Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation
IIntegrals of motion in time periodicHamiltonian systems: The case of the Mathieuequation
A.C. Tzemos and G. Contopoulos Research Center for Astronomy and Applied Mathematicsof the Academy of AthensSoranou Efesiou 4, GR-11527 Athens, Greece
Abstract
We present an algorithm for constructing analytically approximateintegrals of motion in simple time periodic Hamiltonians of the form H = H + εH i , where ε is a perturbation parameter. We apply ouralgorithm in a Hamiltonian system whose dynamics is governed by theMathieu equation and examine in detail the orbits and their strobo-scopic invariant curves for different values of ε . We find the values of ε crit beyond which the orbits escape to infinity and construct integralswhich are expressed as series in the perturbation ε and converge up to ε crit . In the absence of resonances the invariant curves are concentricellipses which are approximated very well by our integrals. Finally weconstruct an integral of motion which describes the hyperbolic stro-boscopic invariant curve of a resonant case. The detection of integrals of motion is an important task in the study ofdynamical systems. The existence of integrals of motion is related to cer-tain symmetries of the equations of motion of the system (through Noether’stheorem) and gives a deep insight into their time evolution. Furthermoreit simplifies significantly the calculations, since it decreases the independentvariables of the system and can be used for the error control of the calcula-tions.The integrals of motion appear in two main categories: a) the exactintegrals of motion and b) the approximate integrals of motion. In the case [email protected] [email protected] a r X i v : . [ m a t h - ph ] J a n ) these integrals are analytical mathematical expressions of the variables ofthe system, that do not depend on time. In the case b) the approximateintegrals (also called formal integrals) are non-convergent series expansions,truncated at a certain order that remain approximately constant in time.Formal integrals of motion have been proved to be very useful in Dynam-ical Astronomy and especially Galactic Dynamics. In particular, their con-struction in the case of autonomous dynamical systems has already been con-sidered in the past, even to high orders using computer algebra ([1, 2, 3, 4]).As regards time-periodic Hamiltonians, it was presented back in 1966 forthe first time a general method for the calculation of formal integrals [5]. Thismethod was applied to some simple Hamiltonians of one and two degrees offreedom. However, due to the lack of computer algebra systems at that time,the calculations were made by hand and were limited to second order withrespect o the perturbation parameter ε .In recent years there has been much interest in periodic in time Hamilto-nians, especially in Russia. Most of this work dealt with stability problemsin one or more degrees of freedom (e.g. Markeev [6, 7, 8, 9, 10], Kholostova[11, 12, 13], Bardin and Lanchares [14]). More recently Bruno ([15, 16])calculated normal forms in particular problems.On the other hand some authors (Kandrup [17, 18] and Terzi´c and Kan-drup [19]) have calculated orbit in time-periodic potentials, with emphasison the generation of chaos.Our interest in this problem was revived because of our need to com-pare the classical with the quantum mechanical results (Efthymiopoulos andContopoulos [20]).In the present paper we exploit the power of the Maple Computer AlgebraSystem in order to construct high order integrals of motion in simple timeperiodic Hamiltonian systems. As a first step towards this direction we applyour algorithm on a simple Hamiltonian system whose equations of motioncan be written in a single second order differential equation of the form: d xdt (cid:48) + (cid:104) a − q cos(2 t (cid:48) ) (cid:105) x = 0 . (1)Although this differential equation is linear in x and there is no chaos inits orbits, it is very useful in Applied Mathematics. It is the well knownMathieu equation (ME) [21, 22] and has applications in various fields ofphysical sciences, such as Acoustics (e.g. in the study of an elliptical drum[23]), Quantum Mechanics (e.g. in the study of the quantum pendulum [24]),2eneral Relativity (e.g. in the study of the solutions of wave equations incurved spaces [25]) and Quantum Chemistry (charged particle in a quadruplefield [26]). Many applications of MEs can be found in [27].As already known from the theory, different parameters in ME can leadto bounded or unbounded motion. We make a detailed study of the orbitsand their invariant curves on a stroboscopic surface of section by solvingnumerically in Python 3.7 the Hamilton equations. In the case of boundedorbits we construct in Maple 2016 an integral of motion which is convergentfor small perturbations and describes very well the invariant curves. Thisintegral becomes divergent and the forms of the orbits change abruptly atthe threshold of the escapes.In section 2 of the present paper we give the Hamiltonian of our modeland describe our algorithm for the calculation of formal integrals of motionup to an arbitrary order in the absence of resonances. Then in Section3 we present our results in the case where the perturbation parameter ε is positive, by calculating both the orbits and their stroboscopic sections(i.e. the distribution of their points after successive periods). We apply ouralgorithm and calculate the integrals at successive orders of the perturbationparameter and show their convergence to the form of the invariant curveson the stroboscopic sections. Then in Section 4 we examine the case of thenegative values of the perturbation parameter and in Section 5 we study aformal integral in a resonant case. Finally in Section 6 we summarize ourresults and draw our conclusion. In the special case of the Mathieu equation we have H = H + εH i = 12 ( y + ω i x ) − εx cos( ωt ) (2)The corresponding equations of motion are: dxdt = y, dydt = d xdt = − (cid:104) ω − ε cos( ωt ) (cid:105) x (3)The second equation takes the form of the Mathieu equation if we set ωt =2 t (cid:48) , a = 4 ω /ω and q = εω . In particular if ω = 2 we have t = t (cid:48) , a = ω and3 = ε . There are resonance conditions when a = 1 , , , . . . . For ω = 2resonances appear if ω = 1 , , , . . . An integral of motionΦ = Φ + ε Φ + ε Φ + · · · + ε s Φ S + . . . (4)must satisfy the equation d Φ dt = ∂ Φ ∂t + [Φ , H ] = 0 , (5)where [Φ , H ] is the Poisson bracket[Φ , H ] ≡ ∂ Φ ∂x ∂H∂y − ∂ Φ ∂y ∂H∂x . (6)We apply Eq.(3) to the terms of successive orders in ε and find ∂ Φ s +1 ∂t + [Φ s +1 , H ] − K s = 0 , (7)where K s = − [Φ s , H ] (8)If we set Φ = H we can calculate successively the terms of various or-ders of Φ expressed in trigonometric terms of multiplicities of ωt . From thecharacteristic curves of Eq. (7) we find dt = dxy = dy − ω x = d Φ s +1 K s (9)The zero order solution is x = √ ω sin( ω t ) , y = (cid:112) cos( ω t ) (10)Consequently from Eqs. (9) we getΦ s +1 = (cid:90) t K s dt (11)where K S is expressed in trigonometric terms of multiples of ωt and ω t . Afterthe integration we use again Eqs. (10) to express back the trigonometric terms4f Φ s +1 which contain ω t in terms of y , x , xy multiplied by trigonometricterms of multiples of ωt . Consequently the integral Φ, which is in practicetruncated at some order, is Φ = Φ( x, y, ωt ).In particular Φ = H andΦ = (cid:90) t ∂ Φ ∂y ∂H ∂x dt = − (cid:90) t xy cos( ωt ) dt (12)The integral Φ is a series in ε and of second order in x, y . Up to second orderin ε it is: Φ = 12 (cid:16) C x ω x + C y y + C xy xy (cid:17) , (13)where C x = 1 + 4 ε (cos ( ωt ) + 1) ω − ω + ε (cid:16)
16 ( ω − ω ) cos ( ωt ) + (cid:16) ω + 7 ω − ω ω (cid:17) cos (2 ωt ) + 12 ω − ω + ω ω (cid:17) ( ω − ω ) ( ω − ω )+ . . . (14) C y = 1 − ε (cos ( ωt ) − ω − ω + ε (cid:16) −
16 ( ω − ω ) cos ( ωt ) + 3 ( ω − ω ) cos (2 ωt ) + 13 ω − ω (cid:17) ( ω − ω ) ( ω − ω ) + . . . (15)and C xy = − εω sin ( ωt ) ω − ω + ε ω (cid:16) −
16 ( ω − ω ) sin ( ωt ) + 6 ( ω − ω ) sin (2 ωt ) (cid:17) ( ω − ω ) ( ω − ω ) + . . . (16)We notice that the first order terms in ε contain cos( ωt ) , sin( ωt ) and aconstant, the second order terms contain also cos(2 ωt ) and sin(2 ωt ) and soon. The constant terms are due to the fact that the integral starts at t = 0.5igure 1: The invariant curves (ellipses) of the stroboscopic surface of sectionin the case ω = 2 , ω = 0 . , ε = 0 . ε m contain cosines of multiples of ωt up toorder mωt together with constant terms in the coefficients of x and y , whilethey contain only sines up to order mωt in the coefficient of xy . The denom-inators contain factors of the form ( ω − ω ) , ( ω − ω ) , ( ω − ω ) , . . . ( ω − m ω ).Using our program for the construction of this integral we have calculatedits terms up to order 28. For ε not large the integral Φ represents an ellipseof the form Φ = Ax + By + 2 Dxy (17)If t = 2 kπ/ω = kT we have A = ω i (cid:104) εω − ω + 2 ε ω ( ω + 8 ω ) + . . .ω ( ω − ω ) ( ω − ω ) (cid:105) , B = 12 , D = 0 (18)This represents an ellipse passing through the initial point ( x = 0 , y =1) with semiaxes a = (cid:113) Φ A , b = (cid:113) Φ B . The points ( x, y ) of an orbit on a6troboscopic Poincar´e surface of section t = kT lie on this ellipse, if theseries giving A converges.We have calculated several orbits and verified that if ε is small and ω isnot close to a resonance ω = ω i , ω i , ω i ... the points of the orbits on thestroboscopic surface of sections lie on such an ellipse. Figure 1 represents aset of such ellipses for ε = 0 . , ω = 2 and ω = 0 .
9. In fact for various initialconditions we have similar concentric ellipses. ε > If we fix ω = 2 and ω = 0 .
9, and initial condition x = 0 , y = 1 we haveΦ = 1 /
2. Then the semiaxes of the ellipses are a = √ A , b = 1. In Fig. 2 wepresent the stroboscopic surface of section for these parameters for variousvalues of ε . We observe that they are all ellipses with one axis from y = 1 to y = −
1. The same holds for other non-resonant values of ω and ω .Figure 2: Invariant curves on the stroboscopic surface of section in the casewith ω = 2 , ω = 0 . ε and the same initial conditions( x = 0 , y = 1). In particular we have given the successive points 1 , , . . . , ε = 0 . ω = 2 , ω = 0 . x = 0 , y = 1) on the stroboscopic surface of section in the cases(a) ε = 0 . ε = 0 .
18. The successive points are joined by straightlines. The points cover roughly for the first time the corresponding invariantcurve. Their numbers are 13 in case (a) and 39 in case (b). The set ofpoints is approached by the theoretical invariant curves truncated at varioussuccessive orders as shown by colors. 8n Figs. 3a,b we mark the successive points after times T, T, T . . . etc.for ε = 0 . ε = 0 .
18. At the same time we draw the ellipses found if wetruncate the integral Φ after the terms of order 2 , ,
6. . . in ε . We see thatthe ellipse of order 2 is far from the points of the numerical solution. Butas we increase the order of the truncation the ellipses approach graduallythe invariant curve formed by the successive points. In the case ε = 0 . ε = 0 .
15 good convergence isreached at order 20, while in the case ε = 0 .
18 we have not yet reached goodconvergence up to order 28. However these figures indicate that the integralΦ in fact converges all the way up to ε = 0 . ε above which we have escapes to infinity.Moreover we see that the angles between the lines joining the successivepoints decrease as ε increases. In particular for ε = 0 the successive points areon a circle on a plane ( ω x, y ) and the successive angles are ∆ φ = πω ω = 0 . π .Thus the number of points required to cover roughly the invariant curve isabout 11 (Fig. 3a). In the case ε = 0 . ε = 0 .
15 itis 17, for ε = 0 .
18 it is about 40 and for ε = 0 .
185 it is almost 110 points. Thefollowing iterations give additional points that fill more densely the ellipseprovided by the integral Φ.These numbers are found also by calculating the distances d = (cid:112) ω x + y of the successive points on the stroboscopic surface of section as functions oftime (Fig. 4). All the orbits start at ( x = 0 , y = 1) with d = 1 and decreasedown to a minimum distance. Then they increase up to the distance d = 1and continue to decrease and increase. We see that for ε = 0 .
01 one comesback to the maximum distance after 11 points, for ε = 0 . ε approaches the critical value ε crit = 0 . d ≡ (cid:112) ω x + y between the origin (0 ,
0) and thesuccessive points of the intersections of the orbits by the stroboscopic surfaceof section, for various values of ε up to t = 200. We have ω = 2 , ω = 0 . , x =0 , y = 1.The motion with initial conditions ( x = 0 , y = 1) for ε < ε crit startsby spiralling inwards until it reaches a minimum distance and then it spiralsoutwards and so on. Thus the orbit fills a ring as in Fig.5a for ε = 0 . ε increases the ring becomesbroader and the empty hole near the center becomes smaller as in Fig. 5bfor ε = 0 . ε → ε crit the ellipse tends to the straight line from y = 1 to y = − ω = 2 , ω = 0 . ε = 0 . x = 0 , y = 1).(b) For ε = 0 .
185 and 100 periods. This orbit fills a ring leaving only a smallhole around the center (0 , ε the orbit comes exactly to its initial point x =0 , y = 1 after a number n of periods i.e. we have a periodic orbit of period nT . Such is the case of Fig. 6 for ε = 0 .
15, where we have a periodic orbitof period 17 T . In this case the energy E increases to a maximum, but thenit returns to its initial value. Similar periodic orbits of other periods appearfor other values of ε .If ε goes beyond a critical value about ε crit = 0 . ε . E.g. for ε = 0 .
19 (Fig.7) the orbit starts close to an initial ellipse but then makes aspiral outwards that extends to about x = ± , y = ±
15 after 50 periods. InFig. 8 we see that the logarithms of the distances for particular values of ε beyond the critical value increase linearly in time. Therefore the distances ofthe escaping orbits increase exponentially in time and the increase is largerfor larger ε . We know from the theory that this motion extends to infinity.11igure 6: A periodic orbit of period 17 T where T = 2 π/ω , together with the17 points of intersection with the stroboscopic surface of section.Figure 7: An escaping orbit for ( ω = 2 , ω = 0 . , ε = 0 .
19) starting at thepoint ( x = 0 , y = 1) for the first 40 periods, together with its intersectionswith the stroboscopic surface of section.12igure 8: The logarithms of the distances r = (cid:112) x + y of the successiveintersections of escaping orbits by the stroboscopic surface of section for ω = 2 , ω = 0 . , x = 0 , y = 1 and various values of ε up to t = 30 T . In allthese cases the logarithms of the distances increase linearly in time thus thedistances icnrease exponentially in time. The distances are longer for larger ε . An interesting aspect of our problem is found if we work in the so calledextended phase space where our system becomes conservative. Namely weextend the phase space to include the ordinary time as a canonical variableand its conjugate momentum E which is minus the energy. Then the newHamiltonian ¯ H ( x, y, E, t ) = H ( x, y, t ) + E (19)is conservative with respect to a fictitious time variable τ = t and the Hamil-ton equations read: dxdt = ∂H∂y , dydt = − ∂H∂xdτdt = ∂ ¯ H∂E = 1 , dEdt = − ∂ ¯ H∂t = − ∂H∂t (20)Then we can calculate the value of the energy E as a function of timeusing the last Eq. (20) and we find that the value of ¯ H is very close to zero13in fact it is of order O (10 − ) which is the accuracy of our calculations). Inthe cases 0 < ε < ε crit for x = 0 , y = 1 the value of E starts at E = − . t = 0, reaches a maximum near E = 0 and then oscillates between thismaximum and a minimum. The values of x and E of the stroboscopic sectionare given by blue dots (Fig. 9a). These dots mark the points of the invariantcurve on the stroboscopic section. The first 40 points are successively onthe right and on the left part of the hyperbolic-like invariant curve which isdirected upwards. Further points fill this curve densely. On the other handfor ε > ε crit the value of E decreases continuously on the average (Fig. 9b)and tends to −∞ . The dots are again on a hyperbola-like curve, but thistime the curve is directed downwards and has no limit. This difference allowsus to find with great accuracy the transition value ε crit for the escapes. At ε = ε crit (cid:39) . x = x = 0 , y = y = 1) on the stroboscopic surface of section. Wehave calculated the monodromy matrix of this periodic orbit and found thatits eigenvalues are all equal to one. Therefore the orbit is marginally unstable.Figure 9: Stroboscopic surfaces of section (blue points). The trajectory of E as a function of x for (a) ε = 0 .
18 up to 41 periods and (b) for ε = 0 .
19 upto 40 periods. We observe that in the case (a) the trajectory is confined andthe blue points oscillate in a certain range of energies, while in the case (b)the trajectory escapes to −∞ and the blue points also tend to −∞ . In theextended space, the total energy ¯ H = H + E is conserved and equal to 0 forour parameters.In the extended phase space we can calculate another integralΨ = Ψ + ε Ψ + ε Ψ + . . . (21)14f we set Ψ = E . Then we findΨ = [(2 ω − ω ) x + 2 y ] cos ( ω t ) + 2 sin ( ωt ) ωxy − ω x + y ) ω − ω (22)But then we notice thatΦ + Ψ = − x cos( ωt ) = H , (23)therefore Ψ = H − Φ (24)If we calculate now Ψ this is equal to − Φ because [ H , H ] = 0. In thesame way we find Ψ = − Φ and so on. ThusΦ + Ψ = 12 ( ω x + y ) + E − εx cos( ωt ) = ¯ H = 0 . (25)Consequently we have only two independent isolating integrals ¯ H and Φ, thusthe transformation of the system to an autonomous Hamiltonian system doesnot give any new results.Figure 10: Invariant curves on a stroboscopic surface of section in the case ω = 2 , ω = 0 . x = 0 , y = 1 and various negative values of ε . As ε decreases and approaches the critical value ε crit (cid:39) − . x increases and tends to ∞ . In the case ε = − .
18 we give alsothe successive points corresponding to successive periods.15igure 11: Orbits in the case ω = 2 , ω = 0 . ε = − . x = 0 , y = 1) (b) For ε = − .
185 and 110 periods. We observe that theorbit extends to much larger distances than in the case (a). ε < If ε < ε = 0 (Fig. 10). The criticalvalue of ε is ε crit = − . ε crit = 0 . ε . The stroboscopic ellipses become longer along the x-axis as ε increases and they tend to infinity as ε tends to the critical value.The orbits for ε < ε >
0. The orbitsare outside the limiting ellipse ω x + y = 1 for ε = 0. The number ofpoints on the stroboscopic ellipse required to cover once the whole ellipseagain increases as | ε | increases. For example for ε = − . ε = − .
185 it is n = 110 (Figs 11a,b). The orbits for ε < x and y but with much smaller ellipticitythan the corresponding stroboscopic ellipse. Finally for | ε | larger than thecritical value | ε crit | = 0 . ε = − . ω we find again ellipses for various valuesof ε given by the integral of motion. However the critical values of ε are nowdifferent. We consider here two examples.16igure 12: An escaping orbit for ω = 2 , ω = 0 . , x = 0 , y = 1 , ε = − . ω = 0 . ω x + y = 1 is just outside the ellipse for ε = 0 . ω = 0 .
9. Furthermore the successive points on the stroboscopic surfaceof section are much closer to each other. The total number of points tocover roughly the ellipse once for ε = 0 . N (cid:39)
16. As ε increases thisnumber increases. In this case the critical value of ε is ε crit = 0 . ω = 0 . ω = 1 .
1, larger than the resonant value ω = 1.In this case the critical value of ε is about ε crit = 0 . ε < ε crit butthese become larger as ε increases (Fig. 14a). The difference betweenthe behaviour of the case ω = 1 . ω = 0 . ω = 0 . ε = 0 the value of thecoefficient A of Eq. (18) is approximately A (cid:39) ω (cid:16) εω − ω (cid:17) (26)17hile Φ = 1 / x = 0 , y = 1) hence a (cid:39) ω (cid:16) − εω − ω (cid:17) (27)Figure 13: Invariant curves (ellipses) on the stroboscopic surface of sectionon the case ω = 2 , ω = 0 . , x = 0 , y = 1, for various values of ε >
0. As ε increases the invariant curves shrink and for ε = 0 . ε = 0 . ε > ω − ω ) > ε/ ( ω − ω ) > a < ω , i.e. the ellipses are inside the ellipse for ε = 0, while for ε < ω − ω ) > a > ω , i.e. the ellipses are outsidethe ellipse for ε = 0. On the other hand for ( ω − ω ) < ε > ε < ω = 1 . ε > ω = 0 . ε . On the other hand for ε < | ε | and for ε approaching the critical value ε = − . ω = 2 , ω = 1 . , x = 0 , y = 1 for various values of ε (a) for ε > ε increases and tend to infinity, as ε tends to thecritical value ε crit = 0 . ε < ε decreases and tend to a straight line from y = − y = 1 as ε tends tothe critical value ε crit = − . ε for the same ω and ω are the samefor positive and negative ε but the limiting ellipse in one case (positive ε for ω <
1, negative ε for ω >
1) tend to the x-axis, while in theopposite cases the ellipse tends to infinity along the x-axis.
The theory of integrals in resonant cases, independent of time, has beendeveloped in detail [28, 29]. However the applicability of the direct methodof Contopoulos to resonant cases periodic in time has not yet been fullyexplored [29].In the present case we will construct a formal integral in a particularresonant case of the Mathieu equation, namely in the case where the periodicterm has a frequency ω which is double of the basic frequency ω of theunperturbed problem.When ω = ω/ x = √ ω sin( ω ( t − t )) , y = √ cos( ω ( t − t ))and find Φ = 18 (cid:34) ( y − x ) cos(2 t ) − xy sin(2 t ) − ( y + x ) cos(2 t ) (cid:35) + 12 ( y + x ) sin(2 t ) t (28)We observe that Φ contains the secular termΦ sec = 12 ( y + x ) sin(2 t ) t (29)However in this resonant case we can construct two more zero order integrals: C = 2Φ cos( ωt − ω ( t − t )) ω =2 ω ===== 2Φ cos(2 ω t ) S = 2Φ sin( ωt − ω ( t − t )) ω =2 ω ===== 2Φ sin(2 ω t ) (30)We start by writing C in the form C = ( y − ω x ) cos( ωt ) + 2 ω xy sin( ωt ) (31)20hich is periodic in time with period T = πω . Then we calculate higher orderterms in C = C + εC + ε C + . . . In particular C is C = (cid:90) t ∂C ∂y ∂H ∂x dt (32)and after some algebra we find C = 12 ω ( ω − ω ) (cid:104) y (1 − cos(2 ωt )) + x (cid:16) cos(2 ωt )(2 ωω − ω ) + ω − ω + 2 ωω (cid:17) − ωxy sin( ωt ) (cid:105) (33)which does not have secular terms. Setting ω = 2 , ω = 1 we have C = 14 (cid:104) (cid:0) y − x (cid:1) (cid:0) − cos (4 t ) (cid:1) − xy sin (4 t ) (cid:105) (34)and for t = kπ we get C = 0. If we calculate C we find C = (cid:90) t ∂C ∂y ∂H ∂x dt (35)and for ω = 2 , ω = 1 we find C = 164 (cid:34) y [ −
10 cos(2 t ) + 53 cos(6 t ) −
13 cos(2 t ) + 643 ]+ x [ −
22 cos(2 t ) −
373 cos(6 t ) −
13 cos(2 t ) + 643 ] − xy [6 sin(2 t ) − t )] (cid:35) −
18 ( y + x ) sin(2 t ) t (36)Therefore C contains the same secular term as Φ (up to a multiplicativeconstant) and we can combine the series of Φ and C in order to avoid theseterms. In particular if we multiply Φ by εq we find an integral¯ C = C + ε ( q Φ + C ) + ε ( q Φ + C ) + . . . (37)that does not have any secular term up to the order ε if we take q = .In a similar way we eliminate the secular terms of order ε by adding anappropriate term ε q (Φ + ε Φ + . . . ) and so on.21he integral (37) describes a stroboscopic surface of section which forsmall ε represents a hyperbola. Namely for t = 0 or t = kπ we find¯ C str = ( y − x ) + ε x + y ) + . . . (38)In Fig. 15a we show the orbits corresponding to ε = 0 .
05 and ε = 0 .
15 andobserve that the larger the ε the faster the escape to infinity. In Fig. 15b wedraw their corresponding stroboscopic sections for times up to t = 15 T andsee that they form two hyperbolae. Then we superimpose the hyperbolaepredicted by the integral (38). We observe the very good agreement betweenthe numerical results and the approximate integral of motion. In a similarway one may construct integrals of motion for every combination of ω, ω that leads to resonances.Figure 15: a) Two orbits in the resonant case ω = 2 , ω = 1 for ε = 0 .
05 (bluecurve) and ε = 0 .
15 (red curve). We observe the faster escape to infinity inthe case of ε = 0 .
15 b) The successive points (blue dots for ε = 0 .
05 andred stars for ε = 0 .
15) of two orbits on the stroboscopic surface of sectionlie, with a very good accuracy, on the correponding blue and red invariantcurves given by Eq. (38). 22
Conclusions
We studied the orbits and the stroboscopic invariant curves in the Hamil-tonian H = H + εH , where H = ( ω x + y ) with y = dx/dt and H = − x cos( ωt ). This Hamiltonian is equivalent to a Mathieu equation.We presented a method for constructing formal integrals of motion which arewritten as series expansions in ε and converge if | ε | is smaller than a criticalvalue ε crit . We studied in detail the forms of the orbits and of the integralsfor various values of ε, ω and ω . Our main results are the following:1. The integral is quadratic in x and y of the formΦ = C x ω x + C y y + C xy xy, (39)where C x , C y , C xy are series in ε . In non-resonant cases the integralforms invariant curves that are similar concentric ellipses on a stro-boscopic Poincar´e surface of section. The points of orbits with t = kT ( k = 1 , , . . . , T = 2 π/ω ) lie on such ellipses when the series con-verge.2. We found the forms of the orbits that fill, in general, elliptical rings,leaving an empty region near the center x = y = 0. We found also thedistribution of the points on the stroboscopic surface of section.3. For particular values of ε the orbits are periodic.4. For values of ω and ω below the resonance ω − ω = 0 the stroboscopicellipses become thinner along x for ε >
0, as ε increases. As ε → ε crit the ellipse tends to a straight line along the y-axis. For ε > ε crit theorbits spiral outwards and tend to infinity. On the other hand for ε < ε decreases and as ε → − ε crit they tend to a straight line along the x-axis.5. For any given set of physical parameters ω, ω and initial conditions x , y , there are two escape values of ε , one positive and one negative,with the same absolute value | ε crit | . We checked that these values arethe same as given for the escape values of q by other authors [21]. Onthe other hand the orbits and the invariant curves are quite differentfor symmetric values of ε . 23. The present system can be considered as a 2-d Hamiltonian systemwith two variables x and t and corresponding momenta y and E where − E is the energy. If | ε | < ε crit the values of E oscillate between amaximum and a minimum, while if | ε | > ε crit the values of | E | extendto infinity.7. If ω − ω > ε > ω − ω <
0. Namely they are more elongated along x as ε increases for ε > y as ε decreases for ε < Acknowledgements
This research was conducted in the framework of the program of the RCAAMof the Academy of Athens “Study of the dynamical evolution of the entan-glement and coherence in quantum systems”. The authors want to thankDr. Christos Efthymiopoulos for his useful comments.