Mathematical analysis of a model for moon-triggered clumping in Saturn s rings

Transcription

1 Mathematial analysis of a model for moon-triggered lumping in Saturn s rings Pedro J. Torres, Prasanna Madhusudhanan, Larry W. Esposito Departamento de Matemátia Apliada, Universidad de Granada, Faultad de Cienias, Granada, Spain. Laboratory of Atmospheri and Spae Physis, University of Colorado, 39 UCB, Boulder, CO 83, USA. ptorres@ugr.es, mprasanna@olorado.edu, larry.esposito@lasp.olorado.edu Abstrat Spaeraft observations of Saturn s rings show evidene of an ative aggregation-disaggregation proess triggered by periodi influenes from the nearby moons. This leads to lumping and break-up of the ring partiles at time-sales of the order of a few hours. A mathematial model has been developed to explain these dynamis in the Saturn s F-ring and B-ring [5], the impliations of whih are in lose agreement with the empirial results. In this paper, we ondut a rigorous analysis of the proposed fored dynamial system for a lass of ontinuous, periodi and zero-mean foring funtions that model the ring perturbations aused by the moon flybys. In speifi, we derive the existene of at least one periodi solution to the dynami system with the period equal to the foring period of the moon. Further, onditions for the uniqueness and stability of the solution and bounds for the amplitudes of the periodi solution are derived. Keywords: Saturn ring, aggregation-disaggregation proess, periodi solution, stability AMS Subjet Classifiation: 34C5, 85A99. Introdution In our solar system, all the giant planets have rings. The rings around the planets are flat disks of innumerable small partiles that orbit the planet in its equatorial plane [4]. The ring partiles range in sizes from dust to boulders to small moonlets. The partiles ontinually ollide, resulting in both fragmentation and aggregation. The partiles will aggregate into temporary units resembling piles of rubble. Dust and smaller partiles ollet on the surfaes of larger ones, and an be knoked off to orbit freely, or be re-aptured by another partile. The dynamial proesses in rings are likely to resemble those in other flattened systems like spiral galaxies, blak-hole aretion disks and in the proto-planetary disks where planets form. The largest and best-studied ring system is Saturn s, whih is now being examined at lose range by the Cassini spae mission. At the time surrounding Saturn s equinox in 9, when the Sun set on the plane of the rings, Cassini s amera found a surprising number of small objets within the rings by the long shadows that they ast. Thus, the low sun angle provided a unique opportunity to detet larger bodies embedded in Saturn s rings. These observations inspired lose analysis of the many star oultations observed by Cassini, when a star s brightness is measured up to a thousand times per seond while the star passes behind the rings. This tehnique yields unpreedented spatial resolution, allowing the detetion of objets bigger than a few meters in size. The starlight flutuations were analyzed by power spetral analysis, indiating lumps -m in size at the outer edge of Saturn s brightest B) ring, the same loation where the shadowasting bodies were seen [6]. Small transient lumps are also seen in Saturn s F ring [6, 3]. In both ases, P.J. Torres is partially supported by projet MTM-365, Spain. P. Madhusudhanan and L. W. Esposito were supported by the NASA Cassini Mission, USA.

2 the lumps are orrelated with the loation of nearby moons. This led Esposito to propose [5] that the moon triggers the lumping, partiularly at the distanes from Saturn where a ring partile s mean orbital motion resonates with that of the moon. The ompliated dynamial interations in planetary rings are sometimes modeled as a fluid and other times as an example of granular flow [9]. Numerial solutions often rely on N-body simulations see for instane [7]). Sine the ring partile dynamis is so ompliated and the N-body solutions are so slow, Esposito et al [5] developed a simplified dynamial model based on some analogies with Plasma Physis. This model solves for the oupled evolution of aggregates and the veloity dispersion at a loation within the rings. The dynamial model proposed in [5] aims to explain the aggregation and disaggregation proesses observed in Saturn s F ring and the B ring outer edge due to the perturbation aused by the Saturn s moon, Prometheus and Mimas, respetively. Their numerial simulations demonstrate that the dynamial model explains ertain spaeraft measurements orresponding to these regions in the Saturn s ring. Further, these numerial results strongly suggest the existene of a limit yle for realisti values of the parameters. In this paper, we provide a rigorous mathematial proof of this fat as well as study other features that are relevant from a dynamial point of view. The model under onsideration relates the mean aggregate mass M and the veloity dispersion v rel of ring partiles at the above mentioned loations in the Saturn s ring, as follows dm dt dv rel dt = M ) v rel M T a v th T oll = v rel ε ) + M v es M ) T oll M ft)..) T stir In the first equation of.), the first term refers to the oagulation of the ring partiles leading to a growth in the mean aggregate mass at an aretion rate T a T a is the aretion period) and the seond term refers to the erosion or break-up) of the larger ring partiles due to ollisions with other ring partiles at ollisional rates T oll T oll is the ollisional period) when their veloity dispersion exeed v th, the threshold veloity for stiking. In the seond equation of.), the first term refers to the dissipative ollisions between the ring partiles leading to the redution in veloity dispersion with ε as the normal oeffiient of restitution. The seond term refers to the inrease in the veloity dispersion to the esape veloity from mass M v es M ) is the esape veloity from an aggregate of mass M ) due to the visous stirring at a rate T stir T stir is the stirring period), aused by the passage of the ring partiles of mass M. Finally, the last term refers to the periodi foring by the Saturn s moon ausing disturbanes in the veloity dispersion, where ft) = πv th β T syn desribes the moon perturbation, where T syn is the foring period, β is the foring amplitude. os πt T syn ) For onveniene, we redue the system to an equivalent dimensionless system with the following substitutions: x M M o, y v rel and t t vth T orb, where T orb is the orbital period of the ring partiles around Saturn, and pass to a more mathematial notation by using d dt =. Then, the system in.) beomes where a = T orb T a, b = T orb x = ax bxy y = y + dx ft).) T oll, = T orb ε ) T oll, d = T orbv es T stir v th are all positive parameters, f t) = A o os ωt), A o = πβt orb T syn, ω = πt orb T syn foring period..3) = π T, and T is the normalized) The above dynamial system has two main differenes when ompared to the predator-prey system. Firstly, in the seond equation, we have the x term instead of the oupling xy. Seondly, the external foring is not standard in eologial models, sine the typial foring is parametrial see for instane the model studied in []).

3 Despite these differenes, the aggregation-disaggregation model represented by this system an be well explained by drawing an analogy with the predator-prey system. Let the mean aggregate mass of the ring partiles orrespond to the prey population, and the veloity dispersion orrespond to the predator population. The veloity dispersion feeds off the aelerations from the aggregates gravity in referene to the seond term of the seond equation). As the veloity dispersion grows too large, it limits the prey as high veloity ollisions lead to to fragmentation of the ring partiles in referene to the seond term of the first equation). In the absene of the interation between the predator and the prey, the prey population mean aggregate mass) grows and the predator population deays in referene to the first terms of the two equations, respetively). In addition, the predator population is periodially ontrolled in a ertain deterministi manner, as represented by the sinusoidal foring funtion. In the following setion, we ondut a thorough qualitative analysis of the above system and omment about the existene and nature of its solution, onditions on uniqueness and asymptoti stability of the solution and finally derive expliit bounds for the solutions. Along with these results, we derive useful insights about the Saturn s ring dynamis based on these results. Further, the results in this paper hold for any general foring funtions ft) as long as it is a ontinuous periodi funtion of minimal period T = π ω with zero mean value, i.e., T ft)dt =. All the results will be presented for this general foring funtion, and the sinusoidal foring funtion will be expliitly studied in Setion 5. Existene of a periodi solution Numerial results in [5] strongly suggest the existene of a limit yle for realisti values of the parameters. In the following theorem, we provide an analytial proof for the same. Further, the result holds for all feasible values of the system parameters. Theorem System.) has at least one T -periodi solution x, y) with xt) > for all t. Proof. By introduing the hange z = ln x, system.) is transformed into z = a by y = y + d expz) ft)..4) The theorem is proved by showing that the above system is equivalent to a seond order differential equation of the Duffing type, and then invoking the Landesman-Lazer onditions for the existene of a periodi solution. See Appendix A for the omplete proof. The above result leads to a physial interpretation that the periodi foring by the moon gives a limit yle in the mean aggregate mass and veloity dispersion with the same period as the foring funtion. It is interesting to remark that the latter result an be extended to a family of foring terms with more ompliated reurrene inluding quasi periodi foring. The main result in [] implies the existene of a quasi periodi solution and the boundedness of all solutions in the future. 3 Uniqueness and asymptoti stability In this setion, we derive onditions for the existene of a unique periodi solution and omment about the stability of the solution. Theorem When at < +, 3.5) system.) has a unique T -periodi solution, whih is asymptotially stable. Proof. See Appendix B. 3

4 Remark Observe that for f, the unique equilibrium of system.) is hyperboli, hene robust to small perturbations. Therefore, if A is small enough,.) presents an asymptotially stable T -periodi orbit. The remarkable feature of the latter result is that stability is proved on a ertain region of the involved parameters independently of the size of the external fore f. As we will see in Setion 5, Theorem is appliable to the Saturn s F-ring, in fat for the F -ring a = 6τ ε ).4 so ondition 3.5) is satisfied. The ondition 3.5) is also satisfied for the Saturn s B-ring outer edge when the optial depth is τ =.5. In the following theorem, a ondition for the existene and stability of the solution is derived that depends on the nature of the foring funtion via its supremum norm, denoted by f. Theorem 3 Under the assumption a + b f < 4 + π T, 3.6) system.) has at least one asymptotially stable T -periodi solution. Proof. The above result is obtained by using the tehnique of lower and upper solutions [8]. See Appendix C for the omplete proof. In the ontext of the Saturn s ring dynamis, Theorem and Theorem 3 show that the fored ring system has a limit yle behavior around the fixed point and when the external foring satisfies the above ondition, it drives a stable limit yle. Further, the above theorems provide a theoretial justifiation for the onentri limit yles that were observed through numerial simulations in [5, Fig. 6 and Fig. 7]. 4 Expliit bounds Theorem provides a general existene result, but more onrete quantitative information about the loation of the periodi orbits is desirable. In this setion, we derive expliit estimates for the periodi solutions. In the next result, f + = max{f, }, f = max{ f, } denotes the positive and negative part respetively of a given funtion f, and f p denotes the usual L p -norm. Theorem 4 Let x, y) with xt) > for all t a T -periodi solution of system.). Then, the following bounds hold a bd exp ) T b f xt) T ) a bd exp b f a b T ) f + yt) a b + T ) + f 4.7) for all t. Proof. The bounds derived above are obtained by working with the seond order differential equation of Duffing type.) that is equivalent to the system in.). See Appendix D for the omplete proof. Sine the lower bound for x t) is non-negative, the T -periodi solution x t) is also non-negative for all t. The above argument ats as an alternate proof for the later part of the statement in Theorem. Further, the bounds derived for the T -periodi solution for y t) may not neessarily be tight. Espeially, for small foring amplitudes, the T -periodi solutions are expeted to be lose to the stable feasible fixed point, but the upper and lower bounds deviate from the stable fixed point by a fator of T about the fixed point. As a result, we derive tighter bounds for y t) in the following theorem. Theorem 5 Alternatively, y satisfies the bounds: y t) max ab T e T ), a exp y t) min ab T a exp e T ), b T f b b T f b ) min v max, ) max f et + ) v min, f, e T ), f + f +, e T ) 4.8), f ) e T, 4.9) ) 4

5 where v max = max Green s funtion. t t+t t+t G t, s) f s) ds, v min = min G t, s) f s) ds, and Gt, s) = exps t) t expt ) is the Proof. The alternative bounds given by 4.8) and 4.9) are found working diretly with the seond equation of system.4). Given a T -periodi solution ht), it is known that the unique T -periodi solution of the first-order linear equation y + y = ht) is y = ˆ t+t Then, from the seond equation of system.4), we have y = ˆ t+t t t Gt, s)hs)ds. Gt, s) [d expz) fs)] ds. Tight upper and lower bounds are derived for eah of the two terms in the above integral onsidered separately. By appropriately ombining these bounds, we obtain 4.8) and 4.9). See Appendix E for the omplete proof. The bounds in 4.8) and 4.9) are tight for small foring amplitudes. Combining these with the seond inequality in 4.7), the bounds are tight for all foring amplitudes. Remark Sine y represents a squared relative veloity, a useful lower bound for y should be positive. In this sense, the seond bound given by 4.8) has the advantage that the lower bound is always positive if the external fore f is not too large. In the model ase ft) = A osωt), it is easy to ompute f + = f = A /ω, f = A π ω, and the bounds are diret. Having ompleted the mathematial analysis of the system.), in the following setion, we onsider examples with realisti values of the parameters of the system representing different regions of the Saturn s ring and derive useful insights pertinent to the ring dynamis. 5 Insights on Saturn s ring dynamis We begin with some orollaries of the theorems presented in the previous setions for the Saturn s ring system. Remark 3 The non-trivial, feasible stable fixed point for the unfored dynami system is M, vrel) = Tstir ε ) v, T oll es Mo)Ta T a ). When externally fored by periodi perturbations from the Saturn s moons, the mean aggregate mass and the veloity dispersion demonstrates periodi aggregation-disaggregation yle with the same period as that of the moon s foring. Corollary of Theorem ) There exists a unique, asymptotially stable limit-yle for the M,v rel ) system if the foring period satisfies the ondition T < + 4τ ε ) 8τ ε, 5.) ) where the right-hand-side depends on the mean optial depth and the normal oeffiient of restitution orresponding to the speifi loation in the Saturn s ring system. The realisti parameters orresponding to the Saturn s ring system are ε =.6, T syn = T orb = T =. The optial depth is τ =. for the F-ring and τ =.5,,.5 for the B-ring outer edge. Hene, our result is appliable to the F-ring and also to the B-ring outer edge when τ =.5. Now, we onsider a systemati study of eah region in the Saturn s rings, and start with the dynamis of the B-ring outer edge. 5

6 Corollary For a sinusoidal foring funtion, ft) = πβ T os ) πt T, the upper and lower bounds of the T -periodi solution for the mass of the aggregate is e πβ ε) For vrel t), the bounds are obtained from Theorems 4 and 5 with a = b = v th = m/se, v max = v min = M t) πβ 4τT stir ε ) e ε). 5.) πβ, f T +4π = T t= f t) dt = πβ β,, f + = T t= f t) dt = β, and f = πβ T. = 4τ, v es M ) =.5m/se, T, f + = T t= f + t) dt = We begin with studying the impat of varying the stirring rate f stir = T stir on the amplitudes of the periodi solutions of M t), vrel t)), and Figures a - plot the maxima and minima of the T -periodi solution of the M, vrel) system obtained using numerial simulations against the derived upper and lower bounds for the T -periodi solution of the M, vrel) system from Theorem 4 and Theorem 5. The system parameters orresponding to the Saturn s ring system are ε =.6, T syn = T orb T = ), β =.5 and τ =.5,, and.5 for Figure a, Figure b and Figure, respetively. The Saturn s B-ring inluding the Janus : density wave, and the B-ring outer edge have.5 and as the typial values for the optial depth. Further, τ =.5 is the typial optial depth value for the Saturn s A ring, Mimas 5:3 density wave and the Janus 6:5 density wave. From 5.), it is lear that the amplitude upper and lower bounds for the periodi solutions of M t) vary log-linearly with f stir, and hene the orresponding urves in the plot are straight lines. It is interesting to note that the atual values for the maximum and minimum amplitudes obtained through simulations) also appear to have a log-linear relationship with f stir with the same slope as that of the upper and lower bounds. As a result, an exponential inrease in f stir auses an exponential deay in the mean aggregate mass of the ring partiles. This effet an be understood as follows. Large f stir auses the rate of hange of y t) to be positive, and one vrel t) > T oll T a same as y > a b in.4)), this auses a negative slope for M t), and essentially ausing the exponential deay of M t). Further, the parameter f stir has a negligible effet on the amplitudes of vrel t), as is evident from the negligible slope in the figure above. Notie that both the axes are in log-sale and the plots for the lower-bound of vrel t) does not appear in the plot beause it takes negative values for all values of f stir. We see that the effet of hanging τ only hanges the magnitudes of maxima, minima, and the bounds on M t) and vrel t). Otherwise, M t) has a log-linear relationship with f stir, with the same slope for all values of τ. Comparing Figures a -, the magnitudes of maxima M t) inreases with inrease in τ, and the minima dereases with inrease in τ. Further, the magnitude of the maxima of vrel t) remains unhanged with inrease in τ and the magnitude of the minima dereases with inrease in τ. The physial interpretation is that larger optial depth leads to a larger mean aggregate mass, but not large veloity dispersion. This is beause the veloity dispersion is pegged near the fixed point value given by the strong erosion for vrel t) > v th, whih reverses the growth of M t). For the same reason, inreasing f stir redues the upper value for M t). Next, we study the impat of varying the foring amplitude β on the periodi solutions of M t), vrel t)), and Figures a - plot the maxima and minima of the T -periodi solution of the M, vrel) system obtained using numerial ) simulations against the derived upper and lower bounds for the T -periodi solution of the M, v rel system from Theorem 4 and Theorem 5. The system parameters are the same as before and the stirring period is assumed to be equal to the orbital period of the ring partiles around Saturn, in other words, f stir =. Notie that for small foring amplitudes, the periodi solutions for M t) and vrel t) has a small amplitude about their average values, whih in this ase, are the asymptotially stable fixed points see Remark 3) of the orresponding autonomous system of differential equations. With inrease in β, we see an inreasing deviations in the amplitude of M t) about the average values, while still non-negative, sine the lower-bound from 5.) is non-negative. Translating to the physial phenomena, the ring partile aggregates tend to ahieve large masses as well as negligibly small masses, with inrease in the foring amplitude, in eah period. The inrease in foring amplitudes auses a similar effet on vrel t) as on M t). But inreasing the foring amplitude beyond a ertain threshold auses vrel t) to take negative values, whih is non-physial. ε 6

7 3 Mass V rel 3 Mass V rel Upper bound Lower bound Mass Sim V rel Sim Upper bound Lower bound Mass Sim V rel Sim log sale log sale stirring frequeny relative to orbital frequeny f stir ) in log sale a) stirring frequeny relative to orbital frequeny f stir ) in log sale b) 3 log sale Mass V rel Upper bound Lower bound Mass Sim V rel Sim 3 stirring frequeny relative to orbital frequeny f stir ) in log sale ) Figure : Derived bounds ompared to numerial simulations for varying f stir : The system parameters orresponds to the B-ring outer edge where the normal oeffiient of restitution ε =.6, the foring period of the moon is equal to the orbital period of the ring partiles about Saturn T = ) and the foring amplitude β =.5. a) The mean optial depth is τ =.5, b) the mean optial depth is τ = and ) the mean optial depth is τ =.5. 7

8 This sets a strit upper limit on the foring amplitude, whih in the ase of the above figure, happens around β =.6. Comparing Figure a and, as τ inreases from.5 to.5, M t) maxima inreases, vrel t) maxima hanges only slightly. Finally, notie that the upper and lower bounds for M t) and vrel t) are tight for small foring amplitudes. This means that the behavior of the system is well-haraterized by the bound derived in Theorem 4 and Theorem 5 for the regions in the Saturn s ring system where the influene of the moons are negligible. The analytial results provide a reasonable idea about the nature of the system and parallel the expliit numerial simulations. 6 Conlusions We onsider the mathematial model developed in [5] for understanding the dynamis of ertain speifi loations in the Saturn s ring system due to the periodi perturbations aused by the nearby moon. This model relates the mean aggregate mass and the veloity dispersion of the ring partiles at a ertain loation in the Saturn s ring system in a manner similar to the popular predator-prey model in eology, although it represents a totally different physial phenomenon. Further, this has been studied purely using numerial simulations so far. In this paper, we ondut a rigorous mathematial analysis of the dynamial system and derive results pertaining to the existene, uniqueness and stability of its solution. We analytially prove that the dynamial system auses a periodi solution for the mean aggregate mass and the veloity dispersion of the ring partiles at a period equal to the foring period of the moon. Further, upper and lower bounds for the solution to the system are derived, that are espeially tight for small perturbations aused by the moons. These bounds well-haraterize the dependene of the system solution on various system parameters. Several useful insights are drawn about the Saturn s ring system based on the theoretial results presented here. Referenes [] S. Ahmad, A Nonstandard Resonane Problem for Ordinary Differential Equations, Transations of the Amerian Mathematial Soiety, Vol. 33, No. Feb., 99), pp [] Z. Amine, R. Ortega, A periodi prey-predator system, J. Math. Anal.Appl ), [3] K. Beurle, C.D. Murray, G.A. Williams, M.W. Evans, N.J. Cooper, C.B. Agnor, Diret evidene for gravitational instability and moonlet formation in Saturn s rings, Astophys. J. 78 ), 76-8 [4] L.W. Esposito, Planetary Rings, Cambridge University Press 6). [5] L.W. Esposito, N. Albers, B.K. Meinke, M. Sremčević, P. Madhusudhanan, J.E. Colwell, R.G. Jerousek, A predator-prey model for moon-triggered lumping in Saturn s rings, Iarus 7 ), 3-4. [6] L.W. Esposito, B. K. Meinke, J.E. Colwell, P.D. Niholson, M.M. Hedman, Moonlets and Clumps in Saturn s F Ring, Iarus Vol 94/ 8), [7] M.C. Lewis, G.R. Stewart, Features around embedded moonlets in Saturn s rings: The role of self-gravity and partile size distributions, Iarus 99 9), [8] F.I. Njoku and P. Omari, Stability properties of periodi solutions of a Duffing equation in the presene of lower and upper solutions, Appl. Math. Comput. 35 3), [9] J. Shmidt, K. Ohtsuki, N. Rappaport, H. Salo, F. Spahn, Dynamis of Saturn s Dense Rings, in Saturn from Cassini-Huygens, by M.K. Dougherty, L.W. Esposito, S.M. Krimigis, ISBN Springer Siene+Business Media B.V., 9, p. 43. [] G. Talenti, Best onstant in Sobolev inequality, Ann. Mat. Pura Appl ) [] P.J. Torres, Existene and stability of periodi solutions of a Duffing equation by using a new maximum priniple, Mediterranean Journal of Mathematis, n.4 4),

9 5 5 linear sale linear sale Mass V rel Upper bound Lower bound Mass Sim V rel Sim 5 Foring amplitude β) in log sale a) Mass V rel Upper bound Lower bound Mass Sim V rel Sim 5 Foring amplitude β) in log sale b) 5 Mass V rel Upper bound Lower bound Mass Sim V rel Sim linear sale 5 Foring amplitude β) in log sale ) Figure : Derived bounds ompared to numerial simulations for varying foring amplitude β: The system parameters orresponds to the B-ring outer edge where the normal oeffiient of restitution ε =.6, the foring period of the moon and the stirring period are both equal to the orbital period of the ring partiles about Saturn T = f stir = ). a) The mean optial depth is τ =.5, b) the mean optial depth is τ = and ) the mean optial depth is τ =.5. 9

10 [] J.R. Ward, Periodi solutions for a lass of differential equations, Pro. Amer.Math.So ), A Proof for Theorem Deriving the first equation and inserting the seond one, we obtain the following seond order differential equation of Duffing type z + z + bd expz) = a + bft)..) Now the periodi problems for.) and.) are ompletely equivalent. If x, y) is a T -periodi solution of system.) with xt) > for all t, then z = ln x is a T -periodi solution of.). Conversely, if z is a T -periodi solution of eq..), then x = expz), y = a z )/b is a T -periodi solution of system.) with xt) > for all t. Sine a >, eq..) is of Landesman-Lazer type. Landesman-Lazer onditions are lassial and implies the existene of a periodi solution, see for instane [, Theorem ]. B Proof for Theorem We know that system.) is equivalent to eq..), hene along this proof we will work diretly with this last equation. Uniqueness. Assume that z, z are two T -periodi solutions of eq..). The differene δt) = z t) z t) is a solution of a seond order linear equation where ht) = bd expz) expz) z z. We onsider two ases: δ + δ + ht)δ =,.3) Case : δt) for every t: without loss of generality, we may take δt) >. By the Mean Value Theorem, ht) = bd expξt)), where z t) < ξt) < z t) for every t. Hene, by using 4.5), one obtains ˆ T ˆ T h = bd expξt))dt < bd expz t))dt = at. Using 3.5), one may hek easily that h Ω,, as defined in [, Setion 3]. Then, by [, Corolllary.5], the operator defined by the left-hand side of.3) is inversely positive, in partiular.3) is non-degenerate and its only periodi solution is the trivial one, whih is a ontradition with the assumption that δt) for every t. Case : δt ) = for some t [, T ]: by periodiity, there exists t > t suh that dt ) = and δt) for all t ]t, t [. Again, we may assume without losing generality that δt) > for all t ]t, t [. Define the trunated funtion ĥt) as ht) if t [t, t ] and if t [, t [ ]t, T ]. By reasoning as before, h Ω,. Then, as a onsequene of the results in [] see Remark. therein), the distane between two onseutive zeroes of a solution of δ + δ + ĥt)δ = is greater than T, but δ is a solution vanishing at t, t, whih is a ontradition. After this analysis, the only remaining possibility is dt) = for every t, so the T -periodi solution is unique. Stability. The proof for asymptoti stability is also based on the results from []. Let zt) be the unique T -periodi solution of eq..). A standard omputation gives that the topologial index of z is γz) =. The variational equation is u + u + ht)u = with ht) = bd expzt)) and again h Ω, as a result of 3.5). Then the asymptoti stability follows diretly from [, Theorem 3.].

11 C Proof for Theorem 3 We use the terminology and notation from [8]. A lower solution is found by fixing a onstant α suh that bd expα) = a + b f. 3.4) An upper solution is easily obtained as follows. Take F the unique T -periodi solution of z + z = ft) with mean value zero. Then, βt) = M + F t) with M big enough is an upper solution with β < α. To apply [8, Theorem.], we have to hek two hypotheses. The fist one is the finiteness of the number of T -periodi solutions, whih is diret from the analitiity of the field on the state variables. The seond hypothesis is ondition.6) in [8]. Fix gt, z) = bd expz) a bft). For s [βt), α], s gt, s) = bd exps) bd expα) = a + b f < 4 + π T by 3.6), whih is just.6) in [8]. The result is done. D Proof for Theorem 4 We work again over the equivalent eq..). Integrating.) over a whole period By the integral mean value theorem, there exists t [, T ] suh that ˆ T bd expz)dt = at. 4.5) expzt )) = a bd. 4.6) Observe that zt) zt ) H t, T + t ), then by the Sobolev inequality with the optimal onstant see for instane []), we get T zt) zt ) z. 4.7) On the other hand, multiplying.) by z and integrating over [, T ] and applying Cauhy-Shwarz inequality, we get ˆ T z = b ft)z dt b f z, hene, z b f. 4.8) Now, ombining 4.6),4.7) and 4.8), zt) = zt ) + zt) zt ) a ) T ln b + bd f and zt) = zt ) + zt) zt ) a ) T ln b bd f. The first inequality of 4.7) arises from here taking z = ln x and finding the value of x. The bound for y is found as follows. Let t suh that zt ) = min t [,T ] zt). Integrating.) on [t, t], ˆ t ˆ t z t) + zt) zt ) + bd expz)dt = a + bfs)ds at + b f +. t t

12 Sine zt) zt ) + bd t t expz)dt, then z t) ˆ t t a + bfs)ds at + b f ) Analogously, Let t suh that zt ) = max t [,T ] zt). Integrating.) on [t, t], Then, Sine zt) is T -periodi, t +T t ˆ t ˆ t z t) + zt) zt ) + bd expz)dt = a + bfs)ds b f. t t ˆ t z t) bd expz)dt b f ˆ t +T bd expz)dt b f. t t expz)dt = T expz)dt, then by using 4.5), z t) at b f. 4.) Finally, inserting 4.9) 4.) into y = b a z ), one obtains the seond inequality of 4.7). E Proof for Theorem 5 Note that expt ) Gt, s) expt ) expt ) for all t, s [, T ]. In the following steps, we derive upper and lower bounds for both the terms in the above expression, whih will be used later to obtain bounds for y t). Bounds for u t) = t+t G t, s) exp z s)) ds: Using the bounds for G t, s), at bd e T ) u t) at et bd e T ). 5.) The next set of bounds are derived by using the bounds derived for z t) in the proof of Theorem 4 and noting that t+t G t, s) ds = ) a bd exp b T f u t) abd exp b ) T f. 5.) Combining 5.) and 5.), we get )) at max bd e T ), a bd exp b T f at e T u t) min bd e T ), a bd exp b )) T f. 5.3) Bounds for v t) = t+t G t, s) f s) ds: Gt,s) Let p t, s) Gt,s)ds. We know, t+t G t, s) ds =. Hene, v t) = f min t+t t+t p t, s) f s) ds f max, we have the first set of bounds t+t p t, s) f s) ds. Sine f min v t) f max. 5.4)

13 Similarly, using the Cauhy-Shwartz inequality, we have t+t t+t p t, s) f s) ds f p t, s) ds and hene e v t) f T + e T ). 5.5) Another set of bounds based on the following inequality are ˆ t+t ˆ t+t G t, s) f s) ds v t) G t, s) f + s) ds f f e T e T ) v t) f + 5.6) v t) f + e T e T ). 5.7) And finally, if the v t) is omputable for all t, then, v min v t) v max. 5.8) Note that 5.4) and 5.6) are equivalent. Hene, ombining 5.5) - 5.8), we get v t) min v min, f, f e T ) e T ) ) e v t) min v max, T + f e T ), f +, f + e T e T. 5.9) ) Using 5.3) and 5.9), 4.8) and 4.9) follow diretly. 3