CHAPTER 4 Averaging and Perturbation from a Geometric Viewpoint In this chapter we describe some classical methods of analysis which are particularly applicable to problems in nonlinear oscillations. While these methods might be familiar to the reader who has studied nonlinear mechanics and perturbation theory, the present geometrical approach and the stress on obtaining approximations to Poincare maps will probably be less familiar. We start with the averaging method, originally due to Krylov and Bogoliubov [1934], which is particularly useful for weakly nonlinear problems or small perturbations of the linear oscillator. We show that, under suitable conditions, global information, valid on semi-infinite time intervals, can be obtained by this approach. Generally, in perturbation methods one starts with an (integrable) system whose solutions are known completely, and studies small perturbations of it. Since the unperturbed and perturbed vector fields are close, one might expect that solutions will also be close, but as we shall see, this is not generally the case, in that the unperturbed systems are often structurally unstable. As we have seen, arbitrarily small perturbations of such systems can cause radical qualitative changes in the structure of solutions. However, these changes are generally associated with limiting, asymptotic behavior and one does usually find that unperturbed and perturbed solutions remain close for finite times. Moreover, in this chapter we shall show that such finite time results, together with ideas from dynamical systems theory, do enable us to make deductions about the asymptotic behavior of solutions and the structure of the nonwandering set for the perturbed system. Averaging is applicable to systems of the form x = ef(x, r); xeU", e < 1, (4.0.1) where / is T-periodic in t. In such a system the T-periodic forcing contrasts 4 1. Averaging and Poincare Maps 167 vvitli the "slow" evolution of solutions on the average due to the (9(e) vector field. In the first four sections we will show how weakly nonlinear oscillators 0f the form (4.0.2) can be recast in the standard form (4.0.1) and averaging applied. In this analysis we essentially deal with small perturbations of the linear oscillator % + qj2x = 0, which is an example of an integrable Hamiltonian system. We continue with a description of Melnikov's [1963] method for dealing with perturbations of general integrable Hamiltonian systems. Here one typically starts with a strongly nonlinear system, x = /(*), x e M2", (4.0.3) and adds weak dissipation and forcing: x = f(x) + eg(x, t). (4.0.4) While we are primarily concerned with periodically forced two-dimensional systems, we state the averaging results in the more general n-dimensional context, since they are no more difficult in that form. For Melnikov's method we restrict ourselves to two-dimensional problems, although some ^-dimensional, and even infinite-dimensional, generalizations are available. We comment on extensions of this nature towards the end of the chapter. We also outline some of the theory of area preserving maps of the plane arising as Poincare maps in time-periodic single degree of freedom Hamiltonian systems and in time-independent two degree of freedom systems. 4.1. Averaging and Poincare Maps There are many versions of the averaging theorem. Our account is based on the versions due to Hale [1969; Chapter V, Theorem 3.2], and Sanders and Verhulst [1982], who give a very full discussion from the viewpoint of asymptotics. We consider systems of the form x = ef(x, t, s); xeU^U", 0 < e < 1, (4.1.1) where /: |R» x U x U+ -* U" is C\ r > 2, bounded on bounded sets, and of period T > 0 in t. We normally restrict ourselves to a bounded set U c: U". The associated autonomous averaged system is defined as 3> = f f(y,t,0)dtdM£f(y). Jo (4.1.2) 168 4. Averaging and Perturbation from a Geometric Viewpoint In this situation we have Theorem 4.1.1 (The Averaging Theorem). There exists a C change of coordinates x = y + ew(y, t, e) under which (4.1.1) becomes where j\ is of period T in t. Moreover (i) // x(t) and y(t) are solutions of (4.1.1) and (4.1.2) based at x0, y0, respectively, at t = 0, and \x0 — y0| = 0(e), then \x(t) — y(t)\ = (9(e) on a time scale t ~ 1/e. (ii) If p0 is a hyperbolic fixed point of (4.1.2) then there exists s0 > 0 such that, for all 0 < e < e0, (4.1.1) possesses a unique hyperbolic periodic orbit yE(t) = p0 + 0(e) of the same stability type as Po* (iii) // xs(t) e Ws(ys) is a solution of (4.1.1) lying in the stable manifold of the hyperbolic periodic orbit yE = p0 + 0(e), ys(t) e Ws(pQ) is a solution of (4.1.2) lying in the stable manifold of the hyperbolic fixed point p0 and \x%0) - v5(0)| = 0(e), then \xs(t) - ys(t)\ = 0(e) for t e [0, oo). Similar results apply to solutions lying in the unstable manifolds on the time interval t e( — oo, 0]. Remarks. Conclusions (ii) and (iii) generalize to more complicated hyperbolic sets. In particular, Hale [1969] shows that if (4.1.2) has a hyperbolic closed orbit T, then (4.1.1) has a hyperbolic invariant torus Tr. Generalizations to almost periodic functions / are also available (Hale [1969]). Conclusion (iii) implies that the averaging theorem can be used to approximate stable and unstable manifolds in bounded sets and generally to study the global structure of the Poincare map of (4.1.1), as our examples will demonstrate. Proof. We will sketch the first two parts of the proof using standard results from differential equations; for the last parts it is more convenient to use the ideas of Poincare maps and invariant manifolds. We start by explicitly computing the change of coordinates. Let without yet choosing w. Differentiating (4.1.5) and using (4.1.1) and (4.1.4) y = tf(y) + e2f(y, u e), (4.1.3) f(x, u e) = /(*) + /(*, t, e) be split into its mean, /, and oscillating part /. Let ^^x = y + sw(y, f, e)^% (4.1.4) (4.1.5) we have dw = tf(y + ew) + ef(y + ew, t, e) - e dw * y, may be a trivial periodic orbit, -/,(?) = p0, cf. Example f on p. 17 1 below. 4 1. Averaging and Poincare Maps 169 □ y = e[i + eDywYl dw f(y + ew) + f(y + ew, t, e) - — (4.1.6) Expanding (4.1.6) in powers of e and choosing w to be the anti-derivative of /: dw a7 = Ry, u o), we obtain y = &f(y) + £: def yr, Dyf(y, t, 0)w(y, t, 0) - D w(y, t, 0)/(y) + f (y, t, 0) = e/(y) + e2/!(y, t, e), as required. To obtain conclusion (i) we use a version of Gronwall's lemma: (4.1.7) + #(e3) (4.1.8) Lemma 4.1.2 (cf. Coddington and Levinson [1955], p. 37). // u, v, and c > 0 on [0, Q, c is differentiable, and v(t) < c(0 + f u(s)v(s) ds, Jo then v(t) < c(0) exp f u(s) ds + f c'(s) Jo Jo exp u(x) dx ds. To prove the lemma, let R(t) = J'0 u(s)t;(s) and first show that R' — uR < uc. After integrating this differential inequality and some manipulation, including integration by parts, one obtains the result. Now consider equations (4.1.2) and (4.1.3). Integrating and subtracting, we have y*(t) - yit) = ye0 - yo + e f UMs)) - /0>(s))] ds + e2 f fi(ye(s), s, e) ds, Jo where yt(t) is a solution of (4.1.3) based at ye0. Letting yt - y — (, L be the Lipschitz constant of/ and C the maximum value of this becomes |C(0l I, Pe: U -+ I are then defined for (4.1.11), (4.1.12) in the usual way, where U s L is some open set. Note that PE is e2-close to P0 since T is fixed independent of e. If p0 is a hyperbolic fixed point for (4.1.2), then it is also a hyperbolic fixed point of DP0(p0) since DP0(p0) = eETDfip). Therefore, lim^0(l/e)[eE7"fl/(po) - Id] = TDf(p0) is invertible. Since PE is e-close to P0, we also have limE^0(l/s)[DPe(p0) - Id] = TDf(p0). The implicit function theorem implies that the zeros of (l/e)[DPs(p0) — Id] form a smooth curve (pr, e) in [RM x 1R. The pE are fixed points of PE, and the eigenvalues of DPE(pr) are r,2 close to those of DP0(p0) since p£ = p0 + (9(e) and DP£(p£) = exp[er(D/(pE) + e^/^p,))] = exp[fi7D/(p0)] + 0(e2). Thus (4.1.12) has a periodic orbit -/£ £-close to p0, and via t^e change of coordinates (4.1.5), equation (4,1.1) has a similar orbit. We remark that all that is required for the existence of a periodic orbit in (4.1.1) is the absence of any eigenvalues equal to one in the spectrum of DPoiPo)- However, the stability types of p0 and ye may not correspond if any eigenvalues of DP0(p0) lie on the unit circle. To prove (iii), suppose that (4.1.2) has a hyperbolic saddle point p0 and consider solutions y(t) e Ws(p0) and the corresponding solutions yE(t) e Ws(ye) of the full system (4.1.3). The cases in which p0 is a sink or source and Ws is replaced by W" can be dealt with similarly. The proof is divided into two parts; an outer region in which the averaged vector field sf(y) is large in comparison with the remainder term &2fi(y, t\ and an inner region in which the "perturbations" e2fx and ef are of comparable order. For more details see Sanders and Verhulst [1982]. We fix a S neighborhood, Ud, of p0, so that, outside Ud, we have \f(y))\ > e\fL(y, t, e)|. As above, standard Gronwall estimates show that | yE - y0 \ = &{&) for times of order 1/e y = ef(y) + £2fi(y, 6, e); 0 = 1, (4.1.11) (4.1.12) * In P0, the subscript "0" indicates that the &{z2) term is removed, not that s = 0 in (4.1.11). The notation, P0 for the 6(c) map and Pr for the full map, will be used throughout this and the following three sections. 4 2. Examples of Averaging 171 w8(P<). 0 wV€)n2: Figure 4.1.1. Validity of averaging on semi-infinite time intervals. outside U6. On the other hand, in Ud the (local) stable manifold theorem guarantees that the stable manifold W\oc(ye) is e, Cr close to W\oc(Po) x [0, T]. Moreover, within W\0i{y^) and W^oc(p0) solutions are contracting towards yE and p0, respectively, this contraction being dominated by an exponential term of the form e~kl. Using this fact, we can prove that, if yB and y0 enter Us within (9(e), they remain within 0(e) for all forward time, Figure 4.1.1. Piecing the two estimates together and using the transformation (4.1.5) as above, we obtain the desired result. □ We note that Sanders [1980] and Murdock and Robinson [1980] (cf. Robinson [1981b]) give proofs of part (iii) of this theorem. In the proof of the last part of the theorem we are using the smooth dependence of (local) invariant manifolds on parameters. Thus statement (iii) also follows directly from the "big" invariant manifold theorem of Hirsch et al. [1977, Theorem 4.1]. 4.2. Examples of Averaging Example 1. Consider the scalar system x = ex sin2 t. (4.2.1) Here f(x, t, e) =/(*)+ /(*, t, e) = x/2 - (x/2) cos It and we have / dw y at 2 or w = - y- sin It. (4.2.2) 172 4. Averaging and Perturbation from a Geometric Viewpoint Note that the f-independent term which could appear in the anti-derivative is generally taken to be zero. From (4.1.8), the transformed system is (i - i cos It) (-^ sin 2t j - ( -1 sin 2t) ^ or y = el + e21- sin At + (9(a3). (4.2.3) 2 16 Here the autonomous averaged equation is simply y = ey-. (4.2.4) The exact solution of (4.2.1) with initial value x(0) = x0 is easily found to be x(t) = Xo^(W2)-sin(20M)> (4^.5) Comparing this with the solution of the averaged equation y(t) = y0eEtl2, (4.2.6) we see that x(t) - y{t) = eet/2l\x0 - y0\ - ex0 sin(2t)/4 + (9(e2)l (4.2.7) in agreement with conclusion (i) of the theorem. Here the hyperbolic source y = 0 of (4.2.4) corresponds to a trivial hyperbolic periodic orbit x = 0 of (4.2.1), and, letting t -> - oo in (4.2.5)-(4.2.7), we see that x(t), y(t) -> 0 and hence | x(t) — y(t) \ 0, in agreement with conclusions (ii) and (iii). Exercise 4.2.1. Study the system x = -ex cos t by the method of averaging. Does it have a hyperbolic limit set ? Compare the averaged and exact solutions. Exercise 4.2.2. Repeat the averaging analysis for x = e( — x + cos21). In particular check the validity of conclusions (ii) and (iii) of the theorem. Exercise 4.2.3. Study the nonlinear systems x — s(x - x2) sin2 t and x = e(x sin21 — x2/2) by the method of averaging. What do you notice about their solutions ? Example 2 (Weakly nonlinear forced oscillations). In many weakly nonlinear oscillator problems, the second-order equation to be studied takes the form x + calx = ef(x, x, t), (4.2.8) 4,2. Examples of Averaging 173 where / is T periodic in t. In particular, iff is sinusoidal with frequency co « /cco0, we have a system close to a resonance of order k. In such a situation, our expectation of finding an almost sinusoidal response of frequency co/k prompts the use of the invertible van der Pol transformation, which recasts (4.2.8) into the form (4.1.1) which can then be averaged. We set A = 'cot' cos co . (cot cos (?) . (cot" --(?) co (cots icos[j k . (cot\ k /cot^ cd \ /c y (4.2.9) under which (4.2.8) becomes w =-- co co ^ - klCD (to1 — k2co\ -jx + e/(x, x, t)j sin^ , x, t) j cot x + e/(x /cot> (4.2.10) in which x, x can be written as functions of u, v, and t via (4.2.9). If co2 — k2wl = 6(e), then (4.2.10) is in the correct form for averaging. As a specific example, we take the standard Duffing equation which is considered in almost every text on nonlinear oscillations: x + coqX = e[y cos cot — dx — ax3], (4.2.11) where co^ — co2 = eQ, i.e., we are close to order one resonance. Setting k = 1 m (4.2.9) we obtain the transformed system it — — [Q(u cos cot — v sin cot) — co3(u sin cot + u cos cot) co + a(u cos cot — v sin cot)3 — y cos cot] sin cot, v — — [Q(u cos cot — v sin cot) — co<5(u sin cot + v cos cot) co + a(u cos cot — v sin cot)3 — y cos cot] cos cot. (4.2.12) 174 4. Averaging and Perturbation from a Geometric Viewpoint Averaging (4.2.12) over one period T = 2n/co, we obtain u = 2co 2w n 3a , , . coou — ilv--(u + i? )v 4 def = $Ti(«, v), 3a Qu — a><5t? + — (u2 + v2)u — y def r / \ or, in polar coordinates; r = yju2 + v2, eft = arctan(i?/u); r — . - [ — ojdr — y sin 0], 2co r0 = 2oj _ 3a , Qr + — r — y cos q> (4.2.14) Perturbation methods carried to 0(e) give precisely the same result (cf. Nayfeh and Mook [1979], Section 4.1.1). Recalling the transformation x = u(t) cos cot — v(t) sin cot = r(t) cos(cot + cp(t)), we see that the slowly varying amplitude, r, and phase, cp, of the solution of (4.2.11) are given, to first order, by solutions of the averaged system (4.2.14). It is therefore important to find the equilibrium solutions of or fixed points of (4.2.14), which, by the averaging theorem and the transformation (4.2.9), correspond to steady, almost sinusoidal solutions of the original equation. Fixing a, <5, and y and plotting the fixed points r, $ of (4.2.14) against Q or ca/co0, we obtain the frequency response curve familiar to engineers: see Figure 4.2.1. We shall consider the "jump" bifurcation phenomenon below. For more details on the Duffing equation see Nayfeh and Mook [1979], or, for bifurcation details, Holmes and Rand [1976]. The stability types of the branches of steady solutions shown in Figure 4.2.1 are obtained by consideration of the eigenvalues of the linearized averaged equation, and we invite the reader to check our assertions. In Figure 4.2.2 we show typical phase portraits for (4.2.13)^(4.2.14), obtained by numerical integration for parameter values for which three hyperbolic fixed points coexist. In Figure 4.2.3(a) we reproduce the stable and unstable manifolds of the saddle point, under the orientation reversing transformation (4.2.9) (with k = 1) applied at t = 0: x = u, x = — cov. According to Theorem 4.1.1, these manifolds should approximate the stable and unstable manifolds of the Poincare map of the full system (4.2.11), which we show in Figure 4.2.3(b). These latter manifolds were also computed numerically. We note that the agreement is good, but remark that it deteriorates as co moves away from oj0 (Q increases). For more examples, see Fiala [1976]. 2. Examples of Averaging 175 Exercise 4.2.4. Carry out averaging for the "original" van der Pol equation a . x H— (x - l)x 4- x = ay cos cot, CO with 1 - co2 = acr = 0(a) <| 1. Show that the averaged equation takes the form given in (2.1.13) with /? = ay. Check as many of the assertions made in Section 2.1, concerning the averaged system, as you can. Exercise 4.2.5. Consider equation (4.2.11) with ey = y = 0(1) and co % 3to0 and apply the method of averaging to study the subharmonics of order three. In this case the transformation (4.2.9) should be replaced by /u\ / x + B cos(a>r — (j>) /' \vj \x — coB sin(cot — 0) where (4.2.15)