Non-autonomous Terms Research Articles

Asymptotic behavior of a semilinear non-autonomous wave equation with distributed delay and analytic nonlinearity

This paper is concerned with a semilinear non-autonomous wave equation with distributed delay and analytic nonlinearity. The distributed delay is represented by an integral operator integrating on the delay interval, which considers a segment of the past dynamic information. Firstly we show the global boundedness of solutions and then prove that every globally bounded solution has a relative compact range in the phase space. By the Łojasiewicz–Simon inequality, we prove the convergence to an equilibrium of globally bounded solutions and then show the convergence rate dependence on the Łojasiewicz exponent and the decay conditions on non-autonomous term. The existence and uniqueness of a global strong solution is finally proved. Note that the result of existence and uniqueness of the solution does not need any restrictions on the coefficients between the damping and the delay. The presence of an integral kernel makes distributed delays more difficult to be analyzed compared to its pointwise counterpart.

Multi-index upper semicontinuity of pullback random attractors for fractional discrete FitzHugh-Nagumo systems with delays driven by Wong-Zakai approximations

In this paper, we consider the global well-posedness and multi-index stability of pullback random attractors for random fractional retarded lattice FitzHugh-Nagumo systems with nonlinear Wong-Zakai noise and non-autonomous forcing term. We use the idea of Caraballo et al. (2014) [2] to prove the global well-posedness. Based on the global well-posedness of the lattice FitzHugh-Nagumo system, we study the existence, uniqueness and upper semicontinuity of pullback random attractors Aϱδ={Aϱδ(τ,ω):τ∈R,ω∈Ω}. More precisely, we first utilize the method of tail-estimates of solution and Ascoli-Arzelà theorem to prove the existence and uniqueness of pullback random attractors, and then investigate the four types of upper semicontinuity of pullback random attractors: (1) The long time stability of pullback random attractors as the time parameter τ approaches negative infinity; (2) The upper semicontinuity of pullback random attractors as the step-length δ of the Wong-Zakai approximation tends to positive infinity; (3) The upper semicontinuity of pullback random attractors as the delay time ϱ→0; (4) The upper semicontinuity of pullback random attractors from non-autonomous to autonomous.

Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonance.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.

Random attractors for three-dimensional stochastic globally modified Navier–Stokes equations driven by additive noise on unbounded domains

Abstract This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 𝒪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball’s idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time.

On Periodic Motions of a Nonautonomous Hamiltonian System at Resonance 2:1:1

This paper presents an analysis of nonlinear oscillations of a near-autonomous two-degree-of-freedom Hamiltonian system, $2\pi$-periodic in time, in the neighborhood of a trivial equilibrium. It is assumed that in the autonomous case, for some set of parameters, the system experiences a multiple parametric resonance for which the frequencies of small linear oscillations in the neighborhood of the equilibrium are equal to two and one. It is also assumed that the Hamiltonian of perturbed motion contains only terms of even degrees with respect to perturbations, and its nonautonomous perturbing part depends on odd time harmonics. The analysis is performed in a small neighborhood of the resonance point of the parameter space. A series of canonical transformations is made to reduce the Hamiltonian of perturbed motion to a form whose main (model) part is characteristic of the resonance under consideration and the structure of nonautonomous terms. Regions of instability (regions of parametric resonance) of the trivial equilibrium are constructed analytically and graphically. A solution is presented to the problem of the existence and bifurcations of resonant periodic motions of the system which are analytic in fractional powers of a small parameter. As applications, resonant periodic motions of a double pendulum are constructed. The nearly constant lengths of the rods of the pendulum are prescribed periodic functions of time. The problem of the linear stability of these motions is solved.

Forward dynamics of 3D double time-delayed MHD-Voight equations

This paper is concerned with the long-term behaviour of 3D double time-delayed magnetohydrodynamics (MHD)-Voight equations with non-autonomous forcing term. First, we establish the existence, forward compactness and forward stability of pullback attractors by the forward equi-continuity and forward pullback asymptotic compactness of solutions. We then study the upper semi-convergence of pullback attractors as the time parameter tends to positive infinity. Since the solution of this equation has no high regularity, we use the spectrum decomposition technique to prove the forward pointwise pullback asymptotic compactness of solutions.

Asymptotic Analysis of Systems with Damped Oscillatory Perturbations

For some class of asymptotically autonomous systems we study the asymptotics of solutions with respect to the independent variable at infinity. It as assumed that the limit system describes nonlinear oscillations, whereas nonautonomous additional terms preserve equilibrium, oscillate with asymptotically constant resonance frequency, and vanish at long times. We study the long-term behavior of the solutions in a neighborhood of the equilibrium state of the limit system. Some modifications of the averaging method are used for constructing asymptotics.

High-order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to generic forcing terms and parametrically excited systems

The direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary orders of expansions are considered for the unknown mappings and the reduced dynamics, which are then solved sequentially through the homological equations for both autonomous and time-dependent terms. It is emphasised that the two problems share a similar structure, which can be used for an efficient implementation of the non-autonomous added terms. Special emphasis is also put on the new resonance conditions arising due to the presence of the external forcing frequencies, which allow predicting phenomena such as parametric excitation and isolas formation. The method is then applied to structures of academic and industrial interest. First, the large amplitude vibrations of a forced-damped cantilever beam are studied. This example highlights that high-order non-autonomous terms are compulsory to correctly estimate the maximum vibration amplitude experienced by the structure. The birth of isolated solutions is also illustrated on this example. The cantilever is then used to show how quadratic coupling creates conditions for the excitation of the parametric instability, and that this feature is correctly embedded in the reduction process. A shallow arch excited with multi-modal forcing is then studied to detail different forcing effects. Finally, the approach is validated on a structure of industrial relevance, i.e. a comb-driven micro-electro-mechanical resonator. The accuracy and computational performance reported suggest that the proposed methodology can accurately predict the nonlinear dynamic response of a large class of nonlinear vibratory systems.

Nonautonomous three soliton interactions in an inhomogeneous optical fiber: Application to soliton switching devices

We investigate the interactional behavior of three nonautonomous solitons in an optical fiber with inhomogeneous nature which can be described by nonlinear Schrödinger equation with nonautonomous term. A pair of matrices constructed with the aid of AKNS procedure for nonautonomous NLS equation with variable coefficients. Based on the Lax pair, three soliton solutions are generated by employing Darboux method. We show the switchable characteristics of nonautonomous three solitons by typical graphical illustration. The remarkable results presented in this study can be used to design the optical logic gate devices for optical computing.

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

The influence of time-dependent perturbations on an autonomous Hamiltonian system with an equilibrium of center type is considered. It is assumed that the perturbations decay at infinity in time and vanish at the equilibrium. In this case the stability and the long-term behaviour of trajectories depend on nonlinear and non-autonomous terms of equations. The paper investigates bifurcations associated with a change of Lyapunov stability of the equilibrium and the emergence of new attracting or repelling states in the perturbed non-autonomous system. The dependence of bifurcations on the structure of perturbations is discussed.

Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of the non-autono-mous stochastic 3D Brinkman-Forchheimer equations on unbounded domains. We first define a continuous non-autonomous cocycle for the stochastic equations, and then prove that the existence of tempered random attractors by Ball's idea of energy equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.</p>

Stochastic dynamics of non-autonomous fractional Ginzburg-Landau equations on $ \mathbb{R}^3 $

<p style='text-indent:20px;'>We investigate a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation with multiplicative white noise in three spatial dimensions. Of particular interest is the asymptotic behavior of its solutions. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in <inline-formula><tex-math id="M2">\begin{document}$ L^2( \mathbb{R}^3) $\end{document}</tex-math></inline-formula>. The existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions is then established. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in <inline-formula><tex-math id="M3">\begin{document}$ L^2( \mathbb{R}^3) $\end{document}</tex-math></inline-formula> is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.</p>

Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.

Smoothing and finite-dimensionality of uniform attractors in Banach spaces

The aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions.As applications, we show that the uniform attractor of the 2D Navier-Stokes equation is finite-dimensional in H and in V, and that of a reaction-diffusion equation is finite-dimensional in L2 and in Lp, with p>2.

LIMITING DYNAMICS OF NON-AUTONOMOUS STOCHASTIC GINZBURG-LANDAU EQUATIONS ON THIN DOMAINS

We examine the limiting dynamics of a class of non-autonomous stochastic Ginzburg-Landau equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. The existence and uniqueness of tempered pullback random attractors are established for the stochastic Ginzburg-Landau systems defined on $ (n+1) $-dimensional narrow domain. In addition, the upper semicontinuity of these attractors is obtained when a family of $ (n+1) $-dimensional thin domains collapses onto an $ n $-dimensional domain.

Attractors of the ‘n + 1’ dimensional Einstein-Λ flow

Here we prove a global existence theorem for sufficiently small however fully nonlinear perturbations of a family of background solutions of the ‘n + 1’ dimensional vacuum Einstein equations in the presence of a positive cosmological constant Λ. The future stability of vacuum solutions in the small data and zero cosmological constant limit has been studied previously for both ‘3 + 1’ and higher dimensional spacetimes. However, with the advent of dark energy driven accelerated expansion of the Universe, it is of fundamental importance in mathematical cosmology to include a positive cosmological constant, the simplest form of the dark energy for the vacuum Einstein equations. Such Einsteinian evolution is here designated as the ‘Einstein-Λ’ flow. We study the background solutions of this ‘Einstein-Λ’ flow in ‘n + 1’ dimensional spacetimes in constant mean curvature spatial harmonic gauge, n ⩾ 3 and establish both linear and non-linear stability of such solutions. In the cases of number of spatial dimensions being strictly greater than 3, the finite dimensional Einstein moduli spaces form the centre manifolds of the dynamics. A suitable shadow gauge condition [Andersson L and Moncrief V 2011 J. Differ. Geom. 89 1–47] is implemented in order to treat these cases. In addition, the autonomous character of the suitably re-scaled Einstein flow breaks down as a consequence of including Λ(>0). We construct a Lyapunov functional (controlling a suitable norm of the small data) similar to a wave equation type energy for the non-linear non-autonomous evolution of the data and prove its decay in the direction of cosmological expansion utilizing the structure of the non-autonomous terms and smallness assumption on the data. Our results demonstrate the future stability and geodesic completeness of the perturbed spacetimes, and show that the scale-free geometry converges to an element of the space of constant negative scalar curvature metrics sufficiently close to and containing the Einstein moduli space (a point for n = 3 and a finite dimensional space for n > 3), which has significant consequences for the cosmic topology while restricting to the case of n = 3.

Random attractors of the stochastic extended Brusselator system with a multiplicative noise

In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.

Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods

We solve the scattering problems for nonlinear Schrödinger equations with an inverse-square potential by applying the energy methods. The methods are optimized to the abstract semilinear Schrödinger evolution equations with nonautonomous terms.

Autonomous and non-autonomous unbounded attractors under perturbations

AbstractPullback attractors with forwards unbounded behaviour are to be found in the literature, but not much is known about pullback attractors with each and every section being unbounded. In this paper, we introduce the concept of unbounded pullback attractor, for which the sections are not required to be compact. These objects are addressed in this paper in the context of a class of non-autonomous semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as the initial time goes to -∞, for each elapsed time. Distinct regimes for the non-autonomous term are taken into account. Namely, we address the small non-autonomous perturbation and the asymptotically autonomous cases.

Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains

We study the asymptotic behavior of a class of non-autonomous non-local fractional stochastic parabolic equation driven by multiplicative white noise on the entire space \begin{document}$\mathbb{R}^n$\end{document} . We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in \begin{document}$L^2({\mathbb{R}} ^n)$\end{document} . We then prove the existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in \begin{document}$L^2({\mathbb{R}} ^n)$\end{document} is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.