact set that contains the long-time dynamics of the derivatives ut, without requiring any assumption concerning monotonicity or decay in time of a(t,x).

Asymptotic flocking dynamics of a relativistic Cucker–Smale flock under singular communications

We study collision avoidance and flocking dynamics for the relativistic Cucker–Smale (RCS) model with a singular communication weight. For a bounded and regular communication weight, RCS particles can exhibit collisions in finite time depending on the geometry of the initial configuration. In contrast, for a singular communication weight, when particles collide, the associated Cucker–Smale vector field becomes unbounded and the standard Cauchy–Lipschitz theory cannot be applied so that existence theory after collisions is problematic. Thus, the collision avoidance problem is directly linked to the global solvability of the singular RCS model and asymptotic flocking dynamics. In this paper, we present sufficient frameworks leading to the nonexistence of finite-time collisions and asymptotic flocking in terms of initial configuration and blow-up rate at the singular point of communication weight.

Pullback attractors for 2D MHD equations on time-varying domains

&lt;p style='text-indent:20px;'&gt;In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.&lt;/p&gt;

Stability and Hopf bifurcation of an SIR epidemic model with density-dependent transmission and Allee effect.

In this paper, an SIR model with a strong Allee effect and density-dependent transmission is proposed, and its characteristic dynamics are investigated. The elementary mathematical characteristic of the model is studied, including positivity, boundedness and the existence of equilibrium. The local asymptotic stability of the equilibrium points is analyzed using linear stability analysis. Our results indicate that the asymptotic dynamics of the model are not only determined using the basic reproduction number ${R_0}$. If ${R_0} < 1$, there are three disease-free equilibrium points, and a disease-free equilibrium is always stable. At the same time, the conditions for other disease-free equilibrium points to be bistable were determined. If ${R_0} > 1$ and in certain conditions, either an endemic equilibrium emerges and is locally asymptotically stable, or the endemic equilibrium becomes unstable. What must be emphasized is that there is a locally asymptotically stable limit cycle when the latter happens. The Hopf bifurcation of the model is also discussed using topological normal forms. The stable limit cycle can be interpreted in a biological significance as a recurrence of the disease. Numerical simulations are used to verify the theoretical analysis. Taking into account both density-dependent transmission of infectious diseases and the Allee effect, the dynamic behavior becomes more interesting than when considering only one of them in the model. The Allee effect makes the SIR epidemic model bistable, which also makes the disappearance of diseases possible, since the disease-free equilibrium in the model is locally asymptotically stable. At the same time, persistent oscillations due to the synergistic effect of density-dependent transmission and the Allee effect may explain the recurrence and disappearance of disease.

Random attractors for dissipative systems with rough noises

&lt;p style='text-indent:20px;'&gt;We provide an analytic approach to study the asymptotic dynamics of rough differential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons' theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity.&lt;/p&gt;

Global Asymptotic Dynamics of the Cubic Galileon Interacting with Dark Matter

Global Asymptotic Dynamics of the Cubic Galileon Interacting with Dark Matter

Understanding approximate Fisher information for fast convergence of natural gradient descent in wide neural networks* *This article is an updated version of: Karakida R and Osawa K 2020 Understanding approximate Fisher information for fast convergence of natural gradient descent in wide neural networks Advances in Neural Information Processing Systems vol 33 ed H Larochelle, M Ranzato, R Hadsell, M F

Natural gradient descent (NGD) helps to accelerate the convergence of gradient descent dynamics, but it requires approximations in large-scale deep neural networks because of its high computational cost. Empirical studies have confirmed that some NGD methods with approximate Fisher information converge sufficiently fast in practice. Nevertheless, it remains unclear from the theoretical perspective why and under what conditions such heuristic approximations work well. In this work, we reveal that, under specific conditions, NGD with approximate Fisher information achieves the same fast convergence to global minima as exact NGD. We consider deep neural networks in the infinite-width limit, and analyze the asymptotic training dynamics of NGD in function space via the neural tangent kernel. In the function space, the training dynamics with the approximate Fisher information are identical to those with the exact Fisher information, and they converge quickly. The fast convergence holds in layer-wise approximations; for instance, in block diagonal approximation where each block corresponds to a layer as well as in block tri-diagonal and K-FAC approximations. We also find that a unit-wise approximation achieves the same fast convergence under some assumptions. All of these different approximations have an isotropic gradient in the function space, and this plays a fundamental role in achieving the same convergence properties in training. Thus, the current study gives a novel and unified theoretical foundation with which to understand NGD methods in deep learning.