Global attraction to the origin in a parametrically-driven nonlinear oscillator

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1 Global attraction to the origin in a parametrically-drien nonlinear oscillator M.V. Bartccelli, J.H.B. Deane, G. Gentile and S.A. Gorley Department of Mathematics and Statistics, Uniersity of Srrey, Gildford, GU2 7XH, UK Dipartimento di Matematica, Uniersità di Roma Tre, Roma, I-146, Italy M.Bartccelli/J.Deane/[email protected] [email protected] Jly 8, 29 Abstract We consider a parametrically-drien nonlinear ODE, which encompasses a simple model of an electronic circit known as a parametric amplifier, whose linearisation has a zero eigenale. By adopting two different approaches, we obtain conditions for the origin to be a global attractor which is approached (a) non-monotonically and (b) monotonically. In case (b), we obtain an asymptotic expression for the conergence to the origin. Some frther nmerical reslts are reported. Keywords: Global attractiity, parametric forcing, asymptotics. Short title: Parametrically-drien ODEs 1 Introdction In this paper we consider the behaior of the following nonlinear, parametrically drien ODE d 2 x dt 2 + γdx dt + f(t)x+1 = (1.1) where γ >, m Z + and f(t) > is bonded aboe and below for all t. This differential eqation arises as a simple model of a nonlinear electronic circit known as a parametric amplifier [1], in which f(t) is the signal to be amplified, and x(t) is the otpt of the circit. We denote the pper and lower bonds of f(t) as f + and f respectiely. Eqation (1.1) can be re-written as a pair of copled, first-order differential eqations as ẋ = y ẏ = γy f(t)x +1 (1.2) where (x,y) R 2 and a dot placed aboe a ariable indicates differentiation with respect to time. We refer to the differential eqation in this form as the original system. It is well known that the phase space strctre of time-dependent dynamical systems is in general ery intricate: there are systems possessing bonded periodic and non-periodic 1

2 soltions, qasi-periodic soltions and nbonded non-periodic soltions of oscillatory type with any prescribed nmber of zeros, see for example the papers by V.M. Aleksee [2]. There exist some reslts on a large class of time-dependent potentials in the ndamped case. For example, it has been proed that for sperqadratic potentials which are parametrically drien by a positie time-periodic fnction, as in (1.1) when γ = and f(t + T ) = f(t), all the soltions are bonded for all time and in fact most of the motion is qasiperiodic [3, 4]. Moreoer the search for periodic soltions in time-dependent dynamical systems has proen ery fritfl by sing a nmber of techniqes sch as the calcls of ariations [5] and the Poincaré-Birkhoff theorem [6, 7, 8]. Some work has also been done on the classification of the asymptotic behaior of soltions of nonlinear differential eqations; see [9] and references therein. Another natral qestion one may ask is: do there exist time-dependent potentials sch that all the soltions of the corresponding differential eqations are bonded in phase space? The answer is natrally of fndamental importance for the stability properties of dynamical systems and there crrently exists a sbstantial body of work on this problem in the zerodamping case [1] [15]. The essential idea sed to show bondedness of soltions for these potentials is to transform the system, by a seqence of canonical transformations, into a near-integrable one, and then to apply KAM theory and the Moser twist theorem. It is not difficlt to proe that a certain amont of damping garantees bondedness of soltions of eqation (1.1), bt we go frther in this paper, and derie conditions nder which the origin is a global attractor. We also derie a frther bond on the damping which forces all soltions to decay to the origin in a non-oscillatory way. The system (1.2) has jst one fixed point, namely the origin, (x,y) = (,). The eigenales of the linearised system at the origin are and γ, so that the origin is marginally stable as a soltion of the linearised system. Or aim is to establish sfficient conditions for the origin to be a global attractor, i.e., for all initial ales (x(),y()) R 2, lim t (x(t),y(t)) = (,). With zero as an eigenale of the linearised problem, een the local stability of the origin is not determined by the linearised eqations and ths the problem of the global attractiity is a particlarly delicate one. This paper will establish the global attractiity of the origin for sfficiently large γ >, and also a condition on γ which garantees that the conergence is monotone. 2 Global attractiity of the origin 2.1 Oscillatory decay We se the following transformation, which is de to Lioille: τ = G(t) = t f(s)ds. Since f(t) > and bonded aboe for all t, G(t) is a monotonically increasing fnction of t and G 1 exists. Introdcing F(τ) := f(t) = f(g 1 (τ)), eqation (1.1) is transformed into the system x = y 2

3 y = y ( ) F F 2 F + γ x +1 (2.1) where prime denotes differentiation with respect to τ. Defining the Hamiltonian, H(x,y), as H = x +2 /( + 2) + y 2 /2 gies ( ) H = y2 F F 2 F + γ. (2.2) Hence, proided γ > min τ ( F (τ) 2 F(τ) ) = min t ( ) f(t) 2f(t) we hae H (τ) so that x and y are both bonded. Also, for all τ, ( ) τ y 2 (s) F (s) H(τ) + F(s) 2 F(s) + γ ds = H(). (2.3) Letting τ and sing (2.3), and also the bondedness aboe and below of F(τ), we conclde that { ( )} 1 F (s) min s F(s) 2 F(s) + γ y 2 (s)ds <. Therefore, y as τ (and hence also as t ). We shall now reert to the system in the form (1.2) to show that x as t. Let k > be arbitrary and U = {(x,y) R 2 : H(x,y) < k}. Let S = {(x,y) U : Ḣ = } and let M denote the largest inariant set in S. Then, by the La Salle s inariance principle [16, Theorem 9.22], eery soltion of (1.2) that starts in U has its ω-limit set in M. In this case, S consists of points with y = and, from (1.2), it follows immediately that M consists solely of the origin. Since k > was arbitrary, we may now state: Proposition 1 Let γ satisfy (2.3). Then all soltions of (1.2) satisfy (x(t), y(t)) (, ) as t. 2.2 Non-oscillatory decay This section aims to discoer frther conditions on γ and on the initial conditions which will garantee that conergence of soltions of (1.2) to the origin is monotone. It will be adantageos to introdce the new state ariables (t) and (t) defined by = x, =. (2.4) The main reslt of this section follows by a carefl analysis of the arios regions of the (,) phase plane, and this analysis is carried ot below. With = x, we hae on differentiating ẋ = x+1 and ẍ = x+1 [ ( + 1) 2 ] ü. 3

4 Sbstitting these into (1.1) and defining = gies = = f(t) γ + ( + 1) 2 (2.5) which we refer to as the transformed system. In the transformed system, (t) >, since = x, bt (t) may be of either sign. We therefore need to condct a thorogh inestigation of the right half > of the (,) phase plane. Note that the transformed system (2.5) cannot hae any fixed points. First, we hae Proposition 2 Let the hypotheses of Proposition 1 hold. Then, trajectories of (2.5) mst enter the open first qadrant in finite time, and then remain there. Proof. It is triial to see that the open first qadrant is positiely inariant and so we simply need to address the fate of a trajectory that starts in the forth qadrant, i.e., has () <. Since, by Proposition 1, x and y are bonded, any crossing of the -axis is impossible so (t) > is bonded below for all t. Also ( + 1) 2 = f(t) γ + f γ. (2.6) By a standard comparison argment, we conclde that (t) becomes positie in finite time. The proof is complete. In the light of the aboe Proposition, it is sfficient to consider trajectories that start in the open first qadrant and ths we may assme from now on that (t),(t) > for all t. The next proposition is important. Of corse, or transformation = x clearly reqires x. It is implicitly assmed at the otset that the conergence of x(t) and y(t) to zero (the main reslt we are aiming for) is a non-oscillatory conergence, i.e., either x(t) > for all t, or x(t) < for all t. As we might expect physically, this will be the case only if the damping parameter γ is sfficiently large (see Proposition 3 below). We address this point below by giing conditions on γ, and on the initial conditions, which ensre that neither (t) nor (t) reaches infinity in finite time. Proposition 3 Sppose (WLOG) that (), () >, that the hypotheses of Proposition 1 hold, that γ 2 > 4f+ (2.7) () and that () ( () < m γ + ) γ 2 4f + /(). Then (t) and (t) remain finite for all times t. Proof. It is sfficient to proe that (t)/(t) is bonded for all t, for then we hae (t) const. (t) 4

5 so that (t) (and hence also (t)) does not blow p. Let Straightforward calclations yield that φ = f(t) (t) φ(t) = (t) (t). γφ + 1 φ2. Now, the open first qadrant is positiely inariant and so = >. Hence (t) is strictly increasing and so (t) > () for all t >. Hence φ f+ () γφ + 1 φ2. Soltions of the aboe differential ineqality are bonded by the soltion of the corresponding differential eqation. Simple argments for one-dimensional ODEs, together with the hypotheses on γ and φ(), immediately yield that φ(t) is bonded for all t. The proof is complete. We may now state the main theorem of this section. Theorem 1 Let the hypotheses of Proposition 3 hold. Then soltions of (1.2) satisfy monotonically, as t. (x(t),y(t)) (,) Proof. To show that x(t) we mst show (t) as t. Also, since y = (+1)/() then, to show that y(t) we shall show that ψ(t) where From ineqality (2.6) we hae that ψ = >. (2.8) (+1)/() lim inf t (t) f /γ. Hence, there exists t 1 > sch that, for all t t 1, Therefore, for t t 1, (t) mf /γ. t (t) = (t 1 ) + (s)ds t 1 (t 1 ) + mf γ (t t 1) 5

6 so that (t). Now, with ψ defined by (2.8) it is straightforward to see that ψ = f(t) γψ. ((t))(+1)/() Let ε > be arbitrary. Since (t) as t, and since f(t) is bonded, there exists t 2 > sch that, for all t t 2, f(t) < ε. ((t))(+1)/() Then, for t t 2, we hae from which it follows that ψ ε γψ lim spψ(t) ε t γ. Since ε > was arbitrary, it follows that lim sp t ψ(t). Bt ψ(t) > for all t. Hence lim t ψ(t) = and so lim t y(t) = also. The proof of the theorem is complete. 3 Asymptotics and nmerical simlations In this section we shall condct some frther examination of the (,) phase space, together with some nmerical simlations, to gain frther insight into the dynamics of the system and, in particlar, the manner of the conergence of (x(t),y(t)) to (,) as t for sitably large γ >. Or nmerical simlations indicate that, as t, trajectories in the (,) phase plane typically become trapped between the horizontal asymptotes of the two cres Γ + and Γ defined below, and shown in Fig. 1. Note that d f(t) ( + 1) = = γ + d and so, in particlar, d/d < in regions where and d/d > in regions where f + γ + ( + 1) < (3.1) f γ + ( + 1) >. (3.2) Recalling that (t) increases in the open first qadrant, any trajectory can be characterised as the graph of a fnction, = (). A trajectory has a trning point at any time when d/d =. This will happen at any time with = mγ + 1 [ 1 ± 1 4( + 1)f(t) γ 2 ]. (3.3) Replacing f(t) in (3.3) by its pper and lower bonds defines two non-intersecting cres (proided f + f ) in the (,) plane; we refer to these cres as Γ + and Γ (they are 6

7 Γ Γ + d/d > = γ /(+1) - f - /γ = γ /(+1) - f + /γ d/d < = f + /γ = f - /γ d/d > Figre 1: The cres Γ + and Γ and their asymptotes. To the left of Γ, d/d is positie and to the right of Γ + it is negatie. Between Γ + and Γ, d/d can be positie, negatie or zero. also defined by replacing the ineqalities in (3.1) and (3.2) by eqalities). Only in the region between Γ + and Γ can a trajectory () hae a trning point. The cres Γ + and Γ each possess two asymptotes, obtained by Taylor expanding (3.3). These are Asymptotes to Γ ± = γ + 1 γ f ±, γ f ± (3.4) Since the horizontal asymptotes both hae >, and the other two asymptotes hae negatie intercept with the -axis, Γ + and Γ mst be confined to the first qadrant, >. Frthermore, in this qadrant, and hence d/d, are positie eerywhere to the left of Γ ; similarly, d/d < eerywhere to the right of Γ +. For, d/d is eerywhere positie. The sitation is sketched in Fig. 1. Typical soltions for arios different initial data satisfying the conditions of Theorem 1 are illstrated in Fig. 2. The obserations we hae made so far indicate that, typically, (t) with (t) remaining bonded, and these are borne ot by the nmerical soltions in Fig. 2. For any ale of γ and any initial data giing sch an otcome, we can formally approximate the differential eqations (2.5) in the limit as t to yield the approximated system = = f(t) γ. (3.5) Soling the second of these eqations and ignoring the transient term gies (t) = t e γ(t s) f(s)ds 7

8 Figre 2: Nmerically-obtained periodic soltions starting from arios points in the, - plane, with f(t) = 3 + 2cos 2πt so f + = 5, and parameters that satisfy the conditions of Theorem 1: m = 1, () = 1, γ = 5. The cres Γ and Γ + are shown dotted. from which it follows that (t) is gien asymptotically by (t) or, on reersing the order of integration, t ξ e γ(ξ s) f(s)ds dξ (t) γ t f(s)(1 e γ(t s) )ds. Ths, for sitably large γ and sitable initial data, the conergence of x(t) to zero is gien asymptotically by 1 x(t) 1. t )ds) f(s)(1 e γ(t s) ( γ Attraction to the origin is not the only dynamics displayed by Eqation (1.1); a ariety of periodic soltions exist as well, as shown in Figre 3. 8

9 Figre 3: Nmerical soltions to (1.1) with f(t) = 3 + 2cos 2πt and γ =.4. In all for graphs, x is plotted horizontally and y ertically. The top left graph shows three periodone (same period as f(t)) soltions; the top right graph shows three period-two soltions (repeating eery two periods of f(t)). Bottom left: three period-for soltions; bottom right, one period-ten soltion. 9

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