Attractors for a non-autonomous Liénard equation

Attractors for a non-autonomous Li´enard equation Mar´ıa Anguiano Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla (Spain) e-mail: [email protected] Abstract In this paper we prove the existence of pullback and uniform attractors for a non-autonomous Li´enard equation. The relation among these attractors is also discussed. After that, we consider that the Li´enard equation includes forcing terms which belong to a class of functions extending periodic and almost periodic functions recently introduced by Kloeden and Rodrigues in [14]. Finally, we estimate the Hausforff dimension of the pullback attractor. We illustrate these results with a numerical simulation: we present a simulation showing the pullback attractor for the non-autonomous Van der Pol equation, an important special case of the non-autonomous Li´enard equation. Keywords: Li´enard Equation, Non-autonomous equation, Pullback Attractor, Uniform Attractor, Hausdorff dimension Mathematics Subject Classifications (2010): 37B55, 35B41, 37L30 1 Introduction and setting of the problem The Li´enard equation was introduced in 1928 by the French physicist A. Li´enard in the paper [25]. The Li´enard equation is a particularly interesting equation as many questions arising in the physical sciences are concerned with it (see for instance [29], [33], [34]). The Li´enard equation, which is often taken as the typical example of nonlinear self-excited vibration problem, can be used to model resistor-inductor- capacitor circuits with nonlinear circuit elements. During the development of radio and vacuum tubes, Li´enard equations were intensely studied as they can be used to model oscillating circuits. It can also be used to model certain mechanical systems which contain the nonlinear damping coefficients and the restoring force or stiffness. The Li´enard equation has been studied for a long time, and there are many results. The general autonomous equation of Li´enard type was first studied by Levinson and Smith in the classical paper [24], and later on several authors have contributed to 1 the theory of this equation with respect to existence and uniqueness of nontrivial periodic solutions. The non-autonomous Li´enard equation has been well-studied (see for instance [10, 17, 18, 19, 27] as well as their lists of references) and there are many results on the dissipative nature of the non-autonomous Li´enard equation (see [4], [20], [28]). Leonov proposed in [21] a method for localizing the attractors of the nonautonomous Li´enard equation based on the construction of special piecewise-linear discontinuous comparison systems. Later, the universality of the construction of the comparison systems examined in [22] by Leonov enables to introduce different variable parameters which improve the localization theorem of [21]. 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 (see [3], [13] for more details). To our knowledge, there does not seem to be in the literature any study of the existence of pullback attractors for non-autonomous Li´enard equations. Let us introduce the model we will be involved with in this paper. We consider the following problem for a non-autonomous Li´enard equation, x00 +f(x)x0+a(t)x=E(t).(1) Denote by y=x0, then equation (1) can be reduced to the following equivalent first order system x0=y, y0=−a(t)x−f(x)y+E(t),   (2) with initial condition x(t0) = x0, y(t0) = y0,(3) where t0∈R. We assume that the following conditions are fulfilled: (H1) fis a locally Lipschitz continuous function with respect to xand there exists f0>0 such that f(x)≥f0for all x. (H2) E:R7→ Ris a continuous function such that satisfies Rt −∞ ef0sE2(s)ds < +∞ for all t∈R. 2 (H3) a:R7→ Ris a continuous function such that 0 < a0≤a(t)≤a1for all t∈R and (a(t)ef0t)0≤0 on [t0,∞). Van der Pol equation is an important special case of the Li´enard equation. This equation was introduced in 1920 by the Dutch physicist Balthasar Van der Pol and is an example of the long-standing interaction between differential equations and the physical and biological sciences. A few years after, in [32] Van der Pol and Van der Mark modeled the electric activity of the heart rate. Later, Fitzhugh [9] and Nagumo [26] extended the Van der Pol equation in a planar field as a model for action potentials of neurons. Recently, this equation has also been utilized in seismology to model the two plates in a geological fault. The non-autonomous Van der Pol equation x00 −µ(1 −x2)x0+a(t)x=E(t),(4) where µis a positive constant, describes the behavior of the Van der Pol oscillator when acted upon by a external disturbance in the presence of a linear restoring force and non-linear damping. When |x|is small, the quadratic term x2is negligible and the system becomes a linear differential equation with a negative damping −µ, i.e., x00 −µx0+a(t)x=E(t). On the other hand, when |x|is large, the term x2becomes dominant and the damping becomes positive and dissipation occurs. Therefore, an important special case of (1) is (4) for large |x|. The first aim of this paper is to show the existence of a pullback and a uniform attractor for the process associated to (2)-(3). The fact that aand Eare nonautonomous is the main novelties of our problem. A temporally global solution, if it exists, of a non-autonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions, which has the property that a bounded temporally global solution of a non-autonomous ordinary differential equation belongs to this class when the forcing term does, is introduced by Kloeden and Rodrigues in [14]. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. The second aim of this paper is to consider that (2) includes forcing terms which belong to this class of functions introduced by Kloeden and Rodrigues. On the other hand, the theory of topological dimension [12], 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 3 manner turns out to be quite poor. For investigating attractors, the Hausdorff dimension of a set is much better. In [23] Lyapunov-type functions are introduced into upper estimates for the Hausdorff dimension of negatively invariant sets of cocycles. In this sense, the third aim of this paper is to estimate the Hausdorff dimension of the pullback attractor of (2)-(3) using the recent method proposed by Leonov et al. in [23]. The structure of the paper is as follows. A brief recall on abstract results about the existence of pullback and uniform attractors is given in Section 2. In Section 3, the main goals of proving the existence of pullback and uniform attractors of (2)-(3) and the relation among them under certain suitable assumption, are established. In Section 4, we consider that (1) includes forcing terms which belong to a class of functions introduced by Kloeden and Rodrigues [14]. We illustrate these results with a numerical simulation: we present a simulation showing the pullback attractor for the non-autonomous Van der Pol equation (4) for large |x|. Finally, in Section 5 we estimate the Hausdorff dimension of the pullback attractor associated to (2)-(3). 2 Abstract results on Pullback and Uniform Attractors In this section we recall some abstract results on the theory of pullback attractors (see [1, 2, 3]) and we establish some results on the theory of uniform attractors (see [3, 7]). 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∈X with U(t0, t0)x=xfor 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). 4 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. 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., [15, 16, 30]), 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 t that 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, 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 [31]: Theorem 6 [Crauel et al. [8], Schmalfuss [30]] 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 , 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. 5 Another approach to the asymptotic dynamics of non-autonomous equations, the uniform attractor, has been developed by Chepyzhov and Vishik [7]. The theory of uniform attractors can be developed for a single non-autonomous process (see [6, 7]). 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.(5) 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 [5, 7], Haraux [11]] 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 cross-section of the kernel (the terminology is due to Chepyzhov and Vishik [6, 7]). 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. 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 [3]). Observe that Theorem 9 implies the existence of a (fixed) 6 compact attracting set Kfor U(·,·), so that, from (5) and Theorem 3.11 in [3] 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 [3]). Lemma 13 If U(·,·)has a uniform attractor A1, then [ t∈R K(t)⊆ A1. 3 Pullback and uniform attractors Since the functions on the right hand side of (2) are locally Lipschitz with respect to xand y, then for any t0∈Rand any (x0, y0)∈R2there exists a unique local solution of the model (2)-(3), denoted by u(t;t0, u0) := (x(t;t0,(x0, y0)), y(t;t0,(x0, y0))), and this solution is a global solution one (7) is proved. 3.1 Pullback Attractor In this section, we will show the existence of a pullback attractor in R2of our problem (2)-(3). First, thanks to the uniqueness of solution of (2)-(3), we can define a process {U(t, t0), t0≤t}in R2, by U(t, t0)u0=u(t;t0, u0)∀u0∈R2. (6) The process defined by (6) is continuous in R2. Proposition 14 Assume (H1)-(H3). Then, for any initial condition u0∈R2, the solution uof (2)-(3) satisfies |u(t;t0, u0)|2≤l1 l0 e−f0(t−t0)|u0|2+1 l0f0 e−f0tZt −∞ ef0sE2(s)ds,(7) for all t≥t0, where l0:= min {1, a0}and l1:= max {1, a1}. Proof. We deduce that d dty2(t) = −2a(t)xy −2f(x)y2+ 2E(t)y. We have 2f(x)y2≥2f0y2, and 2E(t)y≤f0y2+1 f0 E2(t). 7 Then, we can deduce d dty2(t) + f0y2+ 2a(t)xy ≤1 f0 E2(t). (8) Multiplying (8) by ef0t, we obtain that d dt ef0ty2(t)+ 2ef0ta(t)xy ≤1 f0 ef0tE2(t). Integrating between t0and t ef0ty2(t)+2Zt t0 ef0sa(s)x(s)y(s)ds ≤ef0t0y2 0+1 f0Zt t0 ef0sE2(s)ds. (9) Taking into account that y=x0, integrating by parts and using (H3), we have 2Zt t0 ef0sa(s)x(s)y(s)ds =ef0ta(t)x2(t)−ef0t0a(t0)x2 0−Zt t0 x2(s)(a(s)ef0s)0ds ≥ef0ta0x2(t)−ef0t0a1x2 0, so that (9) becomes, ef0t|u(t)|2≤l1 l0 ef0t0|u0|2+1 l0f0Zt t0 ef0sE2(s)ds (10) ≤l1 l0 ef0t0|u0|2+1 l0f0Zt −∞ ef0sE2(s)ds, whence (7) follows. We consider the universe of fixed nonempty bounded subsets of R2. Now, we prove that there exists a pullback absorbing family for the process U(t, t0) defined by (6). Proposition 15 Under the assumptions in Proposition 14, the family b D0={D0(t) : t∈R}defined by D0(t) = BR2(0, ρ0(t)), where ρ0(t)is the nonnegative number given by ρ2 0(t) = 1 + 1 l0f0 e−f0tZt −∞ ef0sE2(s)ds,∀t∈R, is pullback absorbing family for the process Udefined by (6). Proof. Let D⊂R2be bounded. Then, there exists d > 0 such that |u0| ≤ dfor all u0∈D. Thanks to Proposition 14, we deduce that for every t0≤tand any u0∈D, 8 |U(t, t0)u0|2≤l1 l0 e−f0tef0t0|u0|2+1 l0f0 e−f0tZt −∞ ef0sE2(s)ds ≤l1 l0 e−f0tef0t0d2+1 l0f0 e−f0tZt −∞ ef0sE2(s)ds. If we consider T(t, D) := f−1 0log(l0 l1ef0td−2), we have |U(t, t0)u0|2≤1 + 1 l0f0 e−f0tZt −∞ ef0sE2(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) = BR2(0, ρ0(t)) is pullback absorbing for the process Udefined by (6). 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 (6). Theorem 16 Under the assumptions in Proposition 14, the process Udefined by (6) possesses a pullback attractor A, which is given by A(t) = [ b D∈D Λb D, t.(11) 3.2 Uniform Attractor Now, we suppose that Eis translation bounded in L2 loc(R;R), i.e., sup t∈RZt+1 t E2(s)ds < ∞.(12) In this subsection, using Theorem 9, we will prove that, under the assumption (12), the process {U(t, t0)}has a uniform (with respect to t0∈R) attractor. Remark 17 Observe that assumption (12) implies (H2). Proposition 18 Assume (H1) and (H3). Let Esatisfies (12). Then, the process Udefined by (6) is uniformly (with respect to t0∈R) asymptotically compact. Proof. Let D⊂R2be bounded, and as in the proof of Proposition 15, let d > 0 such that |u0| ≤ dfor all u0∈D. Using (10), we have for any u0∈D |u(t;t0, u0)|2≤l1 l0 e−f0tef0t0|u0|2+1 l0f0 e−f0tZt t0 ef0sE2(s)ds ≤l1 l0 e−f0tef0t0d2+1 l0f0 e−f0tZt t0 ef0sE2(s)ds, (13) 9 e A(σtp) for all t≥0. If p6=σsg, then p= limtn→+∞σtngand it is easy to see that ϕ(t, p)e A(p)⊇e A(σtp). On the other hand, we can consider the following compact set e K:= [ t∈R A(t)⊂R2, and we have that [ p∈H(g)e A(p)⊂e K, and, consequently, condition i)-ii) hold. We can now establish our result on the estimate of the Haussdorff dimension of the pullback attractor for our model. We denote by dimHKthe Hausdorff dimension of K. Theorem 23 Assume (H1), (H3)-(H5), and that a, E ∈BUC(R,R)satisfy (H6). Then the pullback attractor of the process Udefined by (6) satisfies dimHA(t)≤2−2f0 m+f0 ,(29) for all t∈R, where mis a positive number given by m:= 5 4+a2 1+k(1 + 1 2˜a1+E0), where kis a positive number depending on the compact set ˜ K. Proof. We need to verify iii). It is easy to see that the eigenvalues of the symmetrized Jacobian matrix of the right-hand side of (25) are 1 2n−f(x)±pf2(x) + (1 −a(t))2o. Hence, condition (27) can be written in the form −f(x)(1 + s) + (1 −s)pf2(x) + (1 −a(t))2+ 2 d dtVp(t, x, y)<0,(30) for all t∈R, (x, y)∈e Kand p∈ H(g). Here, Vp(t, x, y)≡V(σtp, ϕ(t, p)(x, y)) is a function defined for (x, y)∈e K,p∈ H(g), and t∈Rby the relation V(σtp, x, y) := 1 4(1 −s)(a(t)x2+y2). 16 Then d dtVp=−1 2(1 −s)a(t)F(x)x+1 2(1 −s)E(t)y+1 4(1 −s)a0(t)x2, and inequality (30) is equivalent to the following −f0(1 + s) + (1 −s)ϑ(t, x, y)<0,(31) where ϑ(t, x, y) := pf2(x) + (1 −a(t))2−a(t)F(x)x+E(t)y+1 2a0(t)x2. Let us denote m:= max t,x,y ϑ(t, x, y). We have iii) from (31), and due to Theorem 2 in [23] we obtain dimHe A(p)≤1 + −f0+m m+f0 = 2 −2f0 m+f0 ,(32) for all p∈ H(g). We have ϑ(t, x, y) = −γpf2(x) + (1 −a(t))2−1 2γ2 +γ2f2(x) + (1 −a(t))2+1 4γ2 −a(t)F(x)x+E(t)y+1 2a0(t)x2, where γ6= 0 is a varied parameter. Further, ϑ(t, x, y)≤γ2f2(x) + (1 −a(t))2+1 4γ2−a(t)F(x)x+E(t)y+1 2a0(t)x2. Applying the mean value theorem for integrals, we observe that −a(t)F(x)x=−a(t)f(x∗)x2≤0. where 0 < x∗< x. Then, we have that ϑ(t, x, y)≤γ2f2(x) + (1 −a(t))2+1 4γ2+E(t)y+1 2a0(t)x2. 17 If we take the varied parameter γ2= 1 and taking into account (H3), (H4) and (H5), we deduce that there exists a positive constant kdepending on the compact set ˜ K such that ϑ(t, x, y)≤5 4+a2 1+k(1 + E0+1 2˜a1), and (28) and (32) imply (29). Acknoledgements. The idea of this paper arose while the author was visiting the Laboratoire Jacques-Louis Lions (May-July 2013), invited by Dr. Fran¸cois Murat, supported by Ayuda para la Movilidad International del Personal dedicado a la Investigaci´on (V Plan Propio de Investigaci´on de la Universidad de Sevilla, Spain) under grant PP2013-1458 and partly supported by Fondo Europeo de Desarrollo Regional and Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2011-22411. 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