{'fraction_non_alphanumeric': 0.0548331765617219, 'fraction_numerical': 0.015548055109832346, 'mean_word_length': 4.097010716300057, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 16, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Averaging with a time-dependent perturbation parameter

Motivated by recent problems in mathematical cosmology, we present two theorems giving the large-time asymptotics for a general class of systems which we propose to be applicable to a wide range of problems in spatially homogenous cosmology with oscillatory behaviour. Indeed, we improve the standard theory on averaging to one that is applicable to a general dynamical system that involves time-dependent coefficients, irrespective of the application to cosmology. This allows one to obtain the stability of such systems by showing the stability of a simpler averaged system.


Introduction
In mathematical cosmology, in general one examines a coupled Einstein-matter system of PDEs. If one considers the universe to be spatially homogeneous then the system reduces to a system of ODEs in time t; cf. [5,9,11,13]. The concrete matter equations depend on the considered sources. If one takes the model to be a field which satisfies a massive wave equation, then oscillations enter the system via the matter equations. This can hinder the task of establishing the future stability for the respective model; cf. [6]. Stability of prominent spacetimes has been established for a wide range of matter models in mathematical cosmology; cf. [4,7,8,10].
The present paper is motivated by the work [6] which considers spatially homogeneous cosmologies of LRS Bianchi type III. Type III thereby refers to the Bianchi class of the underlying three dimensional Lie-group of spatial homogeneity while LRS refers to 'locally rotationally symmetric', i.e., to the presence of an additional rotational one parameter Lie-group of isometries; cf. [11,13]. The considered matter source is a massive scalar field φ. Consequently, the wave equation which it satisfies is the Klein-Gordon equation g φ = φ, where g denotes the d'Alembertian with respect to the space-time metric g. The spatial homogeneity is inherited by the scalar field such that φ = φ(t), where t denotes coordinate time; cf. [13]. In a symmetry-adapted frame the Klein-Gordon equation for LRS Bianchi type III then readsφ + φ = H(−3φ), (1.1) where a dot denotes the differentiation with respect to t and H = H(t) is the Hubble scalar-a measure of the overall isotropic rate of spatial expansion. Geometrically H is given by the trace of the extrinsic curvature of the spatial slices, up to a factor −1/3. From the point of view of (1.1), H represents a time-dependent damping of the harmonically oscillating φ. Its time dependence is governed by the coupling to the Einstein equations, which can be expressed as a set of constraints and firstorder evolution equations for H and other geometric quantities. The equation for H is commonly called Raychaudhuri equation. 1 From its explicit form it is easy to show that for positive initial values, H is strictly decreasing and going to zero as t → ∞; cf. [6,Lemma 3]. Another important feature of the problem is, that the Raychaudhuri equation is proportional to H 2 , while the other evolution equations are proportional to H.
The theory of averaging in nonlinear dynamical systems dates back as far as the 18th century [12, App A]. At that time perturbation methods for differential equations became important for testing the predictions of Newtons theory of gravitation for celestial mechanics against observational data. Nowadays these methods represent widely used tools in science and engineering, and the mathematical theory behind it is well developed, and increasingly so; cf. [12]. The core idea is to approximate a perturbed (or full) system by an unperturbed (or averaged) system. The latter is thereby obtained from the former by an integration over a time scale associated with the perturbation; e.g. a period in the case of a periodic perturbation. The basic theorems then quantify the accuracy of the averaged approximation in terms of error estimates; cf. e.g. [12, p 31, Thm 2.8.1 or p 101, Thm 5.5.1]. Typically the perturbation is controlled by a constant parameter, and the error estimates together with their range of validity are expressed in terms of orders of it.
The present work is motivated by the recent application to mathematical cosmology [6], which concerns a system that can be brought into a form closely resembling the standard form for problems in first-order periodic averaging [12, Chap 2]. The main difference though is that the quantity which plays the role of the perturbation parameter is time-dependent and itself subject to an evolution equation which is part of the system; cf. [6,Sec 3]. However this perturbation function is also strictly decreasing and vanishing in the limit for time to infinity. The perturbation is thereby guaranteed to be arbitrarily small for arbitrarily large times.
For their system the authors put forward an averaging conjecture [6, Conj 1]. Its statement is analogous to the averaging theorems mentioned above, and provides an error estimate between the full solution and a corresponding averaged solution, expressed in terms of orders of the perturbation function. If the conjecture holds, then one striking feature is that the solution of the full system converges to that of the averaged system in the limit for time to infinity, since the perturbation function goes to zero. Hence the task of finding the global future stable solution of the full system could be reduced to finding that of the simpler averaged system. 1 In the present example of LRS Bianchi type III the only constraint is the Hamiltonian constraint and the only evolution equation in addition to the Raychaudhuri equation is one for the (Hubble normalised) shear; cf. [6]. The latter has only one independent component in this case.
While this conjecture is strongly supported numerically [6, Sec 6] as well as analytically [6,Sec 4], it is still open. In the present work we provide a rigorous proof of two closely related statements. Firstly, we present a result concerning the local-in-time asymptotics. Albeit its error estimate is expressed in terms of orders of a constant and not in terms of orders of a strictly decreasing function, this constant can be made arbitrarily small and the corresponding domain of validity arbitrarily large (Theorem 2.1). Secondly, we also obtain a result concerning the global-in-time asymptotics, stating that the error between the full and the averaged solution indeed goes to zero in the limit for time to infinity (Theorem 2.2). More importantly, we emphasize here that our stability theorems can be applied to a general class of nonlinear dynamical systems even in the case that the systems involve time-dependent coefficients, as long as the coefficients are decreasing in time. We obtain the full asymptotic behavior globally in time of the system that involves time-dependent coefficients.
Closely related techniques have been developed in [1-3], where [1,2] also achieved results comparable to our theorems, with regards to the concrete problems considered in these references respectively. Since problems of this kind appear in many classes of cosmologies, and with different kinds of matter models, it is desirable to make theorems available which are applicable to greater generality. To this end, with this work we present mathematically rigorous theorems with only mild restrictions imposed onto the considered systems, such as a strictly decreasing perturbation function and continuity requirements on the underlying functions. Our theorems are thus applicable to a wide range of problems in mathematical cosmology and potentially beyond.
Finally we note that the proof of conjecture of [6] remains open. Its statement differs to that of the above results in the sense that it gives a time-dependent error estimate. If one requires this, or if a global-in-time limit is sufficient depends on the underlying problem. For the results of [6] the time dependence was crucial.
2. Main theorems, reformulation, and our strategy 2.1. A general class of Van der Pol equations. For our purpose it is instructive to compare (1.1) to a general class of Van der Pol equations with sufficiently smooth function g and where ǫ is a constant, in contrast to H = H(t) in (1.1); cf. [12, Sec 2.2]. For small ǫ, (2.1) represents a perturbed harmonic oscillator, which suggests a parametrisation of the solution by variation of arbitrary constants in terms of an amplitude r = r(t) and phase ϕ = ϕ(t): This transforms (2.1) into two first-order equations of the form [ṙ,φ] T = ǫ f (r, ϕ, t), with f 2π-periodic in t. From [12,Chap 2] this is a periodic averaging problem of the more general standard form 2

2.2.
Averaging methods. For this kind of problem the theory of averaging provides theorems which estimate the error between the solution x(t) of (2.3) and a z(t) which solves the corresponding averaged equatioṅ For instance [12, p 31, Thm 2.8.1] states that the error between x(t) and z(t) is O(ǫ) on timescales of O(ǫ −1 ). Note that both the error estimate and its domain of validity are given in terms of orders of powers of the perturbation constant ǫ.
The similarity between (1.1) and (2.1) suggests to treat the former as a perturbed harmonic oscillator as well, and to apply averaging in an analogous way. Care has to be taken however since in contrast to ǫ, H is time-dependent and itself subject to an evolution equation of the coupled system. If valid, then a striking feature of such an approach is the possibility to exploit the fact that H is strictly decreasing and going to zero. In (1.1) H takes the role of the perturbation parameter ǫ in (2.1), which controls the error and the domain of validity of the error estimate. Hence, with strictly decreasing H the error should get smaller and smaller, and the domain of validity of the estimate larger and larger. In the limit for t → ∞ the error should vanish. In other words, the future stability of the simpler averaged system should also determine the future stability of the more complicated full system.
In [6, Conj 1] a respective averaging conjecture has been coined. In [1,2] globalin-time asymptotics could be shown for the particular analogous systems considered in these sources.
2.3. The system in a general form. The purpose of the present work is to provide theorems which are applicable to a wide range of problems in spatially homogeneous cosmology, and potentially beyond. We therefore define our problem as general as possible and as restrictive as necessary.
The systems that we consider in the following have the general form where H = H(t) is positive, is strictly decreasing in t, and lim t→∞ H(t) = 0.
Remark 1. One example of the system of equations that the general form (2.4) represents is the LRS Bianchi type III Einstein-Klein-Gordon system introduced in [6], which is given byḢ with the deceleration parameter Here H is the Hubble scalar, and we see that in this example the first component of (2.4) is the the Raychaudhuri equation. φ is the scalar field satisfying the Klein-Gordon equation. Σ + is the only independent component of the Hubble normalized shear tensor, and Ω is the Hubble normalized energy density. The Hamiltonian constraint 1 − Σ 2 + − Ω > 0 and the non-vacuum condition Ω > 0 give the uniform bounds of Σ + ∈ (−1, 1) and Ω ∈ (0, 1). The third line (2.5) 3 has the form of a Van der Pol equation, and it can further be written as the following two first-order Klein-Gordon equations [6, Section 2.2]: which also implies q = 2Σ 2 + + Ω(3 cos(t − ϕ) 2 − 1). Then H and x def = (Σ + , Ω, ϕ) ⊤ satisfy the system (2.4); i.e., the components ofẋ stand for both the remaining Einstein evolution equation (2.5) 2 and the two firstorder equations (2.6) corresponding to the Klein-Gordon equation. Note that f 1 and f [2] are smooth with respect to x. Also, both f 1 and f [2] are in L ∞ x,t . Beyond the example from [6] the examples of [1-3] demonstrate that a wide range of problems in spatially homogeneous cosmology are of the form (2.4).

Notations.
We first note that, throughout the paper, the terms in bold cases (e.g. F, x, u, and etc.) are vectors, whereas the terms in a normal case (e.g. f , x, and etc.) are scalars.
We denote the 0 th to i th -order coefficients and the coefficient of the i th -order remainder term in the Taylor expansion of f (x, t, ǫ) near ǫ = 0 as f 0 (x, t), ..., f i (x, t) and f [i+1] (x, t, ǫ), respectively. Equivalently, this means that The norm · denotes the standard discrete ℓ 1 norm We also denote as L ∞ x,t (or simply L ∞ ) the standard L ∞ space in both x and t variables with the corresponding norm defined as

Main results.
We are now ready to formulate our main theorems. Consider the system (2.4) with positive and strictly decreasing H = H(t), and with lim t→∞ H(t) = 0. We then obtain the following local-in-time asymptotics on the system (2.4): Fix any ε > 0 with ε < H(0) and define t * > 0 such that ε = H(t * ). Suppose that x,t , f [2] L ∞ x,t < ∞, and that f 1 (x, t) is Lipschitz continuous and f [2] is continuous with respect to x for all t ≥ t * . Also, assume that f 1 and f [2] are T -periodic for some T > 0. Then for all t > t * with t = t * + O(H(t * ) −γ ) for some γ ∈ (0, 1), we have where x is the solution of the system (2.4) with the initial condition x(0) = x 0 and z(t) is the solution of the averaged equatioṅ with the initial condition z(t * ) = x(t * ) where the averagef 1 is defined as

Remark 2. Several Bianchi type Einstein-Klein-Gordon cosmologies including
Bianchi I, II, and III satisfy the assumptions of the theorem.
Using the local-in-time theorem and the smallness of H(t) for a sufficiently large t > 0, we can further deduce the following global-in-time theorem on the asymptotics: Theorem 2.2 (Global-in-time asymptotics). Assume the same assumptions as in Theorem 2.1. Then we have Remark 3. Here we would like to emphasize that we improve the standard theory on the averaging methods for nonlinear dynamical systems to one that is applicable to a general class of nonlinear dynamical systems whose equations can involve time-dependent coefficients on the differential operators. It is irrespective of the application to cosmology.
2.6. Our strategy and the outline of the proof. The proofs on the local and global asymptotic behaviors of the solutions to the system (2.4) involve the firstorder approximation of the system, the error estimates, and the use of periodic averaging. In Section 3.1, we introduce the first-order approximation of our system and prove the main ingredient Proposition 3.1 for our local-in-time theorem on the asymptotics of the system. In Section 3.2 we introduce a periodic averaging lemma, which closes the proof of the local-in-time asymptotics. In Section 3.3, we use the smallness of H(t) to extend our local theory to a global one.

Proof of the main theorems
We first introduce the first-order approximation of the system and the error estimates, which will be crucial ingredients for the proof of the local-in-time asymptotic behavior of the nonlinear system (2.4).
3.1. The first-order approximation and the error estimates. In order to approximate the solution x(t) of the system (2.4), we consider the following truncated first-order system; for t > t * with some given time t * , we consider where H(t * ) = H(t * ) and y(t * ) = x(t * ); i.e., the initial data coincide. Then sincė H = 0, we have H(t) ≡ H(t * ). We hereby denote H * def = H(t * ). Then we introduce the following proposition on the error estimates for the first-order approximation: In addition, suppose that L ∞ x,t < ∞, and that f 1 (x, t) is Lipschitz continuous in x for all t ≥ t * . Then for all t > t * with t = t * + O(H(t * ) −γ ) for some γ ∈ (0, 1), we have Proof. SinceḢ = 0 in (3.1), we have H(t) = H * , for all t ≥ t * . Now, we subtract the equation for y in (3.1) from that for x in (2.4) to obtain thaṫ By integrating this identity, we have by the initial condition y(t * ) = x(t * ). Now we consider the mean-value theorem for H(τ ) and observe that, for some t c = t c (τ ) ∈ (t * , τ ), we have Then (3.2) implies that because H(·) is decreasing, where c L is the Lipschitz constant and the third inequality is by the Lipschitz continuity of f 1 . Now we apply the Grönwall inequality and obtain Therefore, for the time-scale t = t * + O(H −γ * ) for any γ ∈ (0, 1) we have as long as H * is finite. This completes the proof.
3.2. The periodic averaging. Equipped with the first-order approximation and the error estimates, the next ingredient is a periodic averaging lemma. Throughout this section, we introduce the averaging system of the following initial value problem of the general form (3.3)ẏ = εf 1 (y, t), y(0) = y 0 , for t > 0 with y, f 1 (y, t) ∈ R n , where f 1 is T -periodic and ε is a small parameter. Note that (3.3) coincides with the equation for y in (3.1) if ε = H * . Now we fix a sufficiently large time t * > 0. Then we will consider the first-order associated averaged system (3.4)ż = εf 1 (z), for t > t * with the initial condition z(t * ) = y(t * ) where ε = H(t * ) and the averagef 1 is defined asf f 1 (z, s)ds.
Let D ⊂ R n is a connected, bounded open set (with compact closure) containing the initial value y(t * ) = z(t * ). Fix L > 0 and ε 0 > 0 such that the solutions y(t, ε) and z(t, ε) with 0 ≤ ε ≤ ε 0 remain in D for t * ≤ t ≤ t * + L/ε. Then there is a basic result on the asymptotic error estimates: Lemma 3.2 (Theorem 2.8.1 of [12]). Let f 1 be Lipschitz continuous. Also, assume that f 1 is T -periodic for some T > 0. Let ε 0 , D, L be as above. Then there exists a constant c > 0 such that the solutions y and z of (3.3) and (3.4) satisfy y(t, ε) − z(t, ε) < cε, for 0 ≤ ε ≤ ε 0 and t * ≤ t ≤ t * + L ε , where · denotes the norm Then Proposition 3.1 and Lemma 3.2 together imply Theorem 2.1 by letting ε = H(t * ), using the triangle inequality and noting that the time scale O(H(t * ) −γ ) is smaller than O(H(t * ) −1 ) for a sufficiently large time t * such that H(t * ) = ε ≪ 1. This completes the proof of Theorem 2.1. This completes the proof.