c 2005 Society for Industrial and Applied Mathematics

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1 SIAM J. NUMER. ANAL. Vol. 42, No. 6, pp Soiety for Industrial and Applied Matheatis FLOQUET THEORY AS A COMPUTATIONAL TOOL GERALD MOORE Abstrat. We desribe how lassial Floquet theory ay be utilized, in a ontinuation fraework, to onstrut an effiient Fourier spetral algorith for approxiating periodi orbits. At eah ontinuation step, only a single square atrix, whose size equals the diension of the phasespae, needs to be fatorized; the rest of the required nuerial linear algebra just onsists of baksubstitutions with this atrix. The eigenvalues of this key atrix are the Floquet exponents, whose rossing of the iaginary axis indiates bifuration and hange-in-stability. Hene we also desribe how the new periodi orbits reated at a period-doubling bifuration point ay be effiiently oputed using our approah. Key words. Floquet, Fourier, periodi orbit, ontinuation, period doubling AMS subjet lassifiations. 37C27, 37G5, 65L, 37M2, 65P3, 65T4, 65T5 DOI..37/S Introdution. Floquet theory is the atheatial theory of linear, periodi systes of ordinary differential equations (ODEs) and as suh appears in every standard book on ODEs, e.g., [, 2]. (We espeially reoend, however, the extensive eleentary disussion in [9].) In this paper we wish to utilize Floquet theory in order to effiiently opute approxiations to periodi orbits of nonlinear autonoous systes. The basi equations defining a periodi orbit are nonlinear, but applying a Newton-like ethod for their solution will lead to linear, periodi systes. It is for this reason that Floquet theory is so iportant for us. The nonlinear, autonoous syste we shall onsider is (.) ẋ(t) =F (x(t),λ), F : R n R R n ; i.e., F is a sooth funtion on R n and depends on a paraeter λ. For the rest of this setion (and setions 2 and 4), however, we shall teporarily onsider (.2) ẋ(t) =F (x(t)), F : R n R n. The siplest solutions of (.2) are the stationary points x R n defined by F (x )=. We will only be interested in stationary points at whih periodi orbits are reated, i.e., the faous Hopf bifuration points onsidered in setion 3. It is the next siplest solution of (.2) that this paper is onerned with. Definition. u : R R n is a periodi orbit for (.2), with (inial) period 2πT >, if u (t) =F (u (t)) t R, u () = u (2πT ), u (t) u () t (, 2πT ). Reeived by the editors August 28, 23; aepted for publiation (in revised for) June 9, 24; published eletronially Marh 3, Departent of Matheatis, Iperial College of Siene, Tehnology and Mediine, 8 Queen s Gate, London SW7 2AZ, UK (g.oore@iperial.a.uk). 2522

2 FLOQUET THEORY 2523 In order to set up an appropriate set of equations for oputing a periodi orbit, two key fats should be kept in ind: T that defines the period is also unknown; for any R, u (t + ) desribes the sae periodi orbit. (More preisely, u (t) and u (t + ) differ only by phase.) The proble of working with an unknown period is dealt with by defining Hene v has period 2π and satisfies Thus we should solve v (θ) u (T θ). v (θ) =T F (v (θ)). (.3) v(θ) =T F (v(θ)), v() = v(2π) for both the funtion v and the salar T. Sine there is an extra unknown, i.e., T, we ust have an extra salar equation. This fits in with the fat that v (θ + ) isa solution for any R. The odern way of fixing the phase, i.e., onstruting an extra salar equation whih deterines a unique, is as follows. We assue that we know a nearby periodi orbit v (θ) of period 2π. This is a natural assuption to ake in a ontinuation fraework. We fix the phase by seeking the value of whih akes v (θ + ) as lose as possible to v (θ), e.g., in R 2π v (θ + ) v (θ) 2 2 dθ. Setting the derivative with respet to equal to zero gives 2π v (θ) v (θ + )dθ =, but it is onvenient to integrate-by-parts and write 2π v (θ) v (θ + )dθ =. Hene our final set of equations for a periodi orbit is (.4) v(θ) =T F (v(θ)), 2π v() = v(2π), v (θ) v(θ)dθ =. We shall follow this idea later in setion 4, but using a ore appropriate inner produt. As we shall see, Floquet theory is best obined with a Fourier ethod to approxiate periodi orbits. This leads to the question, why aren t Fourier spetral ethods ore popular, opared to the opletely doinant olloation with pieewisepolynoials [, 4, 23, 24]? (Of ourse, Fourier approxiation of periodi orbits has been onsidered in a few papers, e.g., [9, ], but not using the present approah. For exaple, in [3] it is suggested that an approxiation to the onodroy atrix be

3 2524 GERALD MOORE oputed using an initial-value algorith, whih we regard as generally unaeptable.) So long as the solution one is trying to approxiate is sooth, the standard advantage of spetral ethods has always been their approxiation power, and the standard disadvantage has been that they require the solution of linear equations with nonsparse oeffiient atries. The key task of the present paper is to ephasize that this disadvantage is not present when using Fourier spetral ethods to approxiate periodi orbits in a ontinuation fraework. The underlying reason an be split into three parts. Fourier ode deoupling ours for linear, onstant-oeffiient differential equations, as desribed in setion 2. Floquet theory transfors linear, periodi differential equations into onstantoeffiient for, as desribed in setion 4. The Floquet variables required for this transforation an be updated effiiently as part of the ontinuation proess, as desribed in setion 5. The pratial differene between pieewise-polynoial and Fourier approxiation beoes lear if one onsiders the linear, periodi syste ẏ(θ)+a(θ)y(θ) =f(θ), where f :[, 2π] R n and A :[, 2π] R n n is given periodi data, and we wish to deterine a periodi solution y :[, 2π] R n. Applying a pieewise-polynoial olloation ethod at the N esh-points =θ <θ < <θ N <θ N =2π leads to an Nn Nn oeffiient atrix with the alost blok-bidiagonal struture (.5) , where eah is of size n n. (We assue that ondensation of loal paraeters has been applied, as in [3].) With a spetral ethod based on Fourier odes, however, just a single n n atrix needs to be fatorized; all other atrix operations are just bak-substitutions with quasi-upper triangular atries. We ust adit, however, that it is neessary to use a sall ultiple of Nn 2 loations for storing Floquet inforation, i.e., roughly the sae nuber of nonzero eleents as in (.5). To onlude this introdution, we desribe the ontents of this paper. In setion 2, we show how ode-deoupling ours for Fourier approxiation of onstant-oeffiient probles, not only for 2π-periodi equations but also for another for of periodiity that is (in general) required by the Floquet theory. Soe of these results are then applied in setion 3 in order to effiiently alulate periodi orbits reated at a Hopf bifuration point. In setion 4, we desribe Floquet theory in the pratial for that we shall utilize it. It is then iediately eployed, in setion 5, to advane a periodi orbit over a single ontinuation step. To illustrate the perforane of the previous algoriths, we present soe nuerial results in setion 6. Finally, setion 7 shows how the Floquet theory akes it very easy to ove onto the new periodi orbits reated at a period-doubling point. For a bakground to any of the topis disussed in this paper, we reoend [2, 3].

4 FLOQUET THEORY Constant-oeffiient equations. For periodi, onstant-oeffiient equations, Fourier analysis is espeially siple beause the Fourier odes deouple. We shall require two spaes of periodi funtions, whih we denote by Y + and Y, respetively. The first of these is just the usual spae of 2π-periodi funtions, spanned by, os θ, sin θ, os 2θ, sin 2θ,... The seond of these is the subspae of 4π-periodi funtions whih satisfy and is therefore spanned by Of ourse, the diret su y(θ) = y(θ +2π) os θ 2, sin θ 2, os 3θ 2, sin 3θ 2,... Y + Y gives all 4π-periodi funtions. The key reason why the spae Y is iportant to us is that the produt of two of its eleents lies in Y +, and suh produts will arise naturally in setion 4. Siilarly, the produt of an eleent of Y + with an eleent of Y lies in Y. We ention here that, throughout this paper, Fourier series will be desribed using real trigonoetri funtions rather than the atheatially ore elegant oplex exponentials. This is solely beause we wish to reain lose to the pratial ipleentation of our algoriths. 2.. Equations in Y+ n. For a onstant n n atrix A, onsider the linear, periodi, hoogeneous differential equation ż(θ)+az(θ) =, z Y n +, whih eans that eah oponent of z is in Y +. If the set {i : Z} does not ontain any eigenvalue of A, then this equation only has the trivial solution z(θ). Under this assuption, onsider the linear, periodi, inhoogeneous differential equation (2.) ż(θ)+az(θ) =f(θ), z Y n +, for a given f Y+.If n f(θ) f { + f os θ + f s sin θ }, = then the Fourier oeffiients of z(θ) z { + z os θ + z s sin θ } are given by = (2.2a) Az = f

5 2526 GERALD MOORE and (2.2b) ( )( ) ( ) A I z f =, =, 2,... I A z s f s Sine the eigenvalues of ( ) A I I A are related to those of A through the apping µ µ ± i, the above eigenvalue assuption eans that ( ) A I I A is nonsingular for all N Equations in Y n. Siilarly, if we ask the question whether the linear, periodi, hoogeneous differential equation ż(θ)+az(θ) =, z Y n, has any nontrivial solution, then the answer is that it does not, provided that the set { [ 2 ]i : Z} ontains no eigenvalue of A. Furtherore, if f Y n is given, the linear, periodi, inhoogeneous differential equation (2.3) ż(θ)+az(θ) =f(θ), z Y n, has a unique solution under this assuption. If { f(θ) f os [ 2 ]θ + f s sin [ 2 ]θ}, = then the Fourier oeffiients of { z(θ) z os [ 2 ]θ + zs sin [ 2 ]θ} are given by (2.4) = ( A [ 2 ]I )( ) z [ 2 ]I A z s ( ) f = f s, =, 2, Coputational algorith in Y+ n. Now we show how to effiiently opute spetral approxiations to the solution of (2.). This is ahieved by restriting to a finite nuber of Fourier odes ( M) and using aurate nuerial quadrature to approxiate the Fourier oeffiients of f. Thus f in (2.) is approxiated by (2.5) where f(θ) f + f N f 2 N f s 2 N M = N f(θ j ), j= { f os θ + f s sin θ}, N f(θ j ) os θ j, j= N f(θ j ) sin θ j, j= =, 2,...,M, =, 2,...,M,

6 and N 2M + with θ j = 2πj that (2.6) 2π N, 2π FLOQUET THEORY 2527 j =,...,N. In other words, we are using the fat f(θ)g(θ)dθ = N N f(θ j )g(θ j ) j= when f and g are both in the subspae of Y + spanned by, os θ, sin θ, os 2θ, sin 2θ,..., os Mθ, sin Mθ. Also the above transforation fro funtion values to approxiate Fourier oeffiients an be written in atrix for where with Q + g p = N/2 g, g p (g(θ ),g(θ 2 ),...g(θ N )) T, g ( 2 g, g, g,... g s M, g M s ) T g(θ) g + M { g os θ + g s sin θ }, = and Q + is the N N orthogonal atrix with (i, j)th oponent i = odd i 3 even i / N 2/N sin i 2 θ j 2/N os i 2 θ j Thus it an be applied to the point values of eah oponent of f to obtain (2.5). (For large M, it is ore effiient to ap between point values and approxiate odal values of f using the fast Fourier transfor [7, 29]. In this ase it ay be preferable to hoose N to be slightly greater than 2M +, i.e., so as to be a highly oposite integer. The standard tehniques for dealing with this situation are desribed in [6, 29].) Now the approxiate Fourier oeffiients for z in (2.), z(θ) z + M { z os θ + z s sin θ }, = ay be oputed as in (2.2). Note that this is espeially effiient when A has already been redued to Shur for [7]; i.e.,. (2.7) A = QÛQT, where Û Rn n is a quasi-upper triangular atrix and Q R n n is an orthogonal atrix. Then, if z = Qẑ, f = Q ˆf, and } z /s = Qẑ /s, =,...,M, = Q f /s ˆf /s

7 2528 GERALD MOORE we have only to solve the systes Ûẑ = ˆf and ( Û )(ẑ ) I I Û ẑs = ( ˆf ˆf s ), =,...,M. The first is just bak-substitution, starting with û nn or the 2 2 blok ) (ûn,n û n,n, û n,n while the seond only requires solving 2 2or4 4 systes like ) û n,n û n,n (ûnn or û n,n û nn û nn û n,n û n,n. û n,n û nn 2.4. Coputational algorith in Y n. Now we show how to effiiently opute spetral approxiations to the solution of (2.3). Again this is ahieved by restriting to a finite nuber of Fourier odes ( M) and using aurate nuerial quadrature to approxiate the Fourier oeffiients of f. Thusf in (2.3) is approxiated by (2.8) where f(θ) f 2 N f s 2 N M = û nn { f os [ 2 ]θ + f s sin [ 2 ]θ }, N f(θ j ) os [ 2 ]θ j, j= N f(θ j ) sin [ 2 ]θ j, j= =, 2,...,M, =, 2,...,M, and θ j = 2πj N, j =,...,N. In other words, we are now using the fat that (2.6) also holds when f and g are both in the subspae of Y spanned by os θ 2, sin θ 2, os 3θ 2, sin 3θ 2,..., os [M 2 ]θ, sin [M 2 ]θ. Again the above transforation fro funtion values to approxiate Fourier oeffiients an be written in atrix for where Q g p = N/2 g, g p (g(θ ),g(θ 2 ),...g(θ N )) T, g ( g, g,... g s M, g M s ) T with M g(θ) { g os [ 2 ]θ + gs sin [ 2 ]θ}, =

8 FLOQUET THEORY 2529 and Q is the 2M N atrix with orthonoral rows and (i, j)th oponent i even 2/N sin i 2 θ j the inverse transforation siilarly being i odd 2/N os i 2 θ j ; g p = N/2 Q T g. Thus it an be applied to the point values of eah oponent of f to obtain (2.8). (Again, for large M, the apping between point values and approxiate odal values of f is ore effiiently arried out by a variant of the fast Fourier transfor.) Now the approxiate Fourier oeffiients for z in (2.3), z(θ) M { z os [ 2 ]θ + zs sin [ 2 ]θ}, = ay be oputed as in (2.4) above. Note that this is again espeially effiient when A has already been redued to Shur for as in (2.7). Then, if z /s f /s = Qẑ /s = Q ˆf /s we have only to solve the systes ( Û [ 2 ]I )(ẑ ) [ 2 ]I Û = }, =,...,M, ẑ s ( ) ˆf ˆf s, =,...,M. This just involves bak-substitution, e.g., starting with the 2 2 syste ( ûnn [ 2 ] ) [ 2 ] û nn or the 4 4 syste û n,n û n,n [ 2 ] û n,n û nn [ 2 ] [ 2 ] û n,n û n,n. [ 2 ] û n,n û nn 3. Hopf bifuration. In this setion we look at a straightforward appliation of the ideas in setions 2. and 2.3. We were surprised that this does not see to have appeared in the literature before, the losest exaple we have found being [6], whih does not, however, take full advantage of the ode-deoupling. Hopf bifuration refers to the reation of sall aplitude periodi orbits at a partiular point on a urve of stationary points. For the paraeter-dependent equation (.), we shall onsider a stationary point (x,λ ) satisfying the following properties. For sipliity, we assue that the urve of stationary points is paraetrizable by λ; f. [8].

9 253 GERALD MOORE (a) F (x,λ )= and the Jaobian atrix J(x,λ ) is nonsingular. Thus the ipliit funtion theore tells us that there is a loally unique urve of stationary points passing through (x,λ ). This ay be paraetrized by λ, and so we denote it by (x (λ),λ). (b) J(x,λ ) has a pair of siple purely iaginary eigenvalues ±iω, ω >, and no other eigenvalues of the for {iω : Z}. Hene the right eigenvetor pair ϕ R ± iϕ I satisfies i.e., J(x,λ )[ϕ R ± iϕ I]=±iω [ϕ R ± iϕ I], J(x,λ )ϕ R = ω ϕ I, J(x,λ )ϕ I = ω ϕ R, and the left eigenvetor pair ψ R ± iψ I satisfies i.e., J(x,λ ) T [ψ R ± iψ I]= iω [ψ R ± iψ I], J(x,λ ) T ψ R = ω ψ I, J(x,λ ) T ψ I = ω ψ R. A suitable hoie of noralization is that ϕ R ϕ I =, ϕ R ϕ I 2 2 =, ψ R ϕ R =, ψ R ϕ I =, ψ I ϕ R =, ψ I ϕ I =. () If the n n atrix K is defined by then K d dλ { } J(x (λ),λ) λ=λ, (3.) ψ R K ϕ R + ψ I K ϕ I. This eans that, as λ oves away fro λ, the eigenvalues of J(x (λ),λ) orresponding to ±iω are no longer purely iaginary; i.e., if these eigenvalues are denoted µ (λ) ± iω (λ), then (3.) is equivalent to dµ dλ (λ ), sine (fro [22]) (3.2) γr dµ dλ (λ )= ψ R K ϕ R + ψ I K ϕ I, 2 γi dω dλ (λ )= ψ R K ϕ I ψ I K ϕ R. 2

10 FLOQUET THEORY 253 If, for λ near λ, we look for a periodi orbit of (.) near x, then first we ake our usual hange-of-variable v(θ) u(tθ), whih swithes fro u(t) with unknown period 2πT to v(θ) with period 2π, and (.) transfors to (3.3) v(θ) =T F (v(θ),λ), v Y n +. Sine F (x,λ )=, v(θ) x + z(θ), z Y n +, will be an approxiate periodi orbit (of period 2πT) if z(θ) is sall and satisfies (3.4) ż(θ)+t J(x,λ )z(θ) =. But for T = ω, we know that (3.4) has solutions for arbitrary onstants C &, where z(θ) =Ca (θ + ) a (θ) ϕ R sin θ + ϕ I os θ. Thus we seek solutions of (3.3) in the for (3.5) v(θ) x (λ)+ε[a (θ)+z(θ)] for sall nonzero ε, with (3.6a) and (3.6b) where π π 2π 2π a (θ) z(θ)dθ = p (θ) z(θ)dθ =, p (θ) da dθ (θ) =ϕ R os θ ϕ I sin θ. Equation (3.6a) fixes the aplitude of the periodi orbit, so we are using ε as a paraetrization, and (3.6b) fixes the phase. We shall soon require properties of the onstant oeffiient differential operator (3.7) d dθ + A on Y n +, where A T J(x,λ ) and T ω.

11 2532 GERALD MOORE The right null-spae of (3.7) is spanned by p (θ) and a (θ), while its left null-spae is spanned by ã (θ) ψ R sin θ + ψ I os θ and p (θ) ψ R os θ ψ I sin θ. If we onsider the augented linear equation (3.8) ż(θ)+a z(θ)+ T T p (θ)+λt K a (θ) =, a (θ) z(θ)dθ =, π p (θ) z(θ)dθ = π for unknowns (z(θ),t,λ), as a apping fro Y+ n R 2 to itself; then (3.8) will only have the trivial solution (,, ) if the deterinant of T ã (θ) p (θ)dθ T 2π ã (θ) K a (θ)dθ T p (θ) p (θ)dθ T 2π p (θ) K a (θ)dθ is nonzero. Sine 2π this depends only on whih is equivalent to (3.). If the sooth apping ã (θ) p (θ)dθ = 2π and 2π ã (θ) K a (θ)dθ, G :(Y n + R 2 ) R Y n + R 2 is onstruted by the following: for ε,g(z,t,λ; ε) is defined by p (θ) p (θ)dθ, [ȧ (θ)+ż(θ)] + ε T F (x (λ)+ε[a (θ)+z(θ)],λ) G(z,T,λ; ) is defined by π π 2π 2π a (θ) z(θ)dθ p (θ) z(θ)dθ, [ȧ (θ)+ż(θ)] + T J(x (λ),λ)[a (θ)+z(θ)] a (θ) z(θ)dθ π p (θ) z(θ)dθ; π

12 FLOQUET THEORY 2533 then the zeroes (z(θ),t,λ)ofgfor nonzero ε orrespond to periodi orbits of (3.3) through (3.5). It is iediate, however, that G(,T,λ ;) =, and (3.8) also tells us that, at ε =, the linearization of G with respet to (z,t,λ)at(,t,λ ) has no nontrivial solution. Hene the ipliit funtion theore applies to G at (,T,λ ; ) and tells us there is a loally unique solution urve of periodi orbits for (3.3), paraetrized by ε. (We note that, fro a pratial point of view, it is ore effiient to replae x (λ) above with the first-order approxiation where l is defined by x +[λ λ ]l, J(x,λ )l = F λ (x,λ ). For sipliity, however, we do not inlude this extra trik.) Rearranging the equation for zeroes of G enables us to define the following Newton-hord iteration for obtaining these periodi orbits. Choose sall ε and set y () (θ) =a (θ), T () = T, λ () = λ. Solve (3.9) [ ddθ ] + A z(θ)+ δt T p (θ)+δλ T K a (θ) = ε r(k) (θ), π π 2π 2π a (θ) z(θ)dθ =, p (θ) z(θ)dθ = for z Y n +, δt, and δλ, where Set r k (θ) ẏ (k) (θ) T (k) F (x (λ (k) )+εy (k) (θ),λ (k) ). y (k+) (θ) =y (k) + z(θ), T (k+) = T (k) + δt, λ (k+) = λ (k) + δλ. Note that only the sae augented onstant-oeffiient differential equation, with varying right-hand sides, needs to be solved at eah iteration. 3.. Fourier approxiation. Finally, we show how to effiiently opute aurate approxiations to the periodi orbits of (3.3), using the above Newton-hord iteration and the results of setion 2.3. The key step is how to alulate the approxiate Fourier oeffiients z(θ) z + M { z os θ + z s sin θ } =

13 2534 GERALD MOORE fro the right-hand side f(θ) ε r(k) (θ) M { f + f os θ + f s sin θ} = in (3.9), and we see that athing Fourier oeffiients gives the odal equations for = A z = f, for = A I T ϕ R T K ϕ I z I A T ϕ I T K ϕ f R z s ϕ T I ϕ T R δt = f s, T T δλ ϕ R ϕ I for 2 M [ ] A I ][ z I A = z s [ ] f f s. Hene the extra salar unknowns δt and δλ are solved for as part of the = syste, while the other odal equations reain the sae as in setion 2.3. Thus, by applying ψ T I ψ T ψ R R and ψ I to the = equation, we iediately deterine δt and δλ fro [ ][ ] [ T γr δt T T γ = ψ I f + ψ R f ] s I δλ 2 ψ R f ϕ I f s ; i.e., where δλ = T d λ and δt = T d T, d λ ψ I f + ψ R f s 2γR, ( ) ( ) γr ψ R f ϕ I f s γi ψ I f + ψ R f s d T 2γ R and γr,γ I are defined in (3.2). If A has already been redued to Shur for by A Q = Q U,

14 FLOQUET THEORY 2535 where U R n n is a quasi-upper triangular atrix and Q R n n is an orthogonal atrix, then it an be arranged that U has its eigenvalues ±i in the top left orner, i.e., β... β U = for soe nonzero β, as in the LAPACK standard for [2]. This eans that ϕ R β +β 2 q and ϕ I +β 2 q 2, where q j denotes the jth olun of Q. Hene, under the transforations z = Q ẑ, f = Q ˆf, and z /s f /s we have only to solve the systes for = = Q ẑ /s = Q ˆf /s }, =,...,M, U ẑ = ˆf ; for = U I ˆf d T β I U [ẑ ] e +β 2 d λ e T 2 βe T ẑ s = ˆf s + d T e β +β 2 2 d λ βe T e T 2 +β 2 2 +β 2 where the oponents of and 2 are the oeffiients of K q and K q 2, respetively, with respet to the orthonoral basis of R n fored by the oluns of Q, i.e., Q K q and Q 2 K q 2;, for 2 M [ U ][ẑ ] I I U ẑ s = [ ˆf ˆf s ] by bak-substitution. The systes for = and 2 are the sae as in setion 2.3 and nonsingular beause of the eigenvalue onditions satisfied by J(x,λ ) in (b) on page 253. The syste for = is overdeterined, but onsistent by onstrution,

15 2536 GERALD MOORE and the last n 2 oponents for ẑ /s an again be solved by bak-substitution. We are then left with the siple syste β ẑ β ẑ 2 β ẑ s =, β ẑ2 s where ẑ /s and ẑ/s 2 refer to the first two oponents of ẑ/s. 4. Pratial Floquet theory. Now let A(θ) be a 2π-periodi n n atrix, i.e., A(θ +2π) =A(θ) θ R. At first glane the periodi differential equation (4.) v(θ)+a(θ)v(θ) =f(θ), v Y n +, for a given f Y n +, sees uh ore diffiult to analyze and solve than (2.). It is the fundaental result of Floquet theory, however, that there is a hange-of-variable v(θ) =P(θ)w(θ) whih transfors (4.) to onstant-oeffiient for. The prie one has to pay to reain within real aritheti, however, is that soe of the oponents of the solution to the onstant-oeffiient proble ay lie in Y rather than Y + : i.e., soe of the oponents of w(θ)aybeiny, with the orresponding oluns of the n n atrix P(θ) iny, n but this still eans that the produt P(θ)w(θ) isiny+. n In onlusion then, our onstant-oeffiient equations ay be a obination of (2.) and (2.3). To see how this ours, let X(θ) be the prinipal fundaental solution atrix for the differential operator (4.2) d dθ + A(θ); i.e., X() = I and Ẋ(θ) =A(θ)X(θ) θ R, and thus the jth olun of X(θ) solves the hoogeneous initial value proble fored fro (4.2), with e j as the initial value. The oluns of X(θ) reain linearly independent, and so X(θ) is always nonsingular. (There is, of ourse, no neessity for X to have any periodiity property!) X(2π) is alled the onodroy atrix, and solutions of the fundaental algebrai eigenproble X(2π)y = λy lead to the following three possibilities for solutions of the differential eigenproble with either p Y n + or p Y n. ṗ(θ)+a(θ)p(θ) =µp(θ),

16 FLOQUET THEORY For real λ>, we ay define µ R by λ =e 2πµ and set so p Y n + and p(θ) e µθ X(θ)y, ṗ(θ)+a(θ)p(θ) =µp(θ). 2. For real λ<, we ay define µ R by λ =e 2πµ and set so p Y n and p(θ) e µθ X(θ)y, ṗ(θ)+a(θ)p(θ) =µp(θ). 3. For a oplex onjugate pair λ R ± iλ I and y R ± iy I, so that [ ] λ X(2π)[y R, y I ]=[y R, y I ] R λ I, λ I λ R we an do either of the following. (We desribe below how to ake a sensible hoie!) Define µ µ R +iµ I C by λ =e 2πµ and set [ ] [p R (θ), p I (θ)] e µ Rθ os µi θ sin µ X(θ)[y R, y I ] I θ. sin µ I θ os µ I θ Then p R, p I Y n +, with p R () = y R and p I () = y I, and d dθ [p R(θ), p I (θ)] + A(θ)[p R (θ), p I (θ)] [ =[p R (θ), p I (θ)] µ R µ I µ I µ R ]. Define µ µ R +iµ I C by λ =e 2πµ and again set [ [p R (θ), p I (θ)] e µ Rθ os µi θ sin µ X(θ)[y R, y I ] I θ sin µ I θ os µ I θ ]. Now p R, p I Y n, but we still have p R () = y R and p I () = y I, and d dθ [p R(θ), p I (θ)] + A(θ)[p R (θ), p I (θ)] [ =[p R (θ), p I (θ)] µ R µ I µ I µ R The eigenvalues λ of X(2π) are alled the Floquet ultipliers for (4.2), while the orresponding µ are alled the Floquet exponents. (This is not quite the standard terinology but is ertainly what we require for a pratial algorith!) Note that eah µ, and hene also the orresponding p(θ), is not uniquely defined by the above onstrution; i.e., for any l Z we ay set µ µ +il and p(θ) p(θ)e ilθ. We shall always hoose the size of the iaginary parts of the Floquet exponents to be as sall ].

17 2538 GERALD MOORE I(λ) η Re(λ) Fig.. Splitting of the oplex plane for Floquet ultipliers. as possible in odulus. In onlusion, if the Floquet ultipliers and exponents are denoted by λ λ e iθ and µ µ R +iµ I, respetively, with π <θ π, then the following table gives the appings between the for Y + and Y. Y + λ =e 2πµ µ R = 2π ln λ, µ Y λ =e 2πµ µ R = 2π ln λ, µ I = I = θ { 2π θ π 2π, θ >, θ+π 2π, θ < Instead of onsidering individual eigenvalues and eigenvetors for the onodroy atrix X(2π), the above arguent an be better applied to well-onditioned invariant subspaes. Thus we hoose soe <η<π, suh that no eigenvalue λ of X(2π) has arg λ = η, and split the spetru of X(2π) into two parts (see Figure ), i.e., X(2π)Y + = Y + Λ + and X(2π)Y = Y Λ, where, for soe n ± with n + + n = n, Y + R n n+,λ + R n+ n+, Y R n n, Λ R n n. Here the oluns of Y + span the invariant subspae of X(2π) orresponding to all the eigenvalues λ of X(2π) satisfying η < arg λ < η, while the oluns of Y span the invariant subspae of X(2π) orresponding to all the eigenvalues λ of X(2π) satisfying η π < arg[ λ] < π η. Therefore, defining E + R n+ n+ and P + (θ) R n n+ by Λ + =e 2πE+, P + (θ) =X(θ)Y + e E+θ and E R n n and P (θ) R n n by we finally have Λ =e 2πE, P (θ) =X(θ)Y e E θ, (4.3) Ṗ(θ)+A(θ)P(θ) =P(θ)E,

18 FLOQUET THEORY 2539 where P(θ) =[P + (θ), P (θ)] and ( ) E+ O E =. O E Eah olun of P + (θ) isiny n + and eah olun of P (θ) isiny n, but the n n atrix P(θ) ust be nonsingular for all θ. The hoie of η is not ritial but, in order for our linear algebra probles to be uniforly well-posed, we shall keep η away fro and π, so that the oduli of the iaginary parts of the eigenvalues of E ± are all less than 2, let η orrespond to a gap in the arguents of the Floquet ultipliers, so that the su of the largest iaginary part of an eigenvalue fro E + with the largest iaginary part of an eigenvalue fro E stays below 2. We shall see later that it is easy to adapt η within a ontinuation fraework for periodi orbits. The above differential equation for P, i.e., (4.3), should be regarded as an eigenproble. This akes it lear that there is an indeterinay in the hoie of P(θ) and E, although the eigenvalues of E are invariants. Thus, for any orthogonal atrix Q R n n,wehave d dθ [P(θ)Q]+A(θ)[P(θ)Q] =[P(θ)Q] [ Q T EQ ], and therefore we an hoose ( ) Q+ O Q, O Q + R n+ n+, Q R n n, Q so that the transforations P(θ) P(θ)Q, E Q T EQ ean that we an assue E + and E are in real Shur (i.e., quasi-upper triangular) for. It would also be possible to ensure that 4π P(θ) T P(θ)dθ = I, 4π and so (4.3) ould be regarded as a dynai Shur fatorization; but this is of ore theoretial than pratial interest. If we now return to the proble of solving (4.), then we an use the hange-ofvariable v(θ) =P(θ)w(θ) P(θ) ( ) w+ (θ), w (θ) where w + Y n+ + and w Y n, and obtain { Ṗ(θ)+A(θ)P(θ) } P(θ)ẇ(θ)+ w(θ) =f(θ).

19 254 GERALD MOORE This siplifies to ẇ(θ)+ew(θ) =P(θ) f(θ), and so we erely have to solve (4.4) where ẇ + (θ)+e + w + (θ) =g + (θ), ẇ (θ)+e w (θ) =g (θ), P(θ)g(θ) P(θ) ( ) g+ (θ) = f(θ) g (θ) and thus g + Y n+ + and g Y n ; i.e., we have redued the proble of solving (4.) to the sipler proble of solving (2.) and (2.3). Equation (4.4) is nonsingular so long as E + has no eigenvalues of the for i for Z and E has no eigenvalues of the for ( 2 )i for Z. Hene, by our onstrution of E + and E, singularity an only our when is an eigenvalue of E +, i.e., there is a Floquet ultiplier equal to. 4.. Appliation to periodi orbits. Just as for stationary points, it is iportant to distinguish between regular periodi orbits and singular ones. For stationary points x, we have only to look at the Jaobian atrix J(x ): however, for a periodi orbit u (t), with period 2πT, we ust onsider whether a linear differential equation has any nontrivial solutions. Looking at (.4), we see that (v (θ),t ) satisfies (4.5) v(θ) =T F (v(θ)), v Y n +, 2π v (θ) v(θ)dθ =. Hene we onsider whether the linearization of this equation at the solution (v,t ), i.e., (4.6) v(θ)+t J(v (θ))v(θ)+t F (v (θ)) =, v Y n +, 2π v (θ) v(θ)dθ =, has any nonzero solutions (v(θ),t). We investigate (4.6) by applying the above Floquet theory to the linear periodi differential operator (4.7) v(θ)+a (θ)v(θ), v Y n +, where A (θ) T J(v (θ)); i.e., there exist n n atries P (θ) = [ P +(θ), P (θ) ] ( ) and E E = + O O so that the hange-of-variable v(θ) =P (θ)w(θ) P (θ) ( ) w+ (θ) w (θ) E

20 FLOQUET THEORY 254 transfors (4.6) to Sine we know that ẇ(θ)+e w(θ)+t P (θ) F (v (θ)) =, w ± Y n± ±, 2π v (θ) [P (θ)w(θ)] dθ =. v(θ)+t J(v (θ))v(θ) =, v Y n +, has the nontrivial solution v(θ) = v (θ), we an hoose the first olun of P (θ) to be a noralization of v (θ), i.e., α P (θ)e T v (θ) for soe nonzero α R; hene, E + is in quasi-upper triangular for with zero first olun and P (θ) F (v (θ)) = T P (θ) v (θ) = α e. Thus to answer our question about (4.6), we only have to deterine whether the uh sipler proble (4.8) ẇ + (θ)+e +w + (θ)+tα e =, w + Y n+ ẇ (θ)+e w (θ) =, w Y n, 2π [P (θ)e ] [P (θ)w(θ)] dθ = +, has any nontrivial solutions (w(θ),t). Sine, by our onstrution of E ±, it is only possible for a nontrivial solution to appear in the onstant Fourier ode for w +, this leads us to the following key definition. Definition. u : R R n is alled a nonsingular or regular periodi orbit of (.2) if zero is a siple eigenvalue of E +. The justifiation for this definition is that if zero is a siple eigenvalue of E +, with orresponding right and left eigenvetors ϕ and ψ say, then we know that the onditions [23] 2π ψ e, [P (θ)e ] [P (θ)ϕ ]dθ are both neessary and suffiient for (4.8) to have only the zero solution. However, sine ϕ e, both these onditions are iediately satisfied. Siilarly, if zero is not a siple eigenvalue of E +, then it is siple to hek that (4.8) does have nontrivial solutions. Hene the basis of the above definition is the following onlusion. Conlusion. A neessary and suffiient ondition for (4.6) to have only the trivial solution (v,t)=(, ) is that zero is a siple eigenvalue of the atrix E +. Of ourse, this is in oplete agreeent with the standard definition of a nonsingular periodi orbit, i.e., that is a siple Floquet ultiplier for (4.7) [, 2].

21 2542 GERALD MOORE Sine, however, our algorith in the next setion works expliitly with E +, the above definition is ore appropriate for us. Finally, it is also lear that, when using Floquet theory and working with w(θ) instead of v(θ), we an siplify our phase ondition in (4.5); i.e., the seond equation there an be replaed by v (θ) [P (θ)p (θ) T ] v(θ)dθ =, 2π whih is equivalent to the final equation of (4.8) being replaed by (4.9) 2π 2π e w + (θ)dθ =. (As we shall see in the next setion, the 2π is a onvenient noralization.) It is this phase ondition that we shall use as part of our ontinuation algorith in the next setion, whih onnets with the fat that the bordered atrix [ ] E + α e e T is invertible for a nonsingular periodi orbit. 5. Continuation of periodi orbits. Fro our Floquet point of view, we desribe a standard strategy for following a urve of periodi orbits for (.), oonly alled pseudo-arlength [23]. Thus we assue that the equation (5.) v(θ) =T F (v(θ),λ), v Y n +, has a solution v (θ) with period 2πT at λ = λ. We also assue that the linearization where has invariant subspaes defined by d dθ + A (θ), A (θ) T J(v (θ),λ ), Ṗ (θ)+a (θ)p (θ) =P (θ)e, where P (θ) [ P +(θ), P (θ) ] with eah olun of the n n + atrix P + in Y+ n and eah olun of the n n atrix P in Y ; n E is quasi-upper triangular, with [ ] E E + O, O where E + R n+ n+ and E R n n ; the su of the largest iaginary part of an eigenvalue of E + with the largest iaginary part of an eigenvalue of E is less than 2 ; E

22 FLOQUET THEORY 2543 the first olun of P (θ) is a noralization of v (θ) and the first olun of E is zero. If our periodi orbit v (θ) of period 2πT is nonsingular at λ = λ, aording to the definition in setion 4., then the ipliit funtion theore applies. This follows fro onsidering the linearization of (5.) with respet to v(θ),t at (v (θ),t,λ ) and adding the phase ondition (4.9): thus the equation for (v(θ),t) Y n + R, where v(θ)+a (θ)v(θ)+tp (θ) =, P (θ) T e v(θ)dθ = 2π p (θ) T v (θ) =F (v (θ),λ ), has only the zero solution; or equivalently, the equation ẇ + (θ)+e +w + (θ)+α T e =, e w + (θ)dθ =, 2π ẇ (θ)+e w (θ) = for (w + (θ), w (θ),t) Y n+ + Y n R, where α P (θ)e p (θ), has only the zero solution. Hene there is a unique urve of periodi orbits through (v (θ),t ), and this urve is paraetrizable by λ. We do not, however, wish to restrit ourselves to urves of periodi orbits whih are paraetrizable by λ. Thus we onsider the full linearization of (5.) with respet to v(θ),t,λ at (v (θ),t,λ ), and our basi assuption is that the equation (5.2) v(θ)+a (θ)v(θ)+tp (θ)+λk (θ) =, P (θ) T e v(θ)dθ = 2π for (v(θ),t,λ) Y n + R 2, where k (θ) T F λ (v (θ),λ ), has a one-diensional solution set spanned by (v t (θ),t t,λ t ). (This will be noralized in (5.7) below.) So the augented equation (5.3) 2π 2π v(θ)+tf (v(θ),λ)=, P (θ) T e v(θ)dθ =, 2π v t (θ) [v(θ) v (θ)] dθ + T t [T T ]+λ t [λ λ ] ε =

23 2544 GERALD MOORE has the solution [v (θ),t,λ ] for ε = ; and the full linearization of (5.3) at this point gives the equation (5.4) v(θ)+a (θ)v(θ)+tp (θ)+λk (θ) =, P (θ) T e v(θ)dθ =, 2π v t (θ) v(θ)dθ + Tt T + λ t λ =, 2π whih has only the zero solution. Hene, the ipliit funtion theore tells us that (5.3) has a loally unique solution for all ε suffiiently sall, and this gives us a urve of periodi orbits [v(θ),t,λ] paraetrized by ε. (v,t,λ ) (v n,t n,λ n) Newton orretion Predited point on tangent line Fig. 2. Continuing a urve of periodi orbits. Geoetrially, our basi assuption on (5.2) is saying that the urve of periodi orbits has a unique tangent line. Thus, in setion 5., we use our Floquet transforation to opute a Fourier approxiation to this tangent line. Then, in setion 5.2, we solve (5.3) using a Newton-hord iteration whose starting value is a point on this tangent line; see Figure 2. By using a siplified Newton s ethod, whih keeps the linearization fixed, we sarifie the quadrati onvergene of the full Newton s ethod; this is ore than opensated, however, by only having to apply our Floquet theory at (v (θ),t,λ ). Thus we have an effiient algorith for oputing the Fourier approxiation of a new point, (v n(θ),t n,λ n) say, on the urve of periodi orbits. Having obtained this new point, we require the new Floquet variables P n(θ), E n there; i.e., we ust effiiently update fro P (θ), E to P n(θ), E n. The algorith for obtaining a Fourier approxiation of P n(θ), E n is desribed in setion 5.3. Finally, in setion 5.4 we explain how the ruial bound on the size of the iaginary parts of the Floquet exponents is aintained, while in setion 5.5 we show how to start the ontinuation ethod at a Hopf point. 5.. Tangent preditor. First, we onsider what the assuption (5.2) eans in ters of our Floquet variables; i.e., if we set v(θ) P +(θ)w + (θ)+p (θ)w (θ),

24 FLOQUET THEORY 2545 then we are assuing that (5.5) where ẇ + (θ)+e +w + (θ)+α T e + λk +(θ) =, e w + (θ)dθ =, 2π ẇ (θ)+e w (θ)+λk (θ) =, P (θ) [ ] k + (θ) k k (θ), (θ) has a one-diensional solution spae. It is lear that this assuption rests on the equation for the onstant ode of w + (θ); i.e., that the (n + +) (n + + 2) oeffiient atrix [ E + α e k ] e T, where k 2π has full rank. Using the bordered atrix (5.6) 2π [ E + α e e T k +(θ)dθ, this an arise in three different ways. If (5.6) is nonsingular, then v (θ) is a nonsingular periodi orbit as desribed in setion 4.. If (5.6) has rank n +, with E + having rank n +, then this orresponds to zero being an eigenvalue of E + of geoetri ultipliity one and algebrai ultipliity greater than one; in addition k is not in the range of E +. (Generially, we would expet this zero eigenvalue of E + to have algebrai ultipliity two [5, 2].) If (5.6) has rank n +, with E + having rank n + 2, then this orresponds to zero being an eigenvalue of E + of geoetri ultipliity two, but e is not in the range of E +; in addition k is not in the range of E + and not parallel to e. (Generially, we would not expet these onditions to appear [2].) This analysis also shows that a suitable noralization for the solution of (5.2) or (5.5) is ], (5.7) where ˆt R n+ ˆt 2 +[T t ] 2 +[λ t ] 2 =, is the onstant ode of w + (θ); i.e., ˆt 2π 2π w + (θ)dθ. We shall also require t R n later, whih is just ˆt padded out with zeroes.

25 2546 GERALD MOORE give us (5.8) Disretizing (5.5), our Fourier approxiations w + (θ) w + k +(θ) k + M { w os θ + w s sin θ }, = M { k os θ + k s sin θ } = ( E + α e k e T ) w Tt = λ t ( ) for = and ( )( ) ( ) E + I w I E + w s = λ k (5.9) t k s for =,...,M; siilarly, our Fourier approxiations give us (5.) w (θ) k (θ) M { w os [ 2 ]θ + ws sin [ 2 ]θ}, = M { k os [ 2 ]θ + ks sin [ 2 ]θ} = ( E [ 2 ]I )( ) w [ 2 ]I E w s ( ) = λ k t k s for =,...,M. The one-diensional null-spae for the full-rank strutured atrix in (5.8) ay be effiiently obtained by eploying algoriths fro [7, Chap. 5]. First, we onstrut the n (n + ) quasi-upper triangular atrix ( ) (5.) α e Ê + k, where Ê + is obtained fro E + by reoving the zero first olun. Seond, Givens rotations are used to annihilate the nonzero eleents below the diagonal and thus hange (5.) into an n (n + ) upper triangular atrix. Third, post-ultipliation with Householder atries is used to annihilate the final olun of (5.). We an then solve (5.9) and (5.), sine the restrition on the size of the iaginary parts of the eigenvalues of E ± akes these systes nonsingular Newton orretion. Our Newton orretion will solve the syste v(θ)+tf (v(θ),λ)=, P (θ) T e v(θ)dθ =, 2π (5.2) P (θ) T t [v(θ) v (θ)] dθ 2π + Tt [T T ]+λ t [λ λ ] ε =.

26 FLOQUET THEORY 2547 Thus, for the sae reason as in (4.9), we have replaed the final equation for the tangent line step in (5.3) by the ore onvenient equation above. Our Newton-hord iteration for solving (5.2) is defined fro the starting values v () (θ) =v (θ)+εv t (θ), T () = T + εtt, λ () = λ + ελ t for hosen sall ε, and onsists of v (k+) (θ) =v (k) (θ)+z(θ), T (k+) = T (k) + δt, λ (k+) = λ (k) + δλ, where (z(θ),δt,δλ) Y+ R n 2 satisfy ż(θ)+a (θ)z(θ)+δt p (θ)+δλk (θ) =r (k) (θ), P (θ) T e z(θ)dθ =, (5.3) 2π P (θ) T t z(θ)dθ + Tt δt + λ t δλ =, 2π with r (k) (θ) v (k) (θ) T (k) F (v (k) (θ),λ (k) ). Hene, under the Floquet transforations z(θ) P +(θ)w + (θ)+p (θ)w (θ), r (k) (θ) P +(θ)f + (θ)+p (θ)f (θ), our equation in Y n+ + beoes ẇ + (θ)+e +w + (θ)+δt α e + δλk +(θ) =f + (θ), e w + (θ)dθ =, 2π ˆt w + (θ)dθ + Tt δt + λ t δλ =, 2π and our equation in Y n is ẇ (θ)+e w (θ)+δλk (θ) =f (θ). Hene our Fourier approxiation in Y n+ + is w + (θ) w + f + (θ) f + k +(θ) k + M { w os θ + w s sin θ }, = M { f os θ + f s sin θ }, = M { k os θ + k s sin θ }, =

27 2548 GERALD MOORE leading to (5.4) for = and (5.5) E + α e k e T w δt = f (ˆt ) T Tt λ t δλ ( )( ) E + I w I E + w s ( f = δλk ) f s δλk s for =,...,M; while our Fourier approxiation in Y n is M { w (θ) w os [ 2 ]θ + ws sin [ 2 ]θ}, = M { f (θ) f os [ 2 ]θ + f s sin [ 2 ]θ}, = M k { (θ) k os [ 2 ]θ + ks sin [ 2 ]θ}, = giving us (5.6) ( E [ 2 ]I )( ) w [ 2 ]I E w s ( f = δλk ) f s δλk s for =,...,M. By our onstrution of (ˆt,Tt,λ t ), the oeffiient atrix in (5.4) is nonsingular. Our linear algebra at the end of setion 5. also eans that we an solve (5.4) effiiently. We an then solve (5.5) and (5.6), sine the restrition on the size of the iaginary parts of the eigenvalues of E ± akes these systes nonsingular Floquet ontinuation. After having alulated v n(θ), Tn, and λ n, we need to update fro (5.7) Ṗ (θ)+a (θ)p (θ) =P (θ)e to (5.8) Ṗ n(θ)+a n(θ)p n(θ) =P n(θ)e n, where A n(θ) TnJ(v n(θ),λ n). We seek P n(θ) in the Floquet for P n(θ) P (θ)p(θ), with the noralization (5.9) 2π 2π P(θ)dθ = I.

28 FLOQUET THEORY 2549 Hene we an apply a Newton-hord ethod to solve (5.8) for P(θ) and E n, analogous to that desribed theoretially in [28] and pratially in [7], and ake use of our known Floquet variables for (5.7). Thus our Newton iteration starts fro P () (θ) I and E () E and oputes fro P (k+) (θ) =P (k) (θ)+z(θ), E (k+) = E (k) + δe (5.2) Ż(θ)+E Z(θ) Z(θ)E δe = R (k) (θ), where { d [ ] } R (k) (θ) P (θ) P (θ)p (k) (θ) A dθ n(θ)p (θ)p (k) (θ)+p (θ)p (k) (θ)e (k). We now ake use of the deopositions [ ] [ ] E E + O δe+ O, δe O O δe E and the fat that P (θ) [ P +(θ) P (θ) ] leads to [ ] Z++ (θ) Z Z(θ) + (θ) Z + (θ) Z (θ) and R (k) (θ) [ R (k) ++(θ) R (k) +(θ) ] R (k) + (θ) R (k). (θ) Here Z ++ (θ) R n+ n+ and Z (θ) R n n have oponents in Y +, while Z + (θ) R n+ n and Z + (θ) R n n+ have oponents in Y, and siilarly for the deoposition of R (k) (θ). Hene we an deopose (5.2) into two equations with oponents in Y +, (5.2) and Ż++(θ)+E +Z ++ (θ) Z ++ (θ)e + δe + = R (k) ++(θ) (5.22) Ż (θ)+e Z (θ) Z (θ)e δe = R (k) (θ), and two equations with oponents in Y, (5.23) and Ż+ (θ)+e +Z + (θ) Z + (θ)e = R (k) + (θ) (5.24) Ż +(θ)+e Z + (θ) Z + (θ)e + = R (k) +(θ). Eah of (5.2), (5.22), (5.23), and (5.24) an be replaed by the analogous Fourier approxiation, whih leads to ode-deoupling as we see below. We then obtain Sylvester equations [7], whih an be solved by the Bartels Stewart algorith [4, 7]. Most of the work in this algorith is devoted to reduing the appropriate atries to Shur for, but here E ± already have this for! Otherwise, only bak-substitutions are required. Of ourse the produt to for R (k) (θ) in (5.2) is arried out in pointspae.

29 255 GERALD MOORE To obtain Z ++ (θ), we set Z ++ (θ) and so (5.2) gives us M {Z os θ + Z s sin θ}, = R (k) ++(θ) R + M {R os θ + R s sin θ}, = (5.25) for = and (5.26) [ ][ ] E + I Z I E + δe + = R Z s [ ] Z Z s E + = [ ] R R s for =,...,M. The Sylvester equations in (5.26) are nonsingular, beause the restrition on the size of the iaginary parts of the eigenvalues of E + eans that [ ] E + I I E + and E + have no oon eigenvalue. To obtain Z (θ), we set Z (θ) and so (5.22) gives us M {Z os θ + Z s sin θ}, = R (k) (θ) R + M {R os θ + R s sin θ}, = (5.27) for = and (5.28) [ ][ ] E I Z I E δe = R Z s [ ] Z Z s E = [ ] R R s for =,...,M. The Sylvester equations in (5.28) are nonsingular, beause the restrition on the size of the iaginary parts of the eigenvalues of E eans that [ ] E I and E I have no oon eigenvalue. E

30 To obtain Z + (θ), we set Z + (θ) R (k) + (θ) FLOQUET THEORY 255 M { Z os [ 2 ]θ + Zs sin [ 2 ]θ}, = M { R os [ 2 ]θ + Rs sin [ 2 ]θ}, = and so (5.23) gives us [ E + [ 2 ]I ][ ] [ ] [ ] Z Z [ 2 ]I E + Z s Z s E R (5.29) = R s for =,...,M. The Sylvester equations in (5.29) are nonsingular, beause of the restrition on the size of the iaginary parts of the eigenvalues of E + and E eans that [ E + [ 2 ]I ] [ 2 ]I and E E + have no oon eigenvalue. To obtain Z + (θ), we set Z + (θ) R (k) +(θ) M { Z os [ 2 ]θ + Zs sin [ 2 ]θ}, = M { R os [ 2 ]θ + Rs sin [ 2 ]θ}, = and so (5.24) gives us [ E [ 2 ]I ][ ] [ ] [ ] Z Z [ 2 ]I E Z s Z s E R (5.3) + = R s for =,...,M. The Sylvester equations in (5.3) are nonsingular, beause the restrition on the size of the iaginary parts of the eigenvalues of E and E + eans that [ E [ 2 ]I ] [ 2 ]I and E E + have no oon eigenvalue. Finally, we note that our liit [ E E (k) n li k O + O will not generally be quasi-upper triangular, and so we will have to perfor a last Shur fatorization with ( ) Q+ O Q, Q O + R n+ n+, Q R n n, Q E (k) ]

31 2552 GERALD MOORE so that the transforations P n(θ) P n(θ)q, E n Q T E nq ean that the diagonal bloks of E n are now in real Shur for Controlling the Floquet exponents. On page 2539 we stated the onditions on our Floquet exponents, the eigenvalues of [ ] E+ O E, O that ust be aintained during the ontinuation proess. Basially, this eans that the su of any iaginary part of an eigenvalue of E + with any iaginary part of an eigenvalue of E ust be less than 2. We now show what to do if this ondition is found either to fail or to be dangerously lose to failing at the end of a ontinuation step. Suppose we have E Ṗ(θ)+A(θ)P(θ) =P(θ)E, with [ ] E+ O P(θ) [P + (θ), P (θ)] and E. O E If the iaginary parts of a pair of oplex onjugate eigenvalues of E are too large, we ould [7] ove the to the top left of E, blok-diagonalize E (altering P (θ) in onsequene), so that now E has the for α β β 2 α E = , where β,β 2 are positive. Now, if we denote the first two oluns of P (θ) byp (θ) and p 2 (θ), then transforing the by [ p (θ) p 2 (θ) ] [ p (θ) p 2 (θ) ] [ β os 2 θ β sin 2 θ ] β 2 sin 2 θ β2 os 2 θ will transfor the leading 2 2 blok of E into [ ] α β β 2 2 2, β β 2 α

32 FLOQUET THEORY 2553 thus dereasing the size of the iaginary parts of the eigenvalues by 2. Now the new p (θ), p 2 (θ) are in Y+, n and so we inrease n + by 2 and derease n by 2. Siilarly, if the iaginary parts of a pair of oplex onjugate eigenvalues of E + are too large, we arry out the analogous proedure to transfer the to E ; e.g., oving the to the botto right of E +, blok-diagonalizing E +, transforing the final two oluns of P + (θ), so that the size of the iaginary parts of the dangerous eigenvalues is dereased by 2. We have desribed above the siplest situations, where there is only a single pair of dangerous eigenvalues. We oit the obvious extensions, where a larger blok of eigenvalues has to be ontrolled Starting at a Hopf point. Finally, we show how E and P (θ) aybe onstruted at a Hopf bifuration point in order to start the ontinuation proess; so (x,λ ) is a Hopf bifuration point for (.), satisfying the onditions at the beginning of setion 3. Hene we have a Shur fatorization for. Blok-diagonalize A to obtain A Q = Q U A T J(x,λ ). A S = S D, where D is the n n blok-diagonal atrix [ ] D U + O O with quasi-upper triangular U + R n+ n+ and U R n n. In fat eah of U + and U is itself blok-diagonal with U U U + U U and U U 2 U 3 2 U 5 2, so that the eigenvalues of U j,j, have an iaginary part lose to ±j and the eigenvalues of U j,j, have an iaginary part lose to ± j 2 2.

33 2554 GERALD MOORE 2. If [ α β ] β 2 α isa2 2 diagonal blok of Uj or U j for j, with β,β 2 positive 2 and s l, s l+ denoting the orresponding oluns of S, then we an transfor these oluns by Thus we obtain s l β s l β 2 s l+, s l+ β s l + β 2 s l+. A S = S D, where ] [Ũ D + O O Ũ with U Ũ Ũ + Ũ and Ũ Ũ 2 Ũ 3 2 Ũ 5 2, so that the 2 2 diagonal bloks of Ũj, Ũ j,j, have the for [ α ββ 2 ] 2 β β 2 α. 3. Finally, if [ α ββ 2 ] β β 2 α is now a 2 2 diagonal blok of Ũ j,j, with orresponding oluns s l, s l+ of S, then P +(θ) R n n+ is obtained by replaing these oluns with s l os jθ s l+ sin jθ and s l sin jθ + s l+ os jθ. Siilarly, if [ α ββ 2 ] β β 2 α is now a 2 2 diagonal blok of Ũ j,j, with 2 orresponding oluns s l, s l+ of S, then P (θ) R n n is obtained by replaing these oluns with s l os jθ/2 s l+ sin jθ/2 and s l sin jθ/2+ s l+ os jθ/2. Hene the oluns of P + are in Y n + and the oluns of P are in Y n, with P (θ) [ P +(θ), P (θ) ] satisfying Ṗ (θ)+a (θ)p (θ) =P (θ)e.

34 FLOQUET THEORY Spetral deay of zero eigenvalue Size of eigenvalue N Fig. 3. Approxiating the zero Floquet exponent. Here [ ] E E + O O E is onstruted fro Ũ + and Ũ. The iaginary parts of the eigenvalues of E are lose to zero. 6. Nuerial results. Now we illustrate the above algoriths with soe wellknown exaples. 6.. The Lorenz equations. ẋ = σ(y x), ẏ = λx y xz, ż = xy bz. For σ>b+ there is a subritial Hopf bifuration fro the stationary solution urves ( ± b(λ ), ± ) b(λ ),λ, λ >, at λ H σ(σ + b +3) σ b. We use the paraeter values (σ, b) = (, 8 3 ), whih gives λ H 24.74, and follow the branh of periodi orbits in the range λ H λ 24. For N 2 M +, where M is the nuber of Fourier odes, we plot in Figure 3 the axiu size of the sallest Floquet exponent (whih should be zero) over this range of λ. The exponential deay is lear. For this exaple, it is neessary only to work in Y +.

35 2556 GERALD MOORE 8 7 λ=.245 λ=.223 λ=.26 λ=.93 6 x x Fig. 4. Approxiating four periodi orbits The A B reation equations. ẋ = x + λ( x ) exp x 2, ẋ 2 = x 2 + λa( x ) exp x 2 bx 2. This syste is used as an exaple in [4], in partiular for the paraeter values a = 4 and b = 2. In this ase, the stationary solution urve through (x,λ)=(, ) has two turning points as λ inreases, and then there is a Hopf bifuration point at λ.39, x.895, x The urve of periodi orbits reated here exists, with λ dereasing, until λ.55, where it ends in a hoolini orbit onneted to the stationary solution urve. Again it is only neessary to work in Y +. We use this exaple to illustrate how our algorith ay perfor badly when applied to periodi orbits whih lak soothness or are poorly onditioned. In Figure 4, we show approxiations to periodi orbits at four different values of λ, using the algorith in setion 5 with M = 5. We see osillations appearing in the approxiations as they try to ope with the lak of soothness developing as x =(, ) is approahed. As is shown in [4], this point is reahed at λ.55, when the hoolini orbit appears. (If results for saller λ were shown, the osillations would be ore violent.) We exhibit this lak of soothness differently in Figure 5, where the size of the Fourier odes is shown for =,...,M. It is lear that, as λ dereases, the exponential deay of these odes is gradually being lost. Siilar onlusions an be drawn when we plot the zero Floquet exponent against λ in Figure 6. By the tie λ.93 is reahed, the two exponents are alost equal! Finally, and ost tellingly, we exaine ond (P (θ)) in Figure 7. Theoretially P (θ) an never be singular, and its ondition nuber is a easure of the onditioning of the boundary value proble defining the periodi orbit. Here, however, we see ond (P (θ)) reahing 8! It is therefore no surprise that our algorith, whih expliitly works with P (θ), has diffiulties in this situation. Colloation with pieewise-polynoials and an adaptive esh, as used in [4], will obviously perfor better in suh ases.