The Behaviur f the Flquet Multipliers f Peridic Slutins Near a Hmclinic Orbit Zuchang Zheng Dirk Rse Reprt TW 287, December 998 Department f Cmputer S

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1 The Behaviur f the Flquet Multipliers f Peridic Slutins Near a Hmclinic Orbit Zuchang Zheng Dirk Rse Reprt TW 287, December 998 n Kathlieke Universiteit Leuven Department f Cmputer Science Celestijnenlaan 200A { B-300 Heverlee (Belgium)

2 The Behaviur f the Flquet Multipliers f Peridic Slutins Near a Hmclinic Orbit Zuchang Zheng Dirk Rse Reprt TW 287, December 998 Department f Cmputer Science, K.U.Leuven Abstract In this paper, we describe the behaviur f the mduli f the Flquet multipliers f peridic slutins near a hmclinic rbit. We discuss the relatin between the Flquet multipliers and the eigenvalues f the equilibrium pint at which the hmclinic rbit emanates. Fr tw-dimensinal prblems, fr which the equilibrium pint has tw real eigenvalues, we derive an asympttic estimate fr the nntrivial Flquet multipliers in terms f the perid. Fr mre cmplicated situatins (higher dimensinal prblems, cmplex cnjugate pairs f eigenvalues), we perfrm numerical experiments fr sme mdel equatins, shwing that eigenvalues with large negative real part cause small Flquet multipliers. Keywrds : Flquet Multiplier, Peridic Slutin, Hmclinic Orbit. AMS(MOS) Classicatin : Primary : 65N35, Secndary : 35B32.

3 The Behaviur f the Flquet Multipliers f Peridic Slutins Near a Hmclinic Orbit Zuchang Zheng Dirk Rse Department f Cmputer Science Kathlieke Universiteit Leuven Celestijnenlaan 200A B-300 Heverlee-Leuven, Belgium Dirk.Rse@cs.kuleuven.ac.be December,998 Abstract In this paper, we describe the behaviur f the mduli f the Flquet multipliers f peridic slutins near a hmclinic rbit. We discuss the relatin between the Flquet multipliers and the eigenvalues f the equilibrium pint at which the hmclinic rbit emanates. Fr tw-dimensinal prblems, fr which the equilibrium pint has tw real eigenvalues, we derive an asympttic estimate fr the nntrivial Flquet multipliers in terms f the perid. Fr mre cmplicated situatins (higher dimensinal prblems, cmplex cnjugate pairs f eigenvalues), we perfrm numerical experiments fr sme mdel equatins, shwing that eigenvalues with large negative real part cause small Flquet multipliers. Key Wrds Flquet multiplier, Peridic Slutin, Hmclinic Bifurcatin AMS(MOS) subject classicatins 65N2,35B0,35B32. Intrductin Several numerical methds (Beyn (989, 990), Kuznetsv (990) Dedel et al.(997) ) fr the cmputatin f hmclinic bifurcatins f rdinary dierential equatins have been develped. Typically, they are frmulated as apprximate bundary value prblems with peridic r prjected bundary value cnditins. Fr a large-scale system f ODEs, arising frm the space discretizatin f PDEs, these methds becme prhibitively expensive. Recently, Newtn-Picard schemes (Rse et al.(995), Lust et al.(997)) have been develped t slve large-scale peridic bundary value prblem with lw dimensinal dynamics, i.e, when mst

4 f the Flquet Multiplers f the peridic slutin have small mdulus. This is the case fr many ODE systems arising frm the space discretizatin f a PDE. In this paper, we study the behaviur f the mdulus f the Flquet multipliers when the peridic slutins apprach a hmclinic rbit, and we shw that mst f the Flquet multipliers becme small, such that a Newtn-Picard scheme can be used t cmpute peridic slutins as an apprximatin t a hmclinic rbit. Specically, we suppse that the autnmus ODE system _x(t) = F (x(t); ); x 2 R N ; 2 R; N >> () has a nndegenerate hmclinic rbit (x (t); ) cnnecting p( ), where p() is a hyperblic xed pint f ODE system (). As a starting pint, we recall Therem 3. frm (Beyn (989)) Therem There exits T 0 > 0 large enugh and > 0, such that fr any T > T 0, the apprximate peridic bundary value prblem 8>< >: _x(t) = F (x(t); ); t 2 J = [ T; T ] x( T ) = x(t ) _x (0) T (x(0) x (0)) = 0 has a unique slutin (x(t; T ); (T )) in f(x; ) : jjx x jj + j j < g Further, fr any > 0, the slutin (x(t; T ); (T )) satises the estimates (2) jjx(t; T ) x (t)jj = O(exp ( (min( + ; ) )T )) j j = O(exp ( (2 min( + ; ) )T )) (3) with the critical expnents + = MinfRe() : is an eigenvalue f A( ) with Re() > 0g = Minf Re() : is an eigenvalue f A( ) with Re() < 0g (4) and A() = F x (p(); ). In this paper, we relate the Flquet multipliers f the peridic slutin x(t; T ) t the eigenvalues f the xed pint p( ) in terms f the perid T. We shw that lw dimensinal dynamics f the xed pint (mst f the eigenvalues f F x (p( ); ) have a very large negative real part) prduces lw dimensinal dynamics f the peridic slutins (mst f the Flquet multipliers are very small in mdulus). 2. Lw dimensinal dynamics f the xed pint and peridic slutins appraching an hmclinic rbit We discuss the relatin between the lw dimensinal dynamics f the xed pint p( ) and that f the peridic slutins which apprach the hmclinic rbit. The 2

5 discussin is based n the fact that near the xed pint, the vectr eld f () is essentially linear, and when peridic slutins apprach the hmclinic rbit, the time they spend in the small linear regin ges t innity. The lcal linearizatin vectr eld f the xed pint plays an imprtant rle in ur discussin. In fact, fr three dimensinal systems, Glendinning and Sparrw ([984]) shw that the eigenvalues f the lcal linearizatin vectr eld determines the stability f the peridic slutins when they apprach the hmclinic rbit. Fr higher dimensinal ODE systems, the peridic slutins in a neighburhd f the hmclinic rbit have dierent bifurcatin and stability behaviurs, depending n the relative size f the critical expnents (4), see Kuznetsv Yu.A. [997]. The estimate (3) als illustrate that the eigenvalues f the xed pint determine the apprximate rder f the slutin f the apprximate peridic bundary value prblems. We discuss tw dierent cases: a) all eigenvalues f the xed pint are real; b) there are sme cmplex cnjugate pairs f eigenvalues. Since the tplgy f the phase space f the lcal linearizatin vectr eld is qualitatively dierent in these tw cases, we expect that the mdulus f the Flquet multipliers will als vary in qualitatively dierent ways. In fact, Kuznetsv (997) has shwn that the Flquet multipliers may cntract t zer r expand innitely when the peridic slutins apprach the hmclinic rbit. In this paper, we derive sme mre quanlitative infrmatin, e.g. hw many Flquet multipliers cntract t zer and hw many expand t innity. First, we btain a predictin frm the analytical analysis f the simplest situatins with the help f Sil'nikv's standard analysis methds fr the lcal behaviur near a hmclinic rbit (Sil'nikv (965), (970)); Then the predictin is further veried by numerical experiments. 2. All eigenvalues are real When all eigenvalues are real, the simplest situatin is a tw - dimensinal system whse hyperblic xed pint p( ) has tw simple real eigenvalues ; 2 with 2 > 0 and 2 < 0. Fr tw-dimensinal peridic slutins, n perid-dubling bifurcatins and bifurcatins t trus can ccur. Therem 6.5 in Guckenheimer & Hlmes (983) implies that when the peridic slutins apprach the hmclinic rbit, saddle-nde bifurcatins can nt ccur either. Thus the unique nntrivial Flquet multipliers always stays within r utside the unit circle. T relate their mdulus t the eigenvalues f the xed pint, we cmpute an asympttic estimate fr it by Sil'nikv's standard analysis methd. An imprtant bservatin in Sil'nikv's analysis is that the w near p() is essentially linear. Then the w is divided int tw regins as shwn in Figure (), the linear regin bunded by a bx B=fjxj l; jyj hg and the nnlinear regin utside this bx. In the bx B, there exist lcal crdinates (x; y) s that the lcal stable manifld Wlc s = fy = 0g and the lcal unstable manifld Wlc u = fx = 0g. In the bx B, the vectr eld can 3

6 y 2 Σ λ h Σ λ 0 l x Figure : Within the bx, the w is essentially linear. be written as ( _x = x + f(x; y; ); _y = 2 y + g(x; y; ) where f(x; y; ) and g(x; y; ) are higher rder terms. When =, there is a hmclinic rbit. Thus there exists a glbal nnlinear w which takes trajectries clse t the unstable manifld away frm the linear regin and then back int it. The glbal nnlinear w induces a return map (5) = 2 : (l; y)! (l; (y; )) (6) With jyj < h << and j j <<. maps the plane x = l t the plane y = h and is determined by the lcal linearised w f (5). Thus it is easy t check that : (l; y)! (l(y=h) ; h) (7) with jyj < h << and = j 2 j. The time needed t perfrm the linear map frm x = l t y = h is T (y) = 2 ln h y 2 species hw trajectries leave y = h int the nnlinear regin and then return t the linear regin at x = l, and can be written as (8) 2 : (x; h)! (l; (x; )) (9) 4

7 where jxj < l << and j j <<. At =, we have (0; ) = 0. If the linear regin is taken small enugh, (x; ) will be given by its Taylr expansin until the rst rder term, that is, (x; ) = ax + b( ) (0) here a (0; ) and b (0; ). Cmbining (7) with (0), we get the : y! cy + b( ) () with c = al h, jyj < h << and j j <<. The xed pint y() f the return map satises cy + b( ) = y (2) In equatin (2), ignring the higher rder terms f y, the xed pint can be taken t be 8>< b( ); if > y() = (3) >: ( b c ( )) ; if 0 < < and the eigenvalue () f the return map at the xed pint is () = 8>< >: c(b( )) ; if > c(( b c ( )) ; if 0 < < The relatin (4) gives the stability infrmatin f the peridic slutins ( see Guckenheimer & Hlmes [983] ). T derive an asympttic estimate fr the nntrivial Flquet multiplier in terms f the perid, we ntice that if the linear regin is small enugh, the time spent in the linear regin (8) is the main cntributin t the perid T () f the peridic slutin crrespnding t the xed pint y(), and we can take T () ' T (y()). In view f (8) and (3), we have the fllwing asympttic estimate fr the perid which implies that T () 8>< >: 2 lnj j; if > lnj j; if 0 < < 8>< >: e 2T () ; if > e T () ; if 0 < < Substituting (6) in (4), we btain an asympttic estimate fr the nntrivial Flquet multiplier (4) (5) (6) () e 2( )T () ; if 6= (7) 5

8 Thus the nntrivial Flquet multiplier varies expnentially with expnent 2 ( ) as T!. If >, the nntrivial Flquet multiplier cntracts t zer; in particular, the cntractin is very strng when j j is very large. If 0 < <, the nntrivial Flquet multiplier expands t. Thus depending n the relative size f the eigenvalues f the xed pint, the nntrivial Flquet multiplier wuld either expand innitely r cntract tward zer. Based n the analysis abve, we expect that fr higher dimensinal systems, the mduli f the Flquet multipliers vary in the same way. Mre precisely, a typical case is that nly ne f the eigenvalues i ; i = ; 2; ; N, say, is psitive. Then depending n whether i = j i j; i = 2; 3; ; N is larger r smaller than, a Flquet multiplier will cntract r expand with the expnent ( i ). T verify this, we have dne numerical experiments. T make sure that the numerical cmputatin is accurate enugh and reliable, we give sme details f the numerical methds fr cmputing Flquet multipliers and f ur cmputatinal strategies. Fllwing (Fairgrieve (994)), a numerical methd fr cmputing Flquet multipliers is cnsidered t be rbust if the errr in the cmputed Flquet multipliers is directly prprtinal t the truncatin errr f the cmputed peridic slutin and is nt badly inuenced by the rund errr in the eigenvalue calculatin. The numerical methd implemented in the AUTO97 sftware package is rbust (Fairgrieve et al.(99), (994)). In the AUTO97, a spline cllcatin scheme with adaptive mesh selectin is applied t cmpute the peridic slutin. The truncatin errr (Ascher et al.(979) and Hustis (978)) is determined by the number f mesh subintervals and the number f cllcatin pints per subinterval. In ur numerical cmputatins, we have used set 4 cllcatin pints per subinterval and we have chsen the number f mesh subintervals large but nt t large s that the rund errr des nt dminate the truncatin errr. In additin, we set all the relative cnvergence tlerances in the Newtn/Chrd methds small enugh such that the errr in the cmputed peridic slutin is dminated by the truncatin errr. Mrever, thrughut the numerical cmputatin, we always use adaptive mesh selectin strategy and repeat the cmputatin with dierent number f mesh subintervals. These are eective safeguards t avid spurius slutins. The trivial Flquet multiplier is mnitred and cntrlled within an acceptable accuracy, and in mst runs, is btained t maximal accuracy. Except that sme f the Flquet multipliers becme s small that numerical nise becmes visible, the Flquet multipliers cmputed within AUTO are accurate and reliable. Fr the case f three real eigenvalues i ; i = ; 2; 3, if cnsidering the ODE system in reverse time, we nly need t verify the situatin with > 0; 3 < 2 < 6

9 a b Figure 2: Peridic Slutins f the Lrenz equatins: perid versus lgarithm f the mduli f the Flquet multipliers, indicating that the nntrivial Flquet multipliers vary expnentially with the perid. (a) Flquet multiplier which expands t ; (b) Flquet multiplier which cntracts t 0. 8>< >: 0. As ur cmputatinal mdel, we take the well-knwn Lrenz equatins _x = (y x) _y = rx y xz _z = xy bz (8) As usual, we take = 0; b = 8=3 and 0 < r <. When r = r ' 3:92, there is a hmclinic rbit cnnecting the rigin and the three real eigenvalues f the rigin at r are ( ; 2 ; 3 ) = (7:299; 2:6667; 8:299). We trace a branch f peridic slutins, which bifurcates frm a Hpf pint at r ' 24:74 and terminates at the hmclinic rbit. Since this branch des nt cntain any further lcal bifurcatins, the Flquet multipliers d nt crss the unit circle. Their mdulus shuld change in the same way as tw-dimensinal systems. Mre precisely, since the eigenvalues f the rigin fr r satisfy the relatin = j 3 j >, the tw dimensinal analysis predicts that the mdulus f ne f the tw nntrivial Flquet multipliers shuld cntract t zer with expnent ( ) ' :0. The fast cntractin is due t the large negative real eigenvalue 3 ' 8:3. The mdulus f the ther nntrivial Flquet multiplier shuld expand t with expnent ( 2 ) ' 4:4632 since 2 = j 2 j <. The cmputed nntrivial Flquet multipliers are shwn in Figure 2, demnstrating that the nntrivial Flquet multipliers indeed expand r cntract expnentially with the perid T. The cmputed smaller nntrivial Flquet multiplier quickly be- 7

10 cmes s small that the cmputatinal results are subject t nise fr T > 2. Using least squares tting, we that the expnents fr the smaller and larger nntrivial Flquet multiplier is and 4.02, respectively. Cmpared with the predicted expnents :0 and 4:4632, the cmputed expnent fr the larger ne is mre accurate than that fr the smaller ne. 2. A pair f cmplex cnjugate eigenvalues Fr the case where there are cmplex cnjugate pairs f eigenvalues, the analysis is mre cmplicated. A three-dimensinal system with a pair f cmplex cnjugate eigenvalues is the simplest situatin we need t cnsider. We assume that the three eigenvalues i ; i = ; 2; 3, satisfy Re( 3 ) = Re( 2 ) < 0 and > 0. Using Sil'nikv's standard analysis methd, Glendinning and Sparrw (984) discuss the stability f peridic slutins f 3-dimensinal ODE systems which apprach the hmclinic rbit. Their results state that if = j Re( 2) j >, there is a branch f stable peridic slutins which apprach the hmclinic rbit withut any lcal bifurcatins and the tw nntrivial Flquet multipliers cntract t zer; if = j Re( 2) j <, there is a branch f peridic slutins which apprach the hmclinic rbit thrugh an innite sequence f pairs f tw successive saddle-nde and tw successive perid-dubling bifurcatins. The tw nntrivial Flquet multipliers will scillate arund and, respectively. It is nt easy t cmpute their asympttic estimates in terms f the perid T as we did in the tw dimensinal setting. Instead, fllwing the stability analysis in Glendinning and Sparrw (984), we have perfrmed numerical experiments t shw the relatin between the tw nntrivial Flquet multipliers and the perid. T nd ut hw large r hw small the mdulus f the Flquet multipliers wuld be, we take Arned's equatins fr > and Chua's circuit equatins fr < as mdel prblems. Arned's equatins can be written as 8>< >: _x = y _y = z _z = z by + cx x 2 (9) Arned et al.(982) have lcated hmclinic rbits at sme values f the parameters b and c. We x b = 0:5 and take c > 0 as a free parameter. Fr c = c ' 0:964494, there is a hmclinic rbit cnnecting the xed pint at the rigin. The eigenvalues ( ; 2 ; 3 ) f the rigin at c are (0:629; 0:84 i0:928). The peridic slutins arising frm the Hpf pint d nt underg any lcal bifurcatins when they apprach the hmclinic rbit. Thus the mduli f the Flquet multipliers d nt crss the unit circle, but in this case, the mduli f the Flquet multipliers cntract t zer in an scillatry way in functin f the perid. Fr T > 28, the smallest Flquet multipliers (Figure 3b) is cmputed inaccurately due 8

11 0 2 a b Figure 3: The mdulus f the Flquet multipliers f the peridic slutins f Arned's equatin in functin f the perid. Bth f the Flquet multipliers cntract t zer. The peaks in curve (b) indicate an increasing trend, but they becme mre and mre sharp when the perid!. t numerical nise and is mitted frm the Figure, see Figure 3. The scillatins f the mduli f the Flquet multipliers have peaks. Althugh in Figure 3b, the peaks indicate a increasing trend as the perid ges t innity, bth the mdulus and the width f the peaks becme smaller s that numerically it is nt easy t nd the sharp pints by slving the apprximate peridic bundary value prblem (2). Ignring the sharp pints, we can say that this situatin is similar t the case with three real eigenvalues with bth > and 2 >. Fr the case f cmplex cnjugate eigenvalue with <, we deal with Chua's circuit equatins 8 > < _x = (y (x)) > _y = x y + z (20) : _z = z Fllwing Huang, et al.(996), we take (x) = (x 3 =6 x=6), = 8:0 and t be the free parameter. The system has three xed pints, i.e, P 0 = (0; 0; 0); P = ((8=3) =2 ; 0; (8=3) =2 ) fr any 2 R. When = 4:684658, P are Hpf bifurcatin pints, frm which a branch f peridic slutins bifurcates. The stability analysis in (Glendinning et al.(984)) shws that as the peridic slutins apprach the hmclinic rbit, a saddle-nde bifurcatin and a perid-dubling bifurcatin ccur successively and between them, there is a shrt sectin f unstable peridic slutins. This pattern f stability changes is repeated until the peridic slutins end at the hmclinic rbit when = ' 4:5965. The peridic 9

12 a b Figure 4: If the sharp pints are ignred, the mdulus f the Flquet multiplier f Chua's equatins in (a) expand t innity; the ne in (b) cntracts t zer. They vary in a smewhat similar way t the real eigenvalues case shwn in Figure (2). slutins apprach the hmclinic rbit thrugh a sequence f lcal bifurcatins, cntrary t the situatins in Lrenz's equatins and Arned's equatins. The xed pint n the hmclinic rbit is the rigin, whse eigenvalues at = are ( ; 2 ; 3 ) = (2:2776; :089 :6357i). Since there is an innite sequence f saddle-nde and perid-dubling bifurcatins, the mduli f the Flquet multipliers d nt increase r decrease in a mntnus way as in Lrenz's equatins, but scillate arund. The cmputed mduli f the Flquet multipliers, presented in Figure 4, indeed reects the bifurcatin infrmatin. The peaks f the curves f the cmputed mdulus result frm the saddle-nde and perid-dubling bifurcatins. The intervals between these tw successive bifurcatins are s small that the tw nntrivial Flquet multipliers crss almst fr the same value f the perid; thus the mduli f the Flquet multipliers will change very quickly. In additin, these intervals will becme smaller and smaller when the perid tends t innity. Successive saddle-nde and perid dubling bifurcatins and the unstable peridic slutins between them are nt fund by numerical cmputatin with AUTO when T is larger than 5, even with T = 0 5. Ignring the peaks, caused by the lcal bifurcatins, the general behaviurs f the Flquet multipliers is as fllws: ne f the Flquet multipliers expands t innity, similar t the case f real eigenvalues with < ; the ther Flquet multipliers cntracts t zer, similar t the case f real eigenvalues with >. The discussin abve has shwn that fr a pair f cmplex cnjugate eigenval- 0

13 Eigenvalues f xed pint Real Real part Imaginary part Flquet multipliers E E E E E E E E E E E E E E E E E E E E E E E E+ Table : Kuramt-Sivashinsky equatin: The eigenvalues f the xed pint and the Flquet multipliers f the peridic slutins with = 3:574067E + 0. ues, numerically the mduli f the Flquet multipliers vary in a similar way as in the case f three real eigenvalues, i.e., depending n the relative size f the psitive and negative real part f the eigenvalues, the Flquet multipliers will expand innitely r cntract t zer. Fr higher dimensinal systems, we expect that the same cnclusin is true. A typical case is when there is nly ne pair f cmplex cnjugate eigenvalues with psitive real part, say, i ; i = ; 2, with Re( )=Re( 2 )> 0, and j 3 j j 4 j j N j are negative real eigenvalues. Cnsider this system in reverse time and let i = j i Re j; i = 3; ; N, then we predict that if 3 <, tw f the nntrivial Flquet multipliers expand t innity; if >, ne f nntrivial Flquet multipliers expands innitely and anther nntrivial Flquet multiplier will cntract t zer. Mrever, depending n i (i 3) < r >, ne f the ther nntrivial Flquet multipliers will expand r cntract. When ne eigenvalue has a large real part, ne f the nntrivial Flquet multipliers becmes quickly very small. T verify this further, we cnsider an innite dimensinal mdel, Kuramt-Sivashinsky equatin + 4@4 4 + (@2 2 + ) = 0 (2) subject t spatial peridic bundary cnditins u(x; t) = u(x+2; t) with 0 x 2. Using the traditinal Galerkin methd t discretize the space variable, Jlly et al.(990)) btain the fllwing ODE system where _x k = ( 4k 4 + k 2 )x k m k ; k m (22) m k = =2 mx j= jx j [x j+k + sgn(k j)x jk jj ] (23) The index m represents the number f mdes used in the discretizatin. We set x j = 0 if j < 0 and j > m, respectively. At = 34:299, there are tw Hpf

14 a b Figure 5: The mdulus f the tw nntrivial Flquet multipliers f K-S equatin, which crss thrugh the unit circle. The tw mdulus curves are similar t thse f Chua's equatins as shwn in Figure(4). pints H i ; i = ; 2. Numerical cmputatins (Jlly et al.(990)) indicate that the peridic slutins bifurcating frm bth Hpf pints terminate at a hmclinic rbit at = ' 35: We fcus n ne f the hmclinic rbits. As Jlly et al. (990) have shwn, the peridic slutins apprach t the hmclinic rbit in the same way as Chua's equatins. The xed pint n the hmclinic rbit has a cmplex cnjugate pair f eigenvalues with psitive real part, and the ther eigenvalues are large negative real numbers, as shwn in Table. Due t 3 >, tw f the Flquet multipliers, which crss thrugh the unit circle, shuld vary in a similar way as fr Chua's equatins. Figure 5 presents the cmputed mduli f the Flquet multipliers fr the 8-mde discretizatin. It indeed lks similar t Figure 4. Mrever due t i >, i = 4; ; N, the rest f the nntrivial Flquet multipliers shuld cntract t zer. In fact, since the negative real eigenvalues are very large in mdulus, the rest f the cmputed Flquet multipliers becme quickly s small that they are nt btained with a reasnable accuracy in ur cmputatin where the trivial Flquet multiplier has nly 4 signicant digits. T shw the rder f these small Flquet multipliers, in Table(), we list them fr a peridic slutin clse t the hmclinic rbit. Our cmputatinal results agree with the predictin that eigenvalues with a large negative real part make the Flquet multipliers cntract very quickly. The cmputed results fr discretizatins f the K-S equatin with 4, 6 and 32 mdes als further cnrm the predictin. 2

15 3. Cnclusins We have related the eigenvalues f the xed pint having a hmclinic rbit with the Flquet multipliers f the peridic slutins which apprach t the hmclinic rbit. We have shwn that the Flquet multipliers expand t innity r cntract t zer, depending n the relative size f the psitive and negative real parts f the eigenvalues. Specically, we have studied the fllwing tw typical if all eigenvalues i ; i = ; 2; ; N are real and nly ne f them, say, is psitive, then depending n whether i = j i j; i = 2; 3; ; N is larger r smaller than, a Flquet multiplier will cntract r expand asympttically with expnent ( i ); Nw suppse that there is nly ne pair f cmplex cnjugate eigenvalues with psitive real part, say, i ; i = ; 2, with Re( )=Re( 2 )> 0, and that i (i 3) are negative real eigenvalues. Then if 3 <, tw f the nntrivial Flquet multipliers expand t innity; if 3 >, ne f nntrivial Flquet multipliers expands innitely and anther nntrivial Flquet multiplier will cntract t zer. Mrever, depending n i (i > 3) < r >, ne f the ther nntrivial Flquet multipliers will expand r cntract. Here i = j i j; i = 3; ; N. Re( ) References [] Beyn W.-J., 989, Glbal bifurcatins and their numerical cmputatin, In Cntinuatin and Bifurcatins: Numerical Techniques and Applicatins, edited by D. Rse, B.D. Dier and A.Spence, Kluwer Academic Publisher, pp. 69-8, 990. [2] Beyn W.-J., 990, The numerical cmputatin f cnnecting rbits in dynamical systems, IMA J. Numer. Anal., 9(990), pp [3] Mre G., 995, Cmputatin and parameterizatin f peridic and cnnecting rbits, SIAM J. Numer. Anal., 5(995), pp [4] Dedel E.J., Friedman M.J. and Kunin B.I., 997, Successive cntinuatin fr lcating cnnecting rbits, Numerical Algrithms, 4(997), pp [5] Yu.A. Kuznetsv, 990, Cmputatin f Invariant Manifld Bifurcatins, in Cntinuatin and Bifurcatins: Numerical Techniques and Applicatins, eds by D.Rse, Bart De Dier and A. Spence, Kluwer Academic Publisher, pp ,

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