Complex instability is a generic kind of instability in 
 Hamiltonian systems with three degrees of freedom. In this work,
 some examples of such instability are shown, together with a
 numerical analysis of the dynamics close to the transition from
 stability to comlex instability for a family of periodic orbits

Ollé Torner, Mercè

Pacha Andújar, Juan Ramón

English

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MOTION NEAR THE TRANSITION TOCOMPLEX INSTABILITY SOME EXAMPLESMERCE OLLE JUAN R PACHA Departament de Matematica Aplicada I Universitat Politecnicade Catalunya Diagonal  		 Barcelona EspanaAbstractComplex instability is a generic kind of instability in Hamiltonian systems withthree or more degrees of freedom In this work some examples of such instabilityare shown together with a numerical analysis of the dynamics close to the transitionfrom stability to complex instability for a family of periodic orbits Parametric stability and eigenvalues collision onthe unit circleLet T  IR IR be a symplectic map with T    Then its jacobian matrix atthe origin A  DT  will be a symplectic matrix It is well known see 	 that if   spec A then also 	 	  spec A so the condition for the origin to be anstable xed point is that each eigenvalue lies on the unit circle Eventually the mapT could depend on one parameter  in such a way that for    the two reciprocalpairs of eigenvalues are on the circle for    a pairwise collision of eigenvalues takesplace and at last for positive values of the parameter the eigenvalues leave the unitcircle by reciprocal pairs See gure 	 for an illustration of the global process At thetransition from stability to complex instability the eigenvalues are equal by pairs so bytheir reciprocity also    eik		 eikIn order to study the linear stability of Dsymplectic mappings some quantitiesrelated with the eigenvalues of the matrix A are introduced see   Stability Parameters Dened by b   	 and b   	 Stability Coecients   b b 	    bb Discriminant    b bλµ1/µ1/λλ = µµλ1/λ = 1/µσ = 01/µ1/λσ < 0 σ > 0Figure 	 Transition stable complexunstable-6-4-20246810-6 -4 -2 0 2 4 6P1 P2P3P4P5P6C-USALPHABETAFigure  Transition StableCU on the Broucke DiagramThe xed point is stable if jbj   jbj   or complexunstable if    andb b C while the transition takes place when    or in terms of the stabilitycoecients when   cosk 	     cosk 	and this happens along the parabolic arc from Pto Pon the Broucke diagram gure See Linked to this change of stability there is a characteristic type of bifurcation the socalled Hopftlike bifurcation for which the bifurcating objects are dierent dependingupon the rotation number k at the transition 	 More precisely if k  mn  Qork  IR nQ n periodic orbits or invariant curves surrounding the xed point respectivelymay bifurcate Furthermore if the bifurcating objects unfold on the unstable side thenthey are stable direct bifurcation Otherwise when the bifurcation unfolds towards thestable side the objects are unstable inverse bifurcationThe eect of the direct bifurcation is to conne the phase space so the consequentsTnx of a point xclose to the xed point  remain in a domain surrounding the xedpoint for large values of n The inverse bifurcation occurs on the stable side so theconsequents of the unstable side are not conned The eect of the inverse bifurcation ismore subtle it compresses the stable curves around the xed point so their number isreducedIn fact the phenomenon described in this last paragraph gives us a key to nd outif the system undergoes direct or inverse bifurcation we shall test if the consequents ofpoints on the unstable side are conned If so this indicates that direct bifurcation istaking place Otherwise we may expect at most inverse bifurcationIn the next section we shall study a couple of mappings which exhibit all the featuresrelated with the transition from stability to complex instability In the last two sectionsthese ideas are applied to Hamiltonian systems Froeschles generalized mappingsIn order to describe the generic behaviour linked to the transition let us consider the DTsand Ttmappings see  around x   xed point They are dened byTsxxxxD xKsin x x  Lsin x x x xx xD xKsin x x  Lsin x x x xx xmod andTtxxxxD xKsin x x  Ltan x x x xx xD xKsin x x  Ltan x x x xx xmod Let A  DT   s t thenA BB	  LKLKLL	 	  LL	  LKLK  	 		CCAfor D  	 Then it is easy to see that   	    KK L L LLThe Froeschles mapping corresponds to  with L L but then the discriminant   so no complex instability is possible at all Instead we restrict the parameter spaceby taking L L L K K and K  Now   KKL If   K  we have that for L  K the origin  is an stable xed point and for L  K theorigin is complexunstable The transition takes place at L  LCrit K   Now for each xed K in the interval   K   we may vary L continuouslyfrom L  LCritto some value L  LCritand explore what happens in the neighbourhoodof LCrit D is a dissipative parameter introduced for computational purposes see theexplanation below-0.6-0.4-0.200.20.40.6-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8X1X2-0.6-0.4-0.200.20.40.6-0.6 -0.4 -0.2 0 0.2 0.4 0.6X1X3-0.6-0.4-0.200.20.40.6-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8X1X4Figure  Connement of the stochastic points around the complexunstable xed pointin the mapping Tsat K  	 L  Numerical simulations show that for the Tsmapping the bifurcation is of direct typewhile for the Ttmapping the bifurcation is inverse In  an extensive study of thebifurcation to periodic orbits and invariant curves has been made both for Tsand for TtThe study of the evolution and further bifurcations of such invariant curves was carriedout in  In gure  we reproduce an invariant curve which bifurcate on the unstableside for K  	 and L    LCrit as well as the connement of the stochastic pointsaround the complex unstable xed pointThe invariant curves were found by the dissipative method these become attractinglimit cycles as the parameter D is D  	 so one starts with a value of D less than oneand calculates the consequents of a point The consequents converge rapidly onto aninvariant curve in fact onto a limit cycle while D  	 Hence increasing smoothly thedissipative parameter until D  	 the desired invariant curve is nally reached The restricted three body problemNow we consider the vertical family of periodic orbits of the restricted threebody problemaround the triangular Lagrange point L From the study of the stability of this familysee gure  it is clear that for an interval of the mass parameter Routh   	the vertical family reaches complex instabilityFor the analysis of the stability we reduce the three degrees of freedom Hamiltoniansystem to a D symplectic mapping by the use of the Poincare map T  with fz  g asa surface of section and by xing the energy level of each orbit In this way the studyof the stability of a two parameter  pz family of periodic orbits is reduced to thecorresponding study for a family depending upon the same parameters of xed pointsof the map T In order to nd out what type of bifurcation takes place from the transition weconsider an orbit close to a complexunstable periodic orbit of the family The consequentsby T are conned for a large number of iterates Therefore the bifurcation is direct andthe existence of the invariant curves is expected We computed them by means of thedissipative method and we show one of them in gure 00.20.40.60.810 0.1 0.2 0.3 0.4 0.5CuSuDuSFigure  Change of the linear stability of the vertical family of L the values of z  pzwhen z   are plotted against the mass parameter  The Su and Du in the gurestand for simpleunstable and doubleunstable-0.02-0.015-0.01-0.00500.0050.010.0150.02-0.015 -0.01 -0.005 0 0.005 0.01 0.015X1X2-0.02-0.015-0.01-0.00500.0050.010.0150.02-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01X1X3-0.02-0.015-0.01-0.00500.0050.010.0150.02-0.015 -0.01 -0.005 0 0.005 0.01 0.015X1X4Figure  Connement and invariant curves of the direct bifurcation in the RTBP Barred galactic model potentialWe consider the MiyamotoFerrers galactic model for a barred galaxy given by the HamiltonianH 	px py pz x y z  xpy ypx    D Bwhere Dand Bstands for the contribution of the disk and the bar to the global galacticpotential As in the RTBP this system has ve equilibrium points three collinear andtwo triangular noted also by L L L and L L The stability of the vertical familyof periodic orbits around Lhas been studied in  In fact this study is parallel to theone done for the RTBP Nevertheless for the galactic potential there appear a transitionstable complexunstable with a bifurcation of inverse typeTo see the eect of this bifurcation on the invariant tori surrounding the stable centralfamily an antidissipative parameter D slightly larger than 	 is introduced in such a waythat the system becomes volume dilating but conserving the Hamiltonian If we takeinitial conditions on the stable side we must note that in the original Hamiltonian casethe orbit lies on a torus but due to the presence of the antidissipative perturbation Dthe consequents of an initial point on the surface of section z   if we think in termsof the Poincare map explore larger and larger tori until they reach the last one anda sudden escape takes place So this provides us with an upper bound of the extentof the invariant tori these bounds are easily seen looking at the shell formed by theaccumulated points just before the escape see gure 5.575085.57515.575125.575145.575165.575185.5752-0.0001 -5e-05 0 5e-05 1e-04yx5.525.545.565.585.65.62-0.1 -0.05 0 0.05 0.1yxFigure  Left x y projection of the consequents of an orbit in the Poincare sectionz   pz  Right the envelope formed by the consequents of the antidissipativeperturbed system just before the escape The same initial conditions are taken in bothcasesReferences	 V I Arnold Methodes Mathematiques de la Mecanique ClassiqueEditions MirMoscou 	  R Broucke Stability of periodic orbits in the elliptic restricted three body problem AIAA Journal  	 	~~	~~ M Olle and D Pfenniger Bifurcation at complex instability Proceedings NATOASI Adv Sci Inst Ser C Math Phys Sci en SAgaro Espa!na Junio 	 KluwerAcad Publ Dordrecht Holland M Olle and D Pfenniger The vertical orbital structure around the Lagrangianpoints in barred galaxies to appear in Astron Astrophys D Pfenniger Numerical study of complex instability I Mappings Astron Astrophys  	  			

Motion near the transition to complex instability: some examples

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