Bifurcations in non-autonomous scalar equations

Bifurcations in non-autonomous scalar equations J. A. Langa1, J. C. Robinson2and A. Su´ arez1 1. Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Fac. Matem´aticas, C/ Tarfia s/n C.P. 41012, Univ. Sevilla, Spain. 2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U. K. e-mail: [email protected], [email protected]arwick.ac.uk, [email protected] Running head. Bifurcations in non-autonomous scalar equations The author to whom proofs should be sent: James C. Robinson, Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U. K. Tel: 44 24 7652 4657 Fax: 44 24 7652 4182 e-mail: [email protected] 1 Abstract In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations. Keywords: Non-autonomous differential equations, bifurcation theory, pullback attracting sets. 2 1 Introduction In a previous paper (Langa, Robinson, & Su´arez [15]) we introduced various definitions of stability and instability that seemed to be potentially useful in discussing the dynamics of the solutions of non-autonomous differential equations. In particular we applied these definitions to various simple model problems that exhibited non-autonomous versions of standard autonomous bifurcations: an explicitly solvable pitchfork bifurcation problem, a saddle-node type bifurcation, and a general n-dimensional ‘loss of stability’. In this paper we develop a more general theory, concentrating on the well-known ‘local bifurcations’ from the autonomous theory, and finding conditions for similar bifurcations in the scalar non-autonomous equation ˙x=f(x, t, λ), where λis a parameter. By imposing conditions on the Taylor coefficients in the expansion of fnear x=λ= 0 (which reduce to the standard conditions in the autonomous case) we are able to prove various general theorems guaranteeing transcritical, pitchfork, and saddle node bifurcations. Although we require a strong ‘balance hypothesis’ on the terms in the Taylor expansion, we believe that these results are a further step towards a general non-autonomous theory of bifurcations. We do not present any concrete examples here, instead concentrating on the development of an abstract theory which we believe should be applicable to a wide variety of particular models. Some particular examples have been analysed in various settings: using the framework of skew product flows Johnson [8] and Johnson and Yi [9] have considered a generalised notion of a Hopf bifurcation; Shen and Yi [19] treat almost periodic scalar differential equations (but leave bifurcation phenomena largely untouched); more recently Kloeden [12] has analysed transcritical and pitchfork bifurcations in an almost periodic equation; Johnson, Kloeden, & Pavani [10] have considered a non-autonomous ‘two step bifurcation’; and Kloeden & Siegmund [14] give a nice discussion of the general problem in the context of skew product flows. In this paper we do not adopt the skew product approach and the restrictions on the generality of fthat it would entail, preferring to use the language of processes. 3 2 Non-autonomous equations as processes For the solution of any non-autonomous equation ˙x=f(x, t)x(s) = x0with x∈Rm(2.1) the initial time (s) is as important as the final time (t). In order to treat these equations as dynamical systems we consider a family of solution operators {S(t, s)}t≥s(termed a “process”, see Dafermos [6] or Sell [18]) that depend on both the final and initial times. We can then denote the solution of (2.1) at time tby S(t, s)x0. If fis sufficiently smooth (which it will be in all that follows) then it is clear that S(t, s) : Rm→Rmmust satisfy a) S(t, t) is the identity for all t∈R, b) S(t, τ)S(τ, s) = S(t, s) for all t,τ, and s∈R, and c) S(t, s)x0is continuous in t,s, and x0. There may in fact be solutions of (2.1) that do not exist for all time, and some restrictions to the possible values of sand tmay be necessary, giving rise to only a ‘local process’. Although we pass over them here, we will deal with such technicalities where necessary in what follows. Since in this paper we will only treat scalar equations with unique solutions both forwards and backwards in time, the resulting process will be order-preserving, i.e. xs> ys⇒S(t, s)xs> S(t, s)ysfor all t, s ∈R (allowing S(t, s)xsor S(t, s)ysto be ±∞ if necessary allows us to take values of tand s from all of R). 3 Stability & instability in non-autonomous systems We now recall some of the definitions from Langa et al. [15] which we will use in our bifurcation analysis. The simple notion of a complete trajectory will be central: Definition 1 The continuous map x:R→Rmis a complete trajectory if S(t, s)x(s) = x(t)for all t, s ∈R. 4 We will investigate the appearance and disappearance of complete trajectories that are ‘stable’ or ‘unstable’ in certain senses that appear to be appropriate for non-autonomous systems. Note that complete trajectories are merely particular examples of invariant sets in non-autonomous systems: Definition 2 A time-varying family of sets {Σ(t)}t∈Ris invariant (we say “Σ(·)is invariant”) if S(t, s)Σ(s) = Σ(t)for all t, s ∈R. In what follows we make constant use of the Hausdorff semidistance between two sets Aand B, dist[A, B], which is defined as dist[A, B] = sup a∈A inf b∈Bd(a, b) : note that this only measures how far Ais from B(dist[A, B] = 0 only implies that A⊆B). We also use the notation N(X, ²) to denote the closed ²-neighbourhood of a set X: N(X, ²) = {y:y=x+z, x ∈X, z ∈Rmwith |z| ≤ ²}. 3.1 Notions of attraction First we define formally the familiar notion of a set that is attracting forwards in time, with a specified domain of attraction D. For any choice of Dwe say that Σ(·)⊂Dif Σ(t)⊂Dfor every t∈R. Definition 3 An invariant set Σ(·)is forwards attracting within Dif Σ(·)⊂Dand for each s∈R lim t→∞ dist[S(t, s)K, Σ(t)] = 0 for all compact subsets1Kof D. In a non-autonomous system the notion of being ‘locally forwards attracting’ is a little more subtle; we allow the neighbourhood of Σ that is attracted to depend on the initial time. It is clear that if Σ(·) is forwards attracting within Dthen it is also locally forwards attracting within D. 1Note that the definition implies attraction of every initial condition in Kat a uniform rate. Our definition in Langa et al. [15] only required convergence for each fixed initial condition. Contrary to the statement in the footnote in that paper, the two definitions are most certainly not equivalent, even for finite-dimensional systems. 5 Definition 4 An invariant set Σ(·)is locally forwards attracting within Dif Σ(·)⊂D and for each s∈Rthere exists a δ(s)such that lim t→∞ dist[S(t, s)K, Σ(t)] = 0 for all compact K⊂N(Σ(s), δ(s)) ∩D. We now introduce the notion of pullback attraction Definition 5 An invariant set Σ(·)is pullback attracting within Dif Σ(·)⊂Dand for every t∈Rand every compact set K⊂D, lim s→−∞ dist[S(t, s)K, Σ(t)] = 0. Σ(·)is globally pullback attracting if we can take D=Rm. For a set Σ(·) to be locally pullback attracting, the neighbourhood of Σ(·) that is attracted can depend only on the final time. Note that the definition allows a different collection of compact sets K(·) to be attracted to Σ(t) for each fixed t∈R. Definition 6 We say that Σ(·)is locally pullback attracting within Dif Σ(·)⊂Dand for every t∈Rthere exists a δ(t)>0such that if K(·)⊂Dis compact and lim s→−∞ dist[K(s),Σ(s)] < δ(t) then lim s→−∞ dist[S(t, s)K(s),Σ(t)] = 0.(3.1) If Dis bounded it is once again clear that any set that is pullback attracting within Dis locally pullback attracting within D. However, it is an uncomfortable consequence of our definitions that a set can be globally pullback attracting but not locally pullback attracting if Dis unbounded. Nevertheless, this cannot occur if the set is ‘bounded in the past’, as shown by the following lemma. Lemma 1 If an invariant set Σ(·)is pullback attracting within Dand bounded ‘in the past’, i.e. [ t<T Σ(t) is bounded for some T, then Σ(·)is locally pullback attracting. 6 Proof. We show that Σ(·) is locally pullback attracting for any choice of constant δ (this was called ‘uniformly pullback attracting’ in Langa et al. [15]). If lim s→−∞ dist[K(s),Σ(s)] < δ then for some τ, which we choose to be less than T, we must have dist[K(s),Σ(s)] <2δ for all s < τ. Since Σ(s) is bounded for s < T , all such K(s) are contained in a bounded set Xδ. Since Σ is globally pullback attracting, this bounded set is (pullback) attracted to Σ: there exists a σsuch that dist[S(t, s)Xδ,Σ(t)] < ² for all s≤σ. Since K(s)⊂Xδfor all s < T, it follows that dist[S(t, s)K(s),Σ(t)] < ² for all s≤σ, and so Σ is locally pullback attracting. ¤ 3.2 Stability We now give a definition of ‘stability’ in the pullback sense. Definition 7 Σ(·)is pullback Lyapunov stable if for every t∈Rand ² > 0there exists aδ(t)>0such that for any s < t,xs∈N(Σ(s), δ(t)) implies that S(t, s)xs∈N(Σ(t), ²). The following result, analogous to the fact that attraction implies stability for stationary points of scalar autonomous systems, means that in what follows we need not be concerned with Lyapunov stability properties of complete trajectories, but only their attraction properties. Lemma 2 Let x∗(·)be a complete trajectory in a non-autonomous scalar ODE that is locally pullback attracting; then this trajectory is also pullback Lyapunov stable. Proof. Fix t∈R. Given an ² > 0, we can guarantee that if x±(s) = x∗(s)±1 2δ(t) then lim s→−∞ |S(t, s)x±(s)−x∗(t)|= 0, and so in particular there exists a σsuch that |S(t, s)x±(s)−x∗(t)|< ² for all s≤σ. 7 Since the system is order preserving |xs−x∗(s)|<δ(t) 2⇒ |S(t, s)xs−x∗(t)|< ² for all s≤σ. Now we can use the continuous dependence on initial conditions for s∈[σ, t], along with the invariance of x∗(·), to guarantee that for δσ< δ(t) and sufficiently small |xs−x∗(s)|< δσ⇒ |S(t, s)xs−x∗(t)|< ² for all σ≤s≤t. Thus x∗(·) is pullback Lyapunov stable. ¤ We note here that Kloeden [11] has shown that one can generalise the classical notion of a Lyapunov function to cover many non-autonomous systems in such a way that there is a Lyapunov function associated with any pullback attracting set. In particular his results imply the existence of a Lyapunov function for a bounded locally pullback attracting trajectory for the equation ˙x=f(x, t) provided that f(x, t) is locally Lipschitz in x. 3.3 Notions of instability In Langa et al. [15] we introduced two notions of instability. One is simply the converse of Lyapunov stability, while the other, stronger, property appears to be more useful. Definition 8 We say that Σ(·)is pullback unstable if it is not pullback Lyapunov stable, i.e. if there exists a t∈Rand an ² > 0such that, for each δ > 0, there exists an s < t and an x0∈N(Σ(s), δ)such that dist[S(t, s)x0,Σ(t)] > ². We say that Σ(·) is ‘asymptotically unstable’ if its unstable set UΣ(·), defined below (cf. Crauel [4]), is non-trivial (i.e. if UΣ(t)6= Σ(t)). Definition 9 If Σ(·)is an invariant set then the unstable set of Σ,UΣ(·), is defined as UΣ(s) = {x0: lim t→−∞ dist [S(t, s)x0,Σ(t)] = 0}. We say that Σ(·)is asymptotically unstable if for some twe have UΣ(t)6= Σ(t).(3.2) The power of this definition comes from the following simple result (see Langa et al. [15] for the proof). 8 Proposition 3 If Σ(·)is asymptotically unstable then it is also pullback unstable and cannot be locally pullback attracting. Most notions of instability are related to the behaviour of solutions x(t) as t→ −∞; the notion of ‘asymptotic instability’ defined above is essentially a time-reversed notion of ‘forwards attraction’. It should therefore be unsurprising that it is possible to define an alternative notion of instability based on a time-reversed version of pullback attraction: Definition 10 An invariant set Σ(·)is (locally) pullback repelling within Dif it is (locally) pullback attracting within Dfor the time-reversed system, i.e. if Σ(·)⊂Dand for any compact set K⊂Dand for each t∈R, lim s→+∞dist[S(t, s)K, Σ(t)] = 0. 3.4 An aside: linear stability in non-autonomous systems We mention here that we make little use of linear notions of stability in this paper. There appear to be major problems with deducing anything from such ‘infinitesimal’ behaviour without further constraints. As an example, consider the equation ˙x=x−e−t 1 + t2x2, whose solution can be given explicitly as x(t, s;xs) = et esx−1 s+ tan−1(t)−tan−1(s). It is clear that if xsis fixed then as s→ −∞ x(t, s;xs)→x∗(t) = et tan−1(t) + π/2. The trajectory x∗(t) is globally pullback attracting, and also, since it is bounded as t→ −∞, locally pullback attracting (Lemma 1). Since we are treating a scalar equation, the trajectory is also pullback Lyapunov stable (Lemma 2). However, suppose that we linearise about x∗(t), and obtain ˙ X=·1−2e−t 1 + t2x∗(t)¸X =·1−2 (1 + t2)(tan−1(t) + π/2)¸X 9 and once again the right-hand side is bounded below by mλ/2, this time using (5.17). Thus for each fixed tthere exists a σtsuch that if s≤σtand |xs|is sufficiently small the solution exists on [s, t] and hence the origin is locally pullback attracting. When λ= 0.When λ= 0 the explicit solution is x(t) = 1 x−1 s+Rt sg(r) dr,(5.19) and for xs>0 it follows from the asymptotic positivity of gand a simplified version of the above argument that the origin is locally pullback attracting in R+; and that for xs<0 but sufficiently small (depending on s), x(t, s;xs)→0 as t→ −∞, and so the origin is asymptotically unstable. When λ > 0.This case was treated before the formal statement of the proposition. Only the asymptotic instability of the origin and the convergence of xλto zero remain. We deal first with the asymptotic instability of the origin. Since x(t)≡0 and xλ(·) are solutions and the equation is order-preserving, any solution with 0 < xs< xλ(s) exists for all t≤s. Since 0 <Rs −∞ eλF(r)g(r) dr < +∞and eλF(t)→0 as t→ −∞ it follows that for such a solution x(t, s;xs)→0 as t→ −∞. To show that xλ(t)→0 as λ→0, fix tand ² > 0. Choose Tsuch that Zt T g(r) dr > 2eλF (t)/² (which is possible since gis asymptotically positive). Then Zt −∞ eλF(r)g(r) dr=ZT −∞ eλF(r)g(r) dr+Zt T eλF(r)g(r) dr > Zt T eλF(r)g(r) dr. Now, choose λsufficiently small that sup r∈[T,t]|eλF (r)−1|<eλF (t) ²Rt T|g(r)|dr, and then Zt −∞ eλF(r)g(r) dr > eλF (t)/², which implies that xλ(t)< ². Including the extra ‘forwards’ conditions in (5.13) and (5.12), when λ < 0 the origin is locally forwards attracting when xsis sufficiently small, since (5.12) guarantees that inf t≥sZt s eλF(r)g(r) dr > −∞. 16 When λ= 0 the origin becomes locally forwards attracting. When λ > 0 the trajectory xλ(·) is now locally forwards attracting: to show this we can rearrange the explicit solution into the alternative form µ1 x(t)−1 xλ(t)¶= eλ(F(s)−F(t)) µ1 xs−1 xλ(s)¶.(5.20) Therefore |x(t)−xλ(t)|=xλ(t)x(t) eλF(t) eλF(s) xλ(s)xs|xλ(s)−xs|.(5.21) The balance condition in (5.10) implies that any solution with xs>0 is bounded as t→+∞. To see this, consider eλF(t) x−1 seλF(s)+Rt seλF(r)g(r) dr≤MλRt −∞ eλF(r)g(r) dr x−1 seλF(s)+Rt seλF(r)g(r) dr =MλRs −∞ eλF(r)g(r) dr+Rt seλF(r)g(r) dr x−1 seλF(s)+Rt seλF(r)g(r) dr. Condition (5.12) guarantees that the second terms in the numerator and denominator are positive for tsufficiently large, and so lim sup t→∞ x(t)≤Mλmax µ1,xs xλ(s)¶. It therefore follows from (5.21) that xλ(·) is forwards attracting while solutions exist. To show that solutions do not blow up for xs<(1 + αs)xλ(s), observe that x−1 seλF(s)+Zt s eλF(r)g(r) dr > 1 1 + αsZs −∞ eλF(r)g(r) dr+Zt s eλF(r)g(r) dr =Zt −∞ eλF(r)g(r) dr−αs 1 + αsZs −∞ eλF(r)g(r) dr. Using the asymptotic positivity of gthis expression is positive for αssufficiently small. This implies that xλ(·) is locally forwards attracting. Under the final condition the results follow by making the transformations λ7→ −λ, x 7→ −x, and t7→ −t. ¤ We note here that an alternative to requiring stronger conditions at infinity (such as the asymptotic positivity of g) might be to make assumptions on integrals of fand gthat are uniform in time, e.g. Zt+T t g(s) ds≥γ > 0 and Zt+T t|g(s)|ds≤Γ<+∞for all t∈R. 17 Since (see Bohr [1]) almost periodic functions ϕ(·) have time averages that converge uniformly, sup t∈R¯¯¯¯Zt+T t ϕ(s) ds−¯ϕ¯¯¯¯→0 as T→ ∞ (here ¯ϕis the time average of ϕ), such conditions would naturally include this important class of specific examples. 5.2 Conditions for localised bifurcating solutions We now give stronger, but perhaps more natural, conditions on f(t) and g(t) that ensure that the balance conditions (5.10) and (5.11) hold. Lemma 6 Suppose that lim inf t→−∞ g(t)>0 (5.22) and that 0< m = lim inf t→−∞ f(t) g(t)≤lim sup t→−∞ f(t) g(t)=M < +∞.(5.23) Then for λ > 0 λm ≤lim inf t→−∞ xλ(t)≤lim sup t→−∞ xλ(t)≤λM, (5.24) while for λ < 0we have lim inf s→−∞ eλF(s) Rt seλF(r)g(r) dr≥ −mλ, (5.25) where Fis an antiderivative of f. Proof. For any K > M there exists a Tsuch that for all t≤Twe have g(t)>0 and f(t) g(t)≤K. For such tit follows that Zt −∞ eλF(s)g(s) ds≥1 KZt −∞ eλF(s)f(s) ds ≥1 K·eλF(s) λ¸t s=−∞ =1 λK eλF(t), 18 since F(t)→ −∞ as t→ −∞ by (5.22) and (5.23). Therefore xλ(t) = eλF(t) Rt −∞ eλF(s)g(s) ds≤Kλ for all t≤T, and hence lim sup t→−∞ xλ(t)≤Mλ. For the lower bound the proof is similar, but now using the fact that for any k < m there exists a Tsuch that f(t) g(t)> k for all t≤T. The proof of (5.25) follows the same lines. ¤ 5.3 The general case We will now consider the general equation ˙x=G(t, x, λ), and prove a bifurcation theorem based on assumptions on the Taylor coefficients of G. Since we will impose conditions on these coefficients similar to those in Lemma 6, we will be able to show that the system undergoes a transcritical bifurcation that is a little more akin to its autonomous counterpart than that in Proposition 5. We now give our formal definition of a ‘transcritical bifurcation’ in a non-autonomous system. Note that we insist in the definition that the non-zero trajectory is in some sense ‘localised’ near the origin, and that the required behaviour depends only on the system in the past (pullback attraction and asymptotic instability). In our results we will be able to deduce further details of the behaviour of solutions by making additional assumptions on the system in the future. Definition 12 The system ˙x=f(x, t, λ)undergoes a local transcritical bifurcation at x= 0,λ= 0 if there exists a λ0>0and an ² > 0such that (i) for all −λ0< λ < 0the zero solution is locally pullback attracting within (−², 0] and pullback attracting within [0, ²); and there is another negative complete trajectory xλ(t)within (−², 0) that is asymptotically unstable and satisfies xλ(t)→0as λ→0; (5.26) (ii) for λ= 0 the zero solution is asymptotically unstable but still pullback attracting within [0, ²); and 19 (iii) for 0<λ<λ0the zero solution is asymptotically unstable, and there is another positive complete trajectory xλ(t)within (0, ²)that satisfies xλ(t)→0as λ→0 (5.27) and is pullback attracting within (0, ²). While we only require pointwise convergence in (5.26) and (5.27), we will in fact obtain uniform convergence in Theorem 7, which treats the equation ˙x=G(t, x, λ) whose right-hand side has the Taylor expansion G(t, x, λ) = G+Gxx+Gλλ+1 2Gxxx2+Gxλxλ +1 2Gλλλ2 +1 6Gxxxx3+1 3Gxxλx2λ+1 3Gxλλxλ2+1 6Gλλλλ3+. . . (all expressions involving Gand its derivatives on the right-hand side are evaluated at (t, 0,0)). We assume that G(t, 0, λ) = 0 for all tand λ, and furthermore that Gx(t, 0,0) = 0. This implies that ∂kG/∂λk(t, 0,0) = 0 for all tand k∈Z+. We therefore have G(t, x, λ) = λ£Gxλ +1 3Gxλλλ+. . .¤x+£1 2Gxx +1 6Gxxxx+1 3Gxxλλ+. . .¤x2 and this motivates the following theorem. Theorem 7 Consider ˙x=G(t, x, λ), and assume that G(t, 0, λ) = 0 for all λ∈Rand Gx(t, 0,0) = 0. Set f(t) = Gxλ(t, 0,0) and g(t) = −1 2Gxx(t, 0,0), and rewrite the equation as ˙x=λ[f(t) + λφ(t, λ)]x−[g(t) + γ(t, x, λ)]x2, where φ(t, 0) = 1 3Gxλλ(t, 0,0) and γ(t, 0,0) = 0.(5.28) Assume that lim inf t→±∞ g(t)>0,(5.29) 20 that 0< m = lim inf t→±∞ f(t) g(t)≤lim sup t→±∞ f(t) g(t)=M < +∞,(5.30) and that |φ(t, λ)| ≤ h(t),|γλ(t, x, λ)| ≤ h(t),and |γx(t, x, λ)| ≤ h(t),(5.31) where lim sup t→±∞ h(t) g(t)≤K. Then there is a local transcritical bifurcation as λpasses through zero. Furthermore when λ < 0the ‘unstable’ trajectory is pullback repelling in (−², 0); when λ= 0 the origin is locally forwards attracting in R+; and when λ > 0the pullback attracting trajectory xλ(·) is forwards attracting in (0, ²). Note that the standard conditions for a transcritical bifurcation in the autonomous equation ˙x=f(x, λ) are (see Glendinning, [7]): f(0, λ) = 0, fx(0,0) = 0, fxλ(0,0) >0,and fxx(0,0) <0. If G(t, x, λ) = f(x, λ) then we recover these conditions in our theorem. Proof. We assume throughout that |λ| ≤ ², where ²will be chosen ‘sufficiently small’. Note that it follows from (5.28) and (5.31) that |γ(t, x, λ)| ≤ h(t)[|x|+|λ|].(5.32) The origin is locally pullback attracting in (−², ²)for λ < 0.While 0 < x(t, s;xs)≤²we have 0 ≤x(t, s;xs)≤v(t, s;xs) where v(t) solves ˙v=λ[f(t) + ²h(t)]v−[g(t)−2²h(t)]v2with v(s) = xs. There exists a Tsuch that if s≤t≤Tthen we can neglect the second term; changing the definition of Tif necessary, we can use the bound |h(t)| ≤ K0f(t)/m (for some K0> K) to deduce that ˙v≤λ(1 −(²K0/m))f(t)v, from which it follows that v(t)≤e(1−(²K0/m))λ(F(t)−F(t0))v(t0). 21 Once more decreasing Tif necessary, so that f(t)>0 for all t≤T, it follows that for s≤t≤Twe have v(t, s;xs)≤²provided that 0 < xs≤²and hence, since the comparison x(t, s;xs)≤v(t, s;xs) remains valid, it follows that lim s→−∞ S(t, s)xs= 0 for all t≤T. Since S(τ, t) is continuous and zero is invariant we have lim s→−∞ S(τ, s)xs=S(τ, t)·lim s→−∞ S(t, s)xs¸=S(τ, t)0 = 0 for all τ∈R, and the origin is pullback attracting within [0, ²). While −²≤x(t, s;xs)≤0 we have u(t, s;xs)≤x(t, s;xs)≤0 where u(t) solves ˙u=λ[f(t)−²h(t)]u−[g(t)+2²h(t)]u2with u(s) = xs; therefore while u≥ −² ˙u≥λf(t)[(1 −²K/m)u]−(1 + 2²K)u2/m. For Tchosen such that f(t)>0 for all t≤T, and for 0 ≥xs≥ −λ(m−²K)/m(1 + 2²K) it follows that lim s→−∞ S(t, s)xs= 0 for all t≤T, and arguing as above the origin is locally pullback attracting within (−², 0]. When λ= 0.While |x| ≤ ²we have ˙x≤ −[g(t)−2²h(t)]x2, which immediately gives the pullback attraction of the zero solution within [0, ²), and the asymptotic instability of zero, since for xs<0 we have x(t, s;xs)→0 as t→ −∞. There is a positive trajectory that is pullback attracting in [0,∞)when λ > 0.While |x(t, s;xs)|,|λ|< ² we have u(t, s;xs)≤x(t, s;xs)≤v(t, s;xs),(5.33) where u(t, s;xs) and v(t, s;xs) are the solutions of ˙u=λ[f(t)−²h(t)] | {z } f−(t) u−[g(t)+2²h(t)] | {z } g+(t) u2with u(s) = xs 22 and ˙v=λ[f(t) + ²h(t)] | {z } f+(t) v−[g(t)−2²h(t)] | {z } g−(t) v2with v(s) = xs.(5.34) In particular, we have an explicit form for the solution of (5.34), namely v(t) = eλF+(t) x−1 seλF+(s)+Rt seλF+(r)g−(r) dr. Using the balance condition (5.23) it follows that for λand xssufficiently small, v(t)≤² for all t≤0. In this case the comparison (5.33) remains valid for all such t. Due to the two-sided balance and the balance between hand git follows that we can define the upper and lower solutions x+(t) = eλF+(t) Rt −∞ eλF+(r)g−(r) dr and x−(t) = eλF−(t) Rt −∞ eλF−(r)g+(r) dr, the pullback attractors of the upper and lower equations. We then have x−(t)≤lim inf s→−∞ x(t, s;xs)≤lim sup s→−∞ x(t, s;xs)≤x+(t). Therefore there exists a pullback attractor A(t) within the phase space consisting of the interval (0, ²). Since the system is order-preserving, there are two solutions x1(t) and x2(t) such that A(t) = [x1(t), x2(t)], and so we have x−(t)≤xj(t)≤x+(t) for j= 1,2. If we set z(t) = x1(t)−x2(t) then dz dt≤λ[f(t) + ²h(t)]z−g(t)(x1+x2)z−[γ(t, x1, λ)x2 1−γ(t, x2, λ)x2 2] ≤λf+(t)z−g(t)(x1+x2)z−γ(t, x1, λ)(x2 1−x2 2) +[γ(t, x1, λ)−γ(t, x2, λ)]x2 2 ≤λf+(t)z−g(t)(x1+x2)z+ 2²h(t)[x1+x2]z+²x1h(t)z ≤[λf+(t)−(2g(t)−5²h(t))x−(t)]z. Since 2g(t)−5²h(t)≥2−5²K 1 + ²K g+(t) this gives dz dt≤"λf+(t)−2−5²K 1+2²K g+(t)eλF−(t) Rt −∞ eλF−(r)g+(r) dr#z. 23 We have z(t)≤z(t0)eI(t,t0), where I(t, t0) := Zt t0 λf+(s)−2−5²K 1+2²K g+(s)eλF−(s) Rt −∞ eλF−(r)g+(r) drds =λ(F+(t)−F+(t0)) −2−5²K 1+2²K ·ln Zs −∞ eλF−(r)g+(r) dr¸t s=t0 . Now, f− M≤g+≤1+2²K m−²K f−, and so ·ln Zs −∞ eλF−(r)g+(r) dr¸t s=t0≥ln µ1 λM eλF−(t)¶−ln µ1+2²K λ(m−²K)eλF−(t0)¶ =λ(F−(t)−F−(t0)) + ln m−²K M(1 + 2²K). Therefore I≤λ(F+(t)−F+(t0)) −2−5²K 1+2²K λ(F−(t)−F−(t0)) + C², where C²=−2−5²K 1+2²K ln m−²K M(1 + 2²K)>0. Since f−≥1−(²K/m) 1+(²K/m)f+ we also have F−(t)−F−(t0)≥1−(²K/m) 1+(²K/m)[F+(t)−F+(t0)], and so I≤λ(F+(t)−F+(t0)) ·1−2(1 −5 2²K)(1 −²K/m) (1 + 2²K)(1 + ²K/m)¸+C². It follows that for ²sufficiently small we can guarantee that z(t) = 0, and hence that there is a single pullback attracting positive trajectory x∗(·). Now note that the above argument is in fact valid for any two trajectories x1(·) and x2(·) that are bounded below by x−(t). Now also note that any trajectory x(t, s;xs) with xs>0 has x(t, s;xs)>3 4x−(t) for tlarge enough (cf. argument following (5.20) in the proof of Proposition 5); this is also enough to apply the above argument, and so x∗(·) is attracting in (0, ²) as t→+∞. 24 The origin is unstable ‘downwards’ when λ > 0.We have 0 ≥x(t)≥u(t) where u(t) solves ˙u=λ[f(t)−²h(t)]u. As t→ −∞ we therefore have u(t)→0, and so we have x(t)→0 too. The unstable trajectory when λ < 0.The transformation x7→ −x,t7→ −t, gives the existence of a candidate for the negative unstable trajectory; its instability follows from the fact that x∗(·) is attracting ‘from above’ as t→+∞.¤ 6 Non-autonomous ‘simple pitchfork’ bifurcation The canonical autonomous example of an equation exhibiting a pitchfork bifurcation is ˙y=µy −y3.(6.1) For µ < 0 the only fixed point is the origin, which is stable; while for µ > 0 the origin is unstable and there are two new fixed points at ±√µwhich are stable. We now give a formal definition of what we understand by a ‘pitchfork bifurcation’ for a non-autonomous system. Note that as before all the behaviour in the definition only relies on the properties of the equation ‘in the past’. Definition 13 The system ˙x=f(x, t, λ)undergoes a localised pitchfork bifurcation at x= 0,λ= 0 if there exists a λ0>0and an ² > 0such that (i) for all −λ0< λ ≤0the zero solution is pullback attracting within (−², ²); (ii) when 0< λ < λ0the zero solution is asymptotically unstable, and there exist bounded trajectories x+ λ(t)and x− λ(t)that are pullback attracting in (0, ²)and (−², 0) respectively, and satisfy x± λ(t)→0 as λ↓0 uniformly on compact subsets of R. Since equation (6.1) is invariant under the transformation y7→ −yit is convenient to consider the new variable x= 2y2, which satisfies the equation ˙x= 2µx −x2. 25 it follows that any trajectory with x−:= −δrm−²K 1 + ²K < xs≤² has |x(t, s;xs)| ≤ ²for all t≥s, and hence that δrm−²K 1 + ²K ≤lim s→−∞ x(t, s;xs)≤². Thus the pullback attractor in (x−, ²] consists of the interval [x1(t), x2(t)]. Considering the difference z=x1−x2this satisfies dz dt=λ[φ(t, x1, λ)−φ(t, x2, λ)] −(x1+x2)g(t)z−x2 1ψ(t, x1) + x2 2ψ(t, x2) ≤δ2h(t)z−(x1+x2)g(t)z+ (x2 2−x2 1)ψ(t, x1)+[ψ(t, x2)−ψ(t, x1)]x2 1 ≤δ2h(t)z−(x1+x2)g(t)z+ [²+δ2]h(t)(x1+x2)z+h(t)z²2 ≤C[²2h(t)−²g(t)]z ≤ −C(1 −K²)²g(t)z as ²→0. It follows that for ²chosen sufficiently small, z(t) = 0. Using the same argument for the time-reversed systems gives a saddle-node bifurcation. ¤ 8 The balance hypothesis: examples In this final section we give some examples demonstrating that without some kind of ‘balance’ between successive terms in the Taylor series we cannot expect the type of bifurcation results above. Note that while all these examples are asymptotically autonomous (as t→ ∞), the behaviour of the non-autonomous equation is different from that of its autonomous limit. Our simplest example is ˙x=λx −e−tx2with x(s) = xs≥0, where the exponential term produces very strong dissipativity as t→ −∞. From the explicit solution x(t, s;xs) = eλt x−1 seλs + (λ−1)−1(e(λ−1)t−e(λ−1)s) 32 it is clear that while for λ < 0 the origin is pullback attracting in R+, this is also the case when 0 <λ<1. Thus the ‘one-sided pitchfork’ type bifurcation that we might expect is suppressed. [Note, however, that the complete (but unbounded) trajectory x∗(t) = (λ−1)etis forwards attracting for all λ > 0.] In the previous example we made one of the terms of the Taylor expansion that plays a prime rˆole in the bifurcation blow up as t→ −∞. However, we can also shift this behaviour to the higher-order terms and run into similar problems. For the equation ˙x=λx −x2−e−tx3with x(s) = xs≥0 it is clear that for λ < 0 the origin is globally pullback (and forwards) attracting; while for λ > 0 we have λx −x2−e−tx3≤λx −e−tx3, so that the continued pullback attraction of the zero solution follows the previous example after setting y=x2and µ= 2λ(see Section 6). A similar example, but one in which the higher-order terms produce instability (rather than enhance the stability), is ˙x=λx −2x2+ e−tx3. Given an initial condition xs, whatever the value of λwe can choose Tsufficiently large and negative that λx −2x2+ e−tx3≥1 2e−tx3for all t≤T. It follows that for any xs, lim s→−∞ x(t, s;xs) = +∞, and there is never a pullback attracting trajectory. 9 Conclusion We have tried to develop a general theory for bifurcations in non-autonomous scalar systems, in particular giving a set of possible definitions for transcritical, pitchfork, and saddle-node bifurcations that depend only on properties of the system in the past. There are, of course, many ways in which these results could be improved. The main problem is the restrictive nature of some of the conditions that we have required on the 33 terms in our Taylor expansion. As we remarked at the end of Section 5.1, it should be possible to prove similar results replacing assumptions such as the asymptotic positivity of terms in the equation by time integrated (or perhaps time-averaged) conditions such as Zt+T t g(s) ds≥γ > 0 for all t∈R. There are higher-dimensional bifurcation results for certain systems, in particular in the almost periodic case (see Kloeden [12], for example). We hope to extend the results here to general higher dimensional systems, by considering the scalar systems obtained by restricting attention to an appropriate centre manifold, as is done in the autonomous case. Acknowledgements. We are all most grateful to the Royal Society for their support: our collaboration has been funded by a Royal Society Joint Project grant. JCR is currently a Royal Society University Research Fellow, JAL and ASF have been supported in part by M.E.C. (Spain, Feder), Proyectos BFM2002-03068 and BFM2003-06446, respectively. We would like to thank two anonymous referees for their helpful and detailed suggestions. References [1] H. Bohr. Almost periodic functions. Chelsea Publishing Company, New York, 1947. [2] D.N. Cheban, P.E. Kloeden, & B. Schmalfuß. The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems. Nonlinear Dynamics and Systems Theory,2(2002) pp. 9–28. [3] V.V. Chepyzhov & M.I. Vishik. Attractors of non-autonomous dynamical systems and their dimension. J. Math. Pures Appl. 73 (1994) pp. 279–333. [4] H. Crauel. Random point attractors versus random set attractors. J. London Math. Soc. 63 (2001) pp. 413–427. [5] H. Crauel, A. Debussche, & F. Flandoli. Random attractors. J. Dyn. Diff. Equ. 9 (1997) pp. 397—341. [6] C.M. Dafermos. An invariance principle for compact processes. J. Diff. Eq. 9(1971) pp. 239–252. 34 [7] P.A. Glendinning. Stability, instability and chaos: an introduction to the theory of nonlinear differential equations, Cambridge Texts in Applied Mathematics, CUP 1994. [8] R.A. Johnson. Hopf bifurcation from nonperiodic solutions of differential equations. I. Linear theory. J. Dyn. Diff. Eq. 1(1989) pp. 179–198. [9] R.A. Johnson & Y. Yi. Hopf bifurcation from nonperiodic solutions of differential equations, II. J. Dyn. Diff. Eq. 107 (1994) pp. 310–340. [10] R.A. Johnson, P.E. Kloeden, & R. Pavani. Two-step transition in nonautonomous bifurcations: an explanation. Stoch. Dyn. 2(2002) pp. 67–92. [11] P.E. Kloeden. A Lyapunov function for pullback attractors of nonautonomous differential equations. Electronic J. Diff. Equ. Conf. 5(2000) pp. 91–102. [12] P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure and Applied Analysis 3(2004) pp. 161–173. [13] P.E. Kloeden & B. Schmalfuß. Cocyle attractors of variable time step discretizations of Lorenzian systems. J. Difference Eqns. Applns. 3(1997) pp. 125–145. [14] P.E. Kloeden & S. Siegmund. Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems. International Journal of Bifurcations and Chaos 15 (2005) pp. 743–762. [15] J.A. Langa, J.C. Robinson, & A. Su´arez. Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity 15 (2002) pp. 887– 903. [16] J.A. Langa & A. Su´arez. Pullback permanence for non-autonomous partial differential equations, Electronic J. Diff. Eq. 72 (2002) pp. 1–20. [17] B. Schmalfuß. Attractors for the nonautonomous dynamical systems. In B. Fiedler, K. Gr¨oger and J. Sprekels (eds.). Proceedings of Equadiff 99 Berlin, World Scientific, Singapore (2000) pp. 684–689. [18] G.R. Sell. Non-autonomous differential equations and Topological Dynamics. Trans. Amer. Math. Soc. 127 (1967), pp. 241-283. 35 [19] W. Shen & Y. Yi. Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem. Amer. Math. Soc. 647 (1998). 36