Asymptotic Behaviour of a Non-Autonomous Lorenz-84 System

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume xx, Number 0x, xxxx xxxx pp. xxxx–xxxx ASYMPTOTIC BEHAVIOUR OF A NON-AUTONOMOUS LORENZ-84 SYSTEM Mar ´ ıa Anguiano and Tom´ as Caraballo Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla Campus Reina Mercedes, Apdo. de Correos 1160 41080 Sevilla, Spain (Communicated by Associate Editor) Dedicado a la Memoria de nuestro querido Maestro y Amigo Pepe Real, con toda nuestra Gratitud. Abstract. The so called Lorenz-84 model has been used in climatological studies, for example by coupling it with a low-dimensional model for ocean dynamics. The behaviour of this model has been studied extensively since its introduction by Lorenz in 1984. In this paper we study the asymptotic behaviour of a non-autonomous Lorenz-84 version with several types of nonautonomous features. We prove the existence of pullback and uniform attractors for the process associated to this model. In particular we consider that the non-autonomous forcing terms are more general than almost periodic. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate some examples of pullback attractors by numerical simulations. 1. Introduction and setting of the problem. Weather and climate prediction are difficult tasks, because of the complexity of the atmospheric evolution. Most of the real world models concerning the atmosphere involve a large number of variables and parameters. Therefore, it is practically impossible to perform detailed studies of their dynamical properties. There is experimental evidence [31] that low-dimensional attractors appear in some hydrodynamical flows just after the onset of turbulence. As a consequence, low-dimensional models have attracted the attention of meteorologists, mathematicians and physicists over the last decades. In recent years, a great deal of interest has been focused on studying the complexity of nonlinear dynamical systems. The Lorenz classical model for thermal convection in the atmosphere [23] was the first chaotic system discovered and has been one of the most extensively investigated. As a modification of this model of turbulence generation, in [24] Lorenz derived a simple but powerful model based on the ‘general circulation” of the atmosphere. This model proposed by Lorenz in 1984 is a Galerkin truncation of the Navier-Stokes equations and gives the simplest approximation to the general atmospheric circulation at midlatitude. Lorenz-84 model has been used in climatological studies and its behaviour has been studied extensively. In this paper we analyze the non-autonomous Lorenz-84 model with non-autonomous 2000 Mathematics Subject Classification. 37B55, 35B41, 37L30. Key words and phrases. Lorenz system, Non-autonomous equation, Pullback Attractor, Uniform Attractor, Hausdorff dimensionn. Partly supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2011-22411. 1 2 M. ANGUIANO AND T. CARABALLO forcing terms. The model is defined by the following three ordinary differential equations x0=−ax −y2−z2+aF(t), y0=−y+xy −bxz +G(t), z0=−z+bxy +xz, 9 > > > = > > > ; (1) with initial condition x(t0) = x0, y(t0) = y0, z(t0) = z0,(2) where t0∈R,a > 0, b∈Rand F, G :R7→ Rare continuous functions such that satisfy Zt −∞ elsF2(s)ds < +∞,Zt −∞ elsG2(s)ds < +∞ ∀t∈R, (3) where l:= min{a, 1}. Fand Grepresent thermal forcings: Fis meant to be the symmetric cross-latitude externalheating contrast and Gthe asymmetric heating contrast between oceans and continents, and a and bare positive parameters (a < 1 and b > 1). Conventionally a=1 4and b= 4. The variable xrepresents the intensity of the symmetric globe-encircling westerly wind current and also the poleward temperature gradient which is assumed to be in permanent equilibrium with it. The variables yand zrepresent the cosine and sine phases of a chain of superposed large-scale eddies, which transport heat poleward at a rate proportional to the square of their amplitude, and transport no angular momentum at all. In [24], some early ideas concerning the general circulation of the atmosphere are reviewed. Moreover, Lorenz proved that for different intensities of the axially symmetric and asymmetric thermal forcing, Lorenz-84 model may possess one or two stable steady-state solutions, one or two stable periodic solutions, or irregular (aperiodic) solutions. Numerical and analytical explorations of model introduced by Lorenz in 1984 can be found for instance in [26], [27] and [30], and the bifurcation diagram of this model has been analysed in [29]. In [25], Lorenz pointed out that Fand Gshould be allowed to vary periodically during a year. In particular, Fshould be larger in winter than in summer. However, in his numerical study he kept Gfixed, identifying (F, G) = (6,1), and (F, G) = (8,1), with summer, respectively winter, conditions. He introduced a periodical variation of the parameter F between summer and winter conditions. In [3] the authors considered that the differential equations of Lorenz-84 model are subject to periodic forcing term, where the period is one year and different types of strange attractors are found in four regions (chaotic ranges). Moreover the related routes to chaos are discussed. There are basically two ways to define attraction of a compact and invariant non–autonomous set for a process on a metric space. The first, and perhaps more obvious, corresponds to the attraction in the sense of Lyapunov stability, which is called forward attraction, and involves a moving target, while the second, called pullback attraction, involves a fixed target set with progressively earlier starting time. In general, these two types of attraction are independent concepts, while for the autonomous case, they are equivalent. Physically, the pullback attractor provides a way to assess an asymptotic regime at time t(the time at which we observe the system) for a system starting to evolve from the remote past. The pullback dynamics contains interesting dynamical properties, which allow us to understand the forward attraction. We would like to emphasize, for instance, Theorem 3.20 in [14], which provides some information about a form of forwards convergence of the cocycle mapping, which is different from that in the definition of a forward attractor. In this theorem, within the framework of cocycle dynamical systems, it is proved that the closure of the union of the fibers of the pullback attractor is forward attracting under the assumption that there exists a compact pullback absorbing set Bfor the skew product flow such that the cocycle mapping on Bis contained in B(see [7], [14] for more details). The first aim of this paper is to show the existence of a pullback and a uniform attractor for the process associated to (1)-(2). The fact that Fand Gare non-autonomous are the main novelties of our problem. The dynamics induced by the class of periodic, almost periodic or almost automorphic continuous functions is not robust to small changes in the forcing term in the sense that a bounded entire solution corresponding to a perturbed forcing term may not belong to this class. Then, Kloeden and Rodrigues presented in [15] an alternative extension of periodic and almost periodic functions. Namely, the introduce the class of functions consisting of uniformly continuous functions, defined on the real line and taking values in a Banach space, with the property that a bounded entire solution of a non autonomous ODE belongs to this class when the forcing term does. ASYMPTOTIC BEHAVIOUR OF A NON-AUTONOMOUS LORENZ-84 SYSTEM 3 The fact that the forcing term belongs to the class more general than almost periodic in a nonautonomous systems means that the external force of the phenomena modeled possesses different types of intensity along time. For example, a tornado is a meteorological problem which can have different steps: initially, the intensity of the external force can be periodic for a certain period of time, then the type of intensity may change and can become constant for another period of time, later on, it may become periodic again but with a different period than the first one, and continue in this way recursively. Then, this phenomena can be modeled by a non–autonomous system where the forcing term belongs to this class more general than almost periodic. For this reason, we also consider that (1) includes forcing terms which belong to this class of functions introduced by Kloeden and Rodrigues in [15] . On the other hand, the theory of topological dimension [13], developed in the first half of the 20th century, is of little use in giving the scale of dimensional characteristics of attractors. The point is that the topological dimension can take integer values only. Hence the scale of dimensional characteristics compiled in this manner turns out to be quite poor. For investigating attractors, the Hausdorff dimension of a set is much better. This dimensional characteristic can take any nonnegative value. In [1], a simple estimate for the dimension of attractors of Lorenz system has been obtain. In [19] the authors interpret the Hausdorff measure as an analogue of a Lyapunov function, and they estimate the Hausdorff dimension of strange attractors, particularly the (generalized) Lorenz systems. Recently in [22] Lyapunov-type functions are introduced into upper estimates for the Hausdorff dimension of negatively invariant sets of cocycles. For this purpose, the methods proposed in [2, 19, 20, 21] are further developed. The second aim of this paper is to estimate the Hausdorff dimension of the pullback attractor of (1)-(2) using the recent method proposed by Leonov et al. in [22]. The structure of the paper is as follows. In Section 2 we briefly recall some abstract results about the theory of pullback and uniform attractors. Some sufficient conditions ensuring the existence of such type of attractors for (1)-(2) are collected in Section 3. We consider that the differential equations of non-autonomous Lorenz-84 model are subjected to periodic forcing terms in Section 4. In Section 5 we consider that (1) includes forcing terms which belong to a class of functions more general than almost periodic. We use recent results proposed by Kloeden and Rodrigues [15] to prove that the solution of (1)-(2) belongs to this class when the forcing terms do. Finally, in Section 6 we estimate the Hausdorff dimension of the pullback attractor associated to (1)-(2). We obtain estimates that are similar to those for the autonomous case (cf. [19]) and illustrate these results with some numerical simulations. 2. Abstract results on Pullback and Uniform Attractors. In this section we recall some abstract results on the theory of pullback attractors (see [5, 6, 7]) and we establish some results on the theory of uniform attractors (see [7, 10]). 2.1. Processes and attractors. Let (X, dX) be a metric space, and let us denote R2 d={(t, t0)∈ R2:t0≤t}. A process on Xis a mapping Usuch that R2 d×X3(t, t0, x)7→ U(t, t0)x∈Xwith U(t0, t0)x=x for any (t0, x)∈R×X, and U(t, r)(U(r, t0)x) = U(t, t0)xfor any t0≤r≤tand all x∈X. Definition 1. Let Ube a process on X.Uis said to be continuous if for any pair t0≤t, the mapping U(t, t0) : X→Xis continuous. Let us denote P(X) the family of all nonempty subsets of X, and consider a family of nonempty sets b D={D(t) : t∈R}⊂P(X). Let Dbe a nonempty set of parameterized families of nonempty bounded sets b D={D(t) = D:t∈R} ⊂ P(X), where D⊂Xis a bounded set. In what follows, we will consider a fixed universe of attraction Dand throughout our analysis the concepts of absorption and attraction will be referred to this fixed universe. Definition 2. It is said that b D0⊂ P(X)is pullback absorbing for the process Uon Xif for any t∈Rand any b D∈ D,there exists a b t0(t, b D)≤tsuch that U(t, t0)D(t0)⊂D0(t)for all t0≤b t0(t, b D). We denote by distX(O1,O2) the Hausdorff semi-distance in Xbetween two sets O1and O2, defined as distX(O1,O2) = sup x∈O1 inf y∈O2 dX(x, y) for O1,O2⊂X. 4 M. ANGUIANO AND T. CARABALLO Definition 3. It is said that b D0⊂ P(X)is pullback attracting if lim t0→−∞ distX(U(t, t0)D(t0), D0(t)) = 0 for all b D∈ D, t ∈R. There exists now a wide literature on pullback attractors (see, e.g., [16, 17, 28]), but we would like to emphasize that these notions take the final time as fixed and moves the initial time backwards towards −∞. Note that this does not mean that we are moving backwards in time, but we consider the state of the system at time tthat had begun in earlier and earlier initial instants t0, i.e., t0→ −∞. Definition 4. Let b D0(t)⊂ P(X). This family is said to be invariant with respect to the process Uif U(t, t0)D0(t0) = D0(t)for all t0≤t. Denote the omega-limit set of b Dby Λ( b D, t) := \ s≤t[ t0≤s U(t, t0)D(t0) X for all t∈R,(4) where {···}Xis the closure in X. Definition 5. The family of compact sets {A(t)}t∈Ris said to be a pullback attractor associated to the continuous process Uif is invariant, attracts every {D(t)} ∈ D and minimal in the sense that if {C(t)}t∈Ris another family of closed attracting sets, then A(t)⊂C(t)for all t∈R. The general result on the existence of pullback attractor is a generalization of the abstract theory for autonomous dynamical systems [32]: Theorem 6. [Crauel et al. [11], Schmalfuss [28]] Assume that there exists a family of compact pullback absorbing sets {B(t)}t∈R. Then, the family {A(t)}t∈Rdefined by A(t) = [ b D∈D Λ“b D, t”X ,(5) is the pullback attractor, where Λ“b D, t”is the omega-limit set at time tof b D∈ D, where Dis the universe of fixed nonempty bounded subsets of X. Another approach to the asymptotic dynamics of non-autonomous equations, the uniform attractor, has been developed by Chepyzhov and Vishik [10]. The theory of uniform attractors can be developed for a single non-autonomous process (see [9, 10]). Definition 7. A set K⊆Xis said to be uniformly (with respect to t0∈R) attracting for the process {U(t, t0)}on Xif for all t0∈Rand for any bounded set B⊂X, lim T→+∞ sup t0∈R distX(U(T+t0, t0)B, K)!= 0.(6) Respectively, the process {U(t, t0)}is said to be uniformly asymptotically compact (with respect to t0∈R) if there exists a compact uniformly (with respect to t0∈R) attracting set of {U(t, t0)}. Definition 8. A closed set A1⊆Xis said to be a uniform (with respect to t0∈R) attractor for a process {U(t, t0)}if it is the minimal closed uniformly (with respect to t0∈R) attracting set for this process. Minimality is meant in the sense that any closed attracting set is contained in A1. Theorem 9. [Chepyzhov and Vishik [8, 10], Haraux [12]] If a process {U(t, t0)}is uniformly asymptotically (with respect to t0∈R) compact, then it has the uniform (with respect to t0∈R) attractor A1. The set A1is compact in X. To describe the structures of uniform attractors and to perform a comparison with the pullback attractor we introduce the notions of complete trajectory of a process, kernel of a process and kernel section (the terminology is due to Chepyzhov and Vishik [9, 10]). Definition 10. A map u:R→Xis called a complete trajectory of a process U(t, t0)if U(t, t0)u(t0) = u(t)for all t≥t0, t, t0∈R. ASYMPTOTIC BEHAVIOUR OF A NON-AUTONOMOUS LORENZ-84 SYSTEM 5 Definition 11. The kernel Kof a process U(t, t0)consists of all of its bounded complete trajectories of the process U(t, t0). Definition 12. The set K(s) = {u(s) : u(·)∈K} is said to be the kernel section at a time moment t=s,s∈R. These kernel sections are, essentially, the fibres of the pullback attractor: if U(·,·) is a process that has a pullback attractor A, then any backwards bounded trajectory is contained in A(t), and we can deduce that if A(·) is bounded then A(t) = K(t) (see for instance [7]). Observe that Theorem 9 implies the existence of a (fixed) compact attracting set Kfor U(·,·), so that, from (6) and Theorem 3.11 in [7] it also implies the existence of a pullback attractor, which is then uniformly included in K. Just as A(t) must contain K(t) for each t, the uniform attractor must contain the union of all the kernel sections (see [7]). Lemma 13. If U(·,·)has a uniform attractor A1, then [ t∈RK(t)⊆ A1. 2.2. The structures of attractors for periodic processes. In this subsection we investigate attractors for periodic processes. Let {U(t, t0)}be a periodic process, and let Tbe its period, i.e., U(t+T, t0+T) = U(t, t0)∀t≥t0, t0∈R. We can now state a theorem about attractors of periodic processes. Theorem 14. [Chepyzhov and Vishik [9]] Let {U(t, t0)}be a periodic uniformly (with respect to t0∈R) asymptotic compact and (X×T1)−continuous process, where T1=R(mod T). Then, the process {U(t, t0)}has a uniform (with respect to t0∈R) attractor, A1, and is given by A1=[ σ∈[0,T )K(σ), where K(σ)is the kernel section of the process {U(t, t0)}at time t=σ. 3. Pullback and uniform attractors. Since the functions on the right hand side of (1) are locally Lipschitz with respect to x,y, and z, then for any t0∈Rand any (x0, y0, z0)∈R3there exists a unique local solution of the model (1)-(2), denoted by u(t;t0, u0) := (x(t;t0,(x0, y0, z0)), y(t;t0,(x0, y0, z0)), z(t;t0,(x0, y0, z0))), and this solution is a global solution one (8) is proved. 3.1. Pullback Attractor. In this section, we will show the existence of a pullback attractor in R3of our problem (1)-(2). First, thanks to the uniqueness of solution of (1)-(2), we can define a process {U(t, t0), t0≤t}in R3, by U(t, t0)u0=u(t;t0, u0)∀u0∈R3. (7) The process defined by (7) is continuous in R3. Proposition 15. Assume that a > 0and b∈R. Then for any initial condition u0∈R3, the solution uof (1)-(2) satisfies |u(t;t0, u0)|2≤e−l(t−t0)|u0|2+ae−lt Zt −∞ elsF2(s)ds +e−lt Zt −∞ elsG2(s)ds,(8) for all t≥t0, where l:= min{a, 1}. Proof. We deduce that d dt |u(t)|2=−2(ax2+y2+z2)+2axF(t)+2yG(t). We have 2axF(t)≤ax2+aF2(t), and 2yG(t)≤y2+G2(t). Then, we can deduce d dt |u(t)|2+l|u(t)|2≤aF2(t) + G2(t), (9) where l:= min{a, 1}. 6 M. ANGUIANO AND T. CARABALLO Multiplying (9) by elt, we obtain that d dt “elt |u(t)|2”≤aeltF2(t) + eltG2(t). Integrating between t0and t elt |u(t)|2≤elt0|u0|2+aZt t0 elsF2(s)ds +Zt t0 elsG2(s)ds (10) ≤elt0|u0|2+aZt −∞ elsF2(s)ds +Zt −∞ elsG2(s)ds, whence (8) follows. We consider the universe of fixed nonempty bounded subsets of R3. Now, we prove that there exists a pullback absorbing family for the process U(t, t0) defined by (7). Proposition 16. Assume that a > 0and b∈R. Let F, G satisfy (3). Then, the family b D0= {D0(t) : t∈R}defined by D0(t) = BR3(0, ρl(t)), where ρl(t)is the nonnegative number given by ρ2 l(t) = 1 + ae−lt Zt −∞ elsF2(s)ds +e−lt Zt −∞ elsG2(s)ds,∀t∈R,(11) is pullback absorbing family for the process Udefined by (7). Proof. Let D⊂R3be bounded. Then, there exists d > 0 such that |u0| ≤ dfor all u0∈D. Thanks to Proposition 15, we deduce that for every t0≤tand any u0∈D, |U(t, t0)u0|2≤e−ltelt0|u0|2+ae−lt Zt −∞ elsF2(s)ds +e−lt Zt −∞ elsG2(s)ds ≤e−ltelt0d2+ae−lt Zt −∞ elsF2(s)ds +e−lt Zt −∞ elsG2(s)ds. If we consider T(t, D) := l−1log(eltd−2), we have |U(t, t0)u0|2≤1 + ae−lt Zt −∞ elsF2(s)ds +e−lt Zt −∞ elsG2(s)ds, for all t0≤T(t, D) and for all u0∈D. Consequently the family b D0={D0(t) : t∈R}defined by D0(t) = BR3(0, ρl(t)) is pullback absorbing for the process Udefined by (7). Now, as a direct consequence of the preceding results and Theorem 6, we have the existence of the pullback attractor for the process Udefined by (7). Theorem 17. Under the assumptions in Proposition 16, the process Udefined by (7) possesses a pullback attractor A, which is given by A(t) = [ b D∈D Λ“b D, t”.(12) 3.2. Uniform Attractor. Now, we suppose that Fand Gare translations bounded in L2 loc(R;R), i.e., sup t∈RZt+1 t F2(s)ds < ∞,sup t∈RZt+1 t G2(s)ds < ∞.(13) In this subsection, using Theorem 9, we will prove that, under the assumption (13), the process {U(t, t0)}has a uniform (with respect to t0∈R) attractor. Remark 18. Observe that assumption (13) implies (3). Proposition 19. Assume that a > 0and b∈R. Let F,Gsatisfy (13). Then, the process U defined by (7) is uniformly (with respect to t0∈R) asymptotically compact. ASYMPTOTIC BEHAVIOUR OF A NON-AUTONOMOUS LORENZ-84 SYSTEM 7 Proof. Let D⊂R3be bounded, and as in the proof of Proposition 16, let d > 0 such that |u0| ≤ d for all u0∈D. Using (10), we have for any u0∈D |u(t;t0, u0)|2≤e−l(t−t0)|u0|2+aZt t0 e−l(t−s)F2(s)ds +Zt t0 e−l(t−s)G2(s)ds ≤e−l(t−t0)d2+aZt t0 e−l(t−s)F2(s)ds +Zt t0 e−l(t−s)G2(s)ds, (14) for all t≥t0. We estimate the integrals on the right-hand side of (14), taking into account (13), Zt t0 e−l(t−s)F2(s)ds ≤Zt −∞ e−l(t−s)F2(s)ds ≤X n≥0Zt−n t−(n+1) e−l(t−s)F2(s)ds ≤X n≥0 e−nl Zt−n t−(n+1) F2(s)ds =C1(1 −e−l)−1, where C1:= supt∈RRt+1 tF2(s)ds < ∞. Similarly, we have Zt t0 e−l(t−s)G2(s)ds ≤C2(1 −e−l)−1, where C2:= supt∈RRt+1 tG2(s)ds < ∞. Then, we can deduce that there exists a positive constant Clsuch that |u(t;t0, u0)|2≤e−l(t−t0)d2+Cl. Replacing tby t+t0, we have |u(t+t0;t0, u0)|2≤e−ltd2+Cl, and if we consider t≥T(D) := log d2 l, we obtain |u(t+t0;t0, u0)|2≤1 + Cl, for all t0and for all u0∈D. Then, the set B0:= BR3(0,1 + Cl) is compact and uniformly (with respect to t0∈R) attracting for the process Udefined by (7). Therefore, the process Uis uniformly (with respect to t0∈R) asymptotically compact. We can now state a theorem about the existence of a uniform attractor of (1)-(2). Taking into account Theorem 9 and Lemma 13, we can deduce the following result. Theorem 20. Under the assumptions in Proposition 19, the process Udefined by (7) has a uniform attractor A1, which is compact in R3. Moreover, we have the following relation: [ t∈RA(t)⊆ A1,(15) where A(t)is given by (12). Remark 21. Note that if bdepends continuously on time, the previous results also hold true. 4. Attractors for Periodic Equations. We consider (1)-(2) with a > 0, b∈Rand T-periodic continuous functions F, G :R7→ R. 4.1. Pullback Attractor. In this subsection we show that when we have periodic nonlinear terms we obtain that the pullback attractor is a periodic pullback attractor. We observe that F and Gsatisfy (3). In fact Fand Gsatisfy (13). Then, under the assumptions in Proposition 16, the process defined by (7) has a pullback attractor Awhich is given by (12). Corollary 22. Assume that Fand Gare T-periodic continuous functions. Then the process U defined by (7) is periodic with period T, that is U(t, t0)u0=U(t+T, t0+T)u0, for all t0, t ∈R, and the pullback attractor A(·)is also periodic with period T. 8 M. ANGUIANO AND T. CARABALLO Proof. We can deduce that (X(·;t0, u0), Y (·;t0, u0), Z(·;t0, u0)) := (x(·+T;t0+T, u0), y(·+T;t0+ T, u0), z(·+T;t0+T, u0)) is the unique solution of (1) with initial value u0at t=t0because dX dt (t) = dx dt (t+T) = dx dτ (τ) = −ax(τ)−y2(τ)−z2(τ) + aF(τ) where τ=t+T =−aX(t)−Y2(t)−Z2(t) + aF(t), by T-periodicity of F, dY dt (t) = dy dt (t+T) = dy dτ (τ) = −y(τ) + x(τ)y(τ)−bx(τ)z(τ) + G(τ) where τ=t+T =−Y(t) + X(t)Y(t)−bX(t)Z(t) + G(t), by T-periodicity of G, and dZ dt (t) = dz dt (t+T) = dz dτ (τ) = −z(τ) + bx(τ)y(τ) + x(τ)z(τ) where τ=t+T =−Z(t) + bX(t)Y(t) + X(t)Z(t). Hence, we have U(t, t0)u0=U(t+T, t0+T)u0, for all t≥t0. Replacing t0by t0−t, where t≥0, and tby t0, we thus have U(t0, t0−t)u0=U(t0+T, t0+T−t)u0, so, by (4), Λ( b D, t0) := \ s≤t0[ t0−t≤s U(t0, t0−t)D(t0−t) =\ s≤t0[ t0−t≤s U(t0+T, t0+T−t)D(t0−t) = Λ( b D, t0+T), and then A(t0) = [ b D∈D Λ“b D, t0”=[ b D∈D Λ“b D, t0+T”=A(t0+T). Hence, A(·) is also T-periodic. In Figure 1 we exhibit a simulation showing the pullback attractor for (1)-(2) with Fand G periodic functions. In this simulation, we used the following parameters and initial conditions values: a=1 4,b= 4, F(t) = cos(t), G(t) = cos(t), x(−2000) = 0, y(−2000) = −0.2 and z(−2000) = 0.5. Figure 1. Numerical solution (x(t), y(t), z(t)) ASYMPTOTIC BEHAVIOUR OF A NON-AUTONOMOUS LORENZ-84 SYSTEM 9 Remark 23. Note that if bis a T-periodic continuous function b:R7→ R, the previous results also hold. In Figure 2 we present a simulation showing the pullback attractor for (1)-(2) with b,Fand Gperiodic functions. We used the following parameters and initial conditions values: a=1 4,b(t) = 4 + 0.5cos(t),F(t) = cos(t),G(t) = cos(t),x(−2000) = 0,y(−2000) = −0.2and z(−2000) = 0.5. Figure 2. Numerical solution (x(t), y(t), z(t)) Now, we consider that Fand Ghave different periods. For a=1 4and b= 4, we have the following simulations: •In Figure 3 we present a simulation showing the pullback attractor for (1)-(2) where we used the following non-autonomous forcing terms and initial conditions values: F(t) = cos(t), G(t) = cos(2t), x(−2000) = 0, y(−2000) = −0.2 and z(−2000) = 0.5. •In Figure 4 we consider F(t) = cos(t), G(t) = cos(t/2) and the following initial conditions values: x(−3000) = 0, y(−3000) = −0.2 and z(−3000) = 0.5. •In Figure 5 we consider F(t) = cos(t), G(t) = cos(√2t) and the following initial conditions values: x(−500) = 0, y(−500) = −0.2 and z(−500) = 0.5. Figure 3. Numerical solution (x(t), y(t), z(t)) 16 M. ANGUIANO AND T. CARABALLO [12] A. 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