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Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers andInformation in Engineering Conference

IDETC/CIE 2007September 4-7, 2007, Las Vegas, Nevada, USA

DETC2007-34583

CENTER MANIFOLD REDUCTION OF PERIODIC DELAY DIFFERENTIAL SYSTEMS

Eric A. ButcherDepartment of Mechanical Engineering

New Mexico State UniversityLas Cruces, NM 88003

Venkatesh DeshmukhDepartment of Mechanical Engineering

Villanova UniversityVillanova, PA 19085-1681

Ed BuelerDepartment of Mathematics

University of AlaskaFairbanks

Fairbanks, AK 99775

ABSTRACT A technique for center manifold reduction of nonlineardelay differential equations (DDEs) with time-periodiccoefficients is presented. Perturbation expansion converts thenonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs)with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbationparameter. Each ODE is solved by a Chebyshev spectralcollocation method. Thus we compute a finite approximationto the nonlinear infinite-dimensional map for the DDE. Centermanifold reduction on the map is then carried out. Centermanifold reduction is illustrated via a single invertedpendulum including both a periodic retarded follower forceand a nonlinear restoring force. In this example, the amplitudeof the limit cycle associated with a flip bifurcation is foundanalytically and compared to that obtained from directnumerical simulation.

INTRODUCTION Dimensional reduction of nonlinear DDEs has beenconsidered by researchers in the past using differentapproaches. A center manifold algorithm for constantcoefficient DDEs near Hopf bifurcation points was firstformulated in [1,2]. Unlike the case for center manifoldreduction of ODEs [3], the algorithm for DDEs is necessarilystated in the language of functional analysis and requires the

description of the adjoint system. The method was firstapplied to a practical system (Hopf bifurcation in machine toolvibrations) much later [4,5]. Recently, alternative techniquesfor dimensional reduction of nonlinear DDEs have beenproposed. These include the use of stiff and soft substructures[6], the method of multiple scales [7], and a Galerkinprojection technique which reduces a DDE to a small numberof ODEs [8]. It should be noted that all of these techniques aredesigned for autonomous nonlinear DDEs. We note that thenumerical software available for nonlinear DDEs includes theMATLAB packages: DDE23 [9], a time series integrator, thenumerical bifurcation analysis tool DDE-BIFTOOL [10], andPDDE-CONT [35], a new continuation and bifurcationsoftware for DDEs. The study of nonlinear time-periodic DDEs has includedthe computation of the response [11,12], the use of normalforms [13], and continuation algorithms [14]. Much of thepractical work in the engineering community has focused onthe milling problem which, unlike the case of turning, ismodeled by DDEs with time-periodic coefficients. It has beenobserved in numerous studies that, in addition to the secondaryHopf or Neimark-Sacker bifurcation, such a system can alsoexhibit flip or period-doubling bifurcations. The centermanifold algorithms in [1,2,4,5] cannot be directly applied tothe time-periodic case without additional complexitiesassociated with redefining the adjoint equation. Nevertheless acenter manifold computation for the case of flip bifurcation in

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single DOF low immersion milling was carried out in [15,16]by assuming that the system can be accurately modeled using a2-dimensional map of the cutting tool’s position and velocityfrom one instantaneous cut to the next. For such a system, thecoefficients are essentially periodically-spaced Dirac deltafunctions. The problem of milling with a finite cutting time,on the other hand, requires a calculation using functionalanalytic tools to decompose the dynamics on the centermanifold [17]. This algorithm is based on previous efforts inthe mathematical literature to generalize the adjoint equationand formalize the center manifold theory for the case ofperiodic DDEs [18-21]. The motivation for the present paper isin a possible alternative method which does not involve theexplicit computation of the adjoint. The method here for dimensional or center manifoldreduction of periodic DDEs makes use of a nonlinear extensionof the infinite dimensional monodromy operator defined in[22] for linear periodic DDEs. The nonlinear operator mapsthe initial function onto each subsequent delay interval.Iteration of the nonlinear operator produces the solution viathe ‘method of steps’. The linear part of this operator has beenpreviously approximated by a finite dimensional matrix usinga variety of numerical techniques, including orthogonalpolynomial expansion [23,24] and collocation [25-28] for thepurpose of stability analysis. In this paper we use the methodof Chebyshev spectral collocation which is discussed at lengthby Trefethen [29] and was successfully applied to study thestability of linear periodic DDEs in [27,28]. It is assumed that the periodically modulated nonlinearitiesare multiplied by a perturbation parameter. Using a classicalperturbation technique, the solution in any interval isexpanded using the solution from the previous interval in thein the zeroth order part. The expansion proceeds in a usualway by powers of perturbation parameter. Collecting terms forlike powers of the perturbation parameter leads to a series oflinear non-homogenous ordinary differential equations whichare then solved using Chebyshev spectral collocation.Quadratic and cubic nonlinearities result in similarnonlinearities in the resulting approximate nonlinear DDEsolution map. Increased accuracy is obtained by increasing thenumber of collocation points and by adding more terms to theperturbation expansion. Next, center manifold reduction isapplied to the nonlinear map. An algorithm that performscenter manifold reduction on large nonlinear maps inMATLAB© is developed for the purpose of making theproblem computationally tractable versus the alternativeapproach of carrying out the calculations in MATHEMATICA,for example, which was found to be intractable. A singleinverted pendulum with a periodic retarded follower force andnonlinear restoring force is used as an illustrative example inwhich the amplitude of the limit cycle associated with a flipbifurcation is found analytically and compared to that obtainedfrom direct numerical simulation using DDE23. The resultsshow an excellent match between the two.

DEVELOPMENT OF NONLINEAR MAP Consider a time periodic nonlinear DDE given by

]0,[)()()),(),(()()()(

τφτετ

−∈∀=−+−+=

sssxttxtxftxtBxtAx&

(1)

where )(tx is an n -dimensional state vector and)()( TtAtA += , )()( TtBtB += are nn× coefficient

matrices with principal period T . Here τ is a fixed delay andε is the perturbation parameter. The nonlinear term

)),(),((),,( ttxtxftf τβα −= is a vector-valuedfunction, with n components, which is periodic with period Tin its third input. For the purpose of simplicity, it is assumedthat the delay τ is the same as the principal period T .Writing equation (1) with the summation convention anddenoting )()( ττ −= txtx yields

),,()()( txxfxtBxtAx ijj

ijj

iiττ ε++=& (2-

a)

We now assume that ),,( tf βα is a polynomial of degree 3in α and β . That is, we may write (2-a) in the form

( ) ( ) [ ( ) ( )

( ) ] [ ( ) ( )

( ) ( ) ]

j j jk jki i j i j i j k i j k

jk jkl jkli j k i j k l i j k l

jkl jkli j k l i j k l

x A t x B t x C t x x D t x x

F t x x G t x x x H t x x x

J t x x x K t x x x

τ τ τ

τ τ

τ τ τ τ τ

ε

ε

= + + +

+ + +

+ +

&

(2-b)

For the linear system obtained by setting 0=ε in (2), onemay analyze stability by considering the infinite dimensionalmonodromy operator which maps the solution )()1( tx p− in

],)1[( ττ pp − onto the solution )()( tx p in])1(,[ ττ +pp . The eigenvalues of the monodromy operator,

the Floquet multipliers, determine the stability of the linearsystem. An accurate finite dimensional approximation of themonodromy operator, that is, a “monodromy matrix”, can becomputed using Chebyshev spectral collocation [13-15] andother techniques. For the nonlinear DDE (2-b), we will approximate thecorresponding nonlinear map )()( )()1( txtx pp a− by aperturbation series in ε . Note that (1) typically has parametersbut these are not explicitly included in the analysis for now.Obtaining the nonlinear map in a parameter dependent(“symbolic”) manner would allow for bifurcation analysis ofequation (2-b) at the critical points. Later we re-introduce the



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bifurcation parameter via versal deformation theory as in [16]for the ODE case. Suppose the solution )(tx in the interval ],0[],0[ T=τfor (2-b) is written in the form

....)()()()()()( 33

22

10 +++++= ttttttx ξεξεεξξφ (3)

Here )(tφ is a vector of known functions as in (1) definedover ],0[ τ by translation from ]0,[ τ− . The functions

,..2,1,0),( =itiξ are unknown n -dimensional vectors foreach power of ε . The initial conditions for these unknownvectors are

0 1 2 3(0) ( ) (0), (0) 0, (0) 0, (0) 0,...ξ φ τ φ ξ ξ ξ= − = = = (4)

so that x(0)=φ(τ). Substituting (3) in (2-b) and retaining termsup to third order, the following sequence of linear time-periodic ODEs is obtained after collecting like powers of ε :

ijj

ij

ijj

ii BAA φφξξε && −++= )(: 000 (5-

a)

1

1 1 0 0 0

0 0 0

0 0 0

: ( )( ) ( )

( )( )( )

( )( ) ( )

j jk jk

i i j i j j k k i j j k

jk jkl

i l n i j j k k l l

jkl jkl jkl

i j j k k l i j j k l i j k l

A C F

D G

H J K

ε ξ ξ ξ φ ξ φ ξ φ φ

φ φ ξ φ ξ φ ξ φ

ξ φ ξ φ φ ξ φ φ φ φ φ φ

= + + + + +

+ + + + +

+ + + + + +

&

(5b)

2

2 2 0 1 1 0

1 0 0 1 1 0 0

0 1 0 0 0 1 0

1

: [( ) ( )]

[( ) ( )

( )] [( ) ( ) ]

j jk

i i j i j j k j k k

jk jkl

i j k i j j k l j k k l

jkl

j k l l i j j k l j k k l

jkl

i j k l

A C

F G

H

J

ε ξ ξ ξ φ ξ ξ ξ φ

ξ φ ξ φ ξ ξ ξ ξ φ ξ

ξ ξ ξ φ ξ φ ξ φ ξ ξ φ φ

ξ φ φ

= + + + +

+ + + + +

+ + + + + +

+

&

(6-a)

3

3 3 2 0 1 1 0 2

2 0 0 0 2 0

0 0 2 1 1 0 1 0 1

0 1 1 0 0 2 0

: [ ( ) ( ) ]

[ ( )( ) ( ) ( )

( )( ) ( ) ( )

( ) ] [( )( ) ( ) ]

j jk

i i j i j k k j k j j k

jkl

i j k k l l j j k l l

j j k k l j k l l j k k l

jkl

j j k l i j j k l j k k l

A C

G

H

ε ξ ξ ξ ξ φ ξ ξ ξ φ ξ

ξ ξ φ ξ φ ξ φ ξ ξ φ

ξ φ ξ φ ξ ξ ξ ξ φ ξ ξ φ ξ

ξ φ ξ ξ ξ φ ξ φ ξ ξ φ φ

= + + + + +

+ + + + + + +

+ + + + + + +

+ + + + +

&

2

jkl

i j k lJ ξ φ φ

+

(6b)

We now consider a sequence of infinite-dimensional mapscorresponding to equations (5,6) of the form

0

0

1 1 1 1

1 2 3

2 2 1 2 1 2

2 2 3

3 3 1 2

3 2

3 1 2 3

3

:

: ( , ) ( , ) ( )

: ( , , ( )) ( , , ( )) ( )

: ( , , ( ), ( ))

( , , ( ), ( )) ( )

m Vm

m P m Vm P m Vm P m

m P m Vm P m P m Vm P m P m

m P m Vm P m P m

P m Vm P m P m P m

ξ φ

ξ φ φ φ φ φ

ξ φ φ φ φ φ φ φ

ξ φ φ φ φ

φ φ φ φ φ

ε

ε

ε

ε

=

= + =

= + =

=

+ =

(7)

Here jiP denotes a polynomial of homogeneous degree i in its

j+1 coefficients. Equation (7) is exactly equivalent to (5,6)until one chooses a particular finite representation for theinfinite-dimensional vector wm corresponding to an arbitrary

function )(tw . Using equations (3,7), we express the

solution )()( )( txtx p= in an interval ])1(,[ ττ +pp as anonlinear transformation of the solution in the previousinterval ( 1)( ) ( )pt x tφ −= in ],)1[( ττ pp − as

1 2 2 3 3( ) ( ) ( ) ( ) ( )xm p I V m P m P m P mφ φ φ φ

ε ε ε= + + + + (8)

which is exactly equivalent to equation (1) with theperturbation expansion of (3). In the next section we discussthe numerical construction of this nonlinear map in detailusing the method of Chebyshev spectral collocation, in whichcase the vector wm is represented by a particular finite vectorconsisting of the solution values at the Chebyshev collocationpoints [29]. (Such vectors will be called collocation vectors.)It is important to note that such a finite representation is notunique, and introduces another approximation (besides theperturbation expansion) into the proposed method.

CHEBYSHEV SPECTRAL COLLOCATIONGauss-Chebyshev-Lobatto points, or Chebyshev extreme

points [25], or merely Chebyshev points [29], for brevity, arethe points in the interval ],[ ** τ+tt defined by

* ( )[cos( /( 1)) 1] 0,1, 2, ..., 12

jt t j N j Nτ

π= + − + = − (9)

Note that *110

* ... ttttt N =>>>=+ −τ , a standardordering for this method [29]. For this set of N collocationpoints we also have an N x N Chebyshev spectral differentialmatrix DN [29], obtained by interpolating a polynomialthrough the collocation points, differentiating that polynomial,and then evaluating the resulting polynomial at the collocationpoints [29]. The entries of this matrix are given, for any N ,by formulas in [29].



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Let us approximate the linear part of equation (1) using theChebyshev spectral collocation method. Let )( pmw denote

the collocation vector for the variable )(tw at shifted

Chebyshev collocation points in the thp interval of length τgiven as

TN

TTTw twtwtwpm )](....)()([)( 110 −= (10)

For 0p = , the collocation vector )0(xm of )(tx is given by

substituting )(tφ in the equation (12) for )(tw . Then in the

first interval of time ],[ 11 τ+tt with 01 =t , the linear (ε =0) part of equation (1) can be written as

φφξφξτ mBmAmD += ++ )1()1(00

(11)

The matrix τD is obtained from ND by scaling to account forthe interval shift [-1,1] → [0,T] by multiplying the resultingmatrix by 2/T and then modifying the last n rows as[0 0 ]n n nIL . The patterns of the matrices A , B are

0

2

0

2

( ) 0 .. 00 : :: ( ) :0 0. .. 0( ) 0 .. 0: : :: : ( ) :

0 .. 0

N

N

n

A t

AA t

B t

BB t

I

−

−

= =

(12)

where )(),( ii tBtA are calculated at the thi point on the

interval of length T, In is the identity matrix and n0 is the zeromatrix of dimension nn× . Here the bar above the operatorrefers to the fact that the matrices are modified by altering thelast n rows to account for the matching condition betweensuccessive intervals [27]. Therefore, we get the approximationto the monodromy operator (i.e. the linear stability matrix) as

BADU 1)( −−= τ

(13)

If N is the number of points considered in each interval and nis the order of the original delay differential equation, then thesize of the U matrix will be nNnN × . Proceeding further, since 0 ( ) ( ) ( )t x t tξ φ= − for 0ε inthe perturbation expansion of equation (3), we have

φφξ VmmIUm =−= )()1(0

(14)

For orders 1ε , 2ε and 3ε the spectral collocationrepresentation is obtained as

1

1 1 1 12 3(1) ( ) [ ( ) ( )] ( )m D A m m P mξ τ φ φ φ

−= − Γ + Γ = (15-a)

2

1 2 2 22 3(1) ( ) [ ( ) ( )] ( )m D A m m P mξ τ φ φ φ

−= − Γ + Γ = (15-b)

3

1 3 3 32 3(1) ( ) [ ( ) ( )] ( )m D A m m P mξ τ φ φ φ

−= − Γ + Γ = (15-c)

where )()()( 1 ji

ji PAD −−=Γ τ (comparing with (7)) is an

thi order nonlinear function of the collocation vector φm for

order jε . Now, we can express the collocation vector )( pmx

of the solution ( ) ( )px t (the state vector) at Chebyshev pointsin an interval ])1(,[ ττ +pp as a nonlinear transformation

of the collocation vector )1( −pmx of the solution ( 1) ( )px t−

in ],)1[( ττ−p by equation (8) where )1( −= pmm xφ .

Equation (8) thus becomes an nN-dimensional nonlinear mapwhich, as an extension of the linear monodromy matrix inequation (15), can be used to approximately solve equation (1)via the method of steps. In the next section, dimensionalreduction is performed on this map via a center manifoldreduction algorithm.

CENTER MANIFOLD REDUCTION The nonlinear map in equation (8) can be written for aparticular value of ε as

32 )]([)]([)()1( kmCkmCkUmkm xCxQxx ++=+ (16)

where )1( +kmx is the collocation vector in a particular

interval, )(kmx is the collocation vector in the preceding

interval, ix km )]([ is the vector of all the possible independent

nonlinear terms of the order 3,2=i of the collocation vector

)(kmx and, QC and CC are the quadratic and cubic



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coefficient matrices. The order of nonlinearities in ix km )]([ is

a lexicographic order. The coefficient matrix QC will have

appropriate entries for each of the nonlinearities appearing inthe map. In equation (16), only up to cubic powers of nonlinearitiesare retained, even though the powers in equation (8) are ofmuch higher order. If n is the dimension of the state space inequation (1) and N is the number of Chebyshev points used in

collocation, then there are )( 2

1nNoi

nN

i=∑

=

independent

quadratic and )(2/)1( 3

1nNoii

nN

i=+∑

=

independent cubic

terms. Equation (16) can be written in the modal coordinates bymeans of a modal transformation of the state

)()( kmMkm zUx =(17)

1 2 2

1 3 3

( 1) ( ) [ ( )]

[ ( )]z U z U Q z

U C z

m k J m k M C M m k

M C M m k

−

−

+ = +

+ (18)

or

2 3( 1) ( ) [ ( )] [ ( )]M Mz U z Q z C zm k J m k C m k C m k+ = + + (19)

Here, UM is the modal decoupling matrix, UJ is the Jordan

canonical form of U and, 2M and 3M are the matricesresulting from the quadratic and cubic terms in equation (18)because of the state transformation (17). Now the states can bepartitioned according to the eigenvalues of UJ into center andstable states as

2 3

( 1) ( )0( 1) ( )0

[ ] [ ][ , ] [ , ]

[ ] [ ]

czc zc

szs zs

M c M cQ C

zc zs zc zsM s M sQ C

m k m km k m k

C Cm m m m

C C

µ

µ

+=

+

+ +

(20)

and a modified method of center manifold reduction fornonlinear maps [30] is applied to equation (20). A nonlineartransformation is defined of the form

33

22 )]([)]([)( kmUkmUkm zczczs +=

(21)

where 2U and 3U are undetermined coefficient matrices ofquadratic and cubic terms. Substituting equation (21) intoequation (20), a nonlinear algebraic equation of the form

2 3 22

2 3 33

2 3

( [ ] [ , ] [ ] [ , ] )

( [ ] [ , ] [ ] [ , ] )

[ ] [ , ] [ ] [ , ]

c M c M czc Q zc zs C zc zs

s M c M czs Q zc zs C zc zs

s M s M szs Q zc zs C zc zs

U m C m m C m m

U m C m m C m m

m C m m C m m

µ

µ

µ

+ + +

+ +

= + +

(22-a)

is obtained with the independent variable k suppressed forbrevity. Only quadratic and cubic terms are retained inequation (21) and then coefficients of the like powers arecollected for determining 2U and 3U . The resultingcoefficient equations for quadratic and cubic terms are

)]([:][ 222 sM

Qsc

zc CsUUm += µµ (22-

b)

with1. =)]([ sM

QCs surviving coefficients from matrixsM

QC ][ due to order 2 truncation

2. =cµ a coefficient matrix resulting from squaring

each element of vector zcc mµ

and3 2

3 2

23

[ ] : ( [ ] [ , ] )

([ ] ) ([ ] [ , ] )

s c M czc zc Q zc zs

s M s M cC Q zc zs

m U U cs m C m m

U s C cs C m m

µ µ

µ

+ ×

= + +(22-c)

with3. =)]([ sM

QCs surviving coefficients of matrixsM

QC ][ due to order 3 truncation

4. =× )],[][( 2zszc

cMQzc

c mmCmcs µ surviving

coefficients of the coefficient matrix resulting fromthe product in the parentheses, due to order 3truncation5. =)],[]([ 2

zszccM

Q mmCcs surviving coefficients

of the coefficient matrix resulting from the term in theparentheses due to order 3 truncation6. =sµ a coefficient matrix resulting from cubing

each element of vector zcsmµ

Equations (22) are linear generalized Lyapunov equationswhich are solved for unknown coefficient matrices 2U and



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3U by using Kronecker products. Solving (22-b) first for 2Uand (22-c) later for 3U using 2U , the original map (20)yields a one- (for fold and flip bifurcations) or two- (forsecondary Hopf bifurcations) dimensional map on the centersubspace given by

33

22 )]([)]([)()1( kmakmakmkm zczczc

czc ++=+ µ (23)

We summarize the process of computing the complete solutionwith the following Center Manifold Reduction Algorithm:

1. The collocation vector of the initialfunction is transformed into the modalcoordinates φmMmM UxU

11 )0( −− =2. The partitioned collocation vector

)(kmzc is computed from the modaltransformation

3. )1( +kmzc is computed using (23)

4. )1( +kmzs is computed from the centermanifold relationship (22)

5. Inverse modal transformation is appliedto T

zszc kmkm )]1(),1([ ++ to evaluate

)1( +kmx in equation (17). Thetrajectory represents the steady statesolution of equation (1) for a particularparameter set

EXAMPLE Consider the single inverted pendulum in the horizontalplane, as shown in Figure 1, with a linear and a quadratictorsional spring at the base, a linear torsional damper at thebase, and acted upon by a T-periodic follower forceproportional to the delayed (with delay period T) angulardisplacement. The stability chart for the version with aconstant force is shown in [34]. The equation of motion isgiven as

2 2

1 2[ cos( )] sin( ( ) ( )) 0l nml q cq k q k q

P P t l q t q t Tω η+ + +

+ + − − =

&& &

(24)

where ωπ /2=T and m is kilograms, l is in meters, c istorsional damping in N.m.s/rad, lk is a linear torsional

stiffness and nk is nonlinear torsional stiffness in N.m/rad.Expanding in a Taylor series about the zero equilibrium

position and retaining up to cubic terms, equation (24) can bewritten in the state space form as

2 2

2 2 3

0 1 0 0 ( )

/ / [ ] 0 ( )

0

( / ) ( ) [ ]( ( ) ( ))

( ) 0.001 ( ) 0 [ , 0]

l

n

x x x t Td

k ml P c mlx x P x t Tdt

k ml x t P x t x t T

x s x s s T

η

η

−= −

− + − −

−+ − −

= = ∀ ∈ −

& & &

&

(25

with mltPPP /))cos(( 21 ω+= and )()( tqtx = . Also,

)6/()/( 32 mlmlkn η= can be designated as ε , so thatequation (25) is in the form of equation (1). At a particularparameter set given by

2 2

1

2 3

2

/ 1.75, / 0.0482979, / 0.4025,

/ 0.734, / / 6 1/ 6, 1, 2.4

l

n

k ml c ml P ml

P ml k ml ml Tη η

= = =

= = = = = (26)

the system undergoes a flip bifurcation which is evidenced by a-1 eigenvalue of the monodromy matrix )( VI + in equation

(8). We choose 2/ mlc as the bifurcation parameter. The numerical center manifold algorithm is programmed inMATLAB and applied to system (25) with parameter set (26)with N=32 (collocation points) and n=2 (states) whichtranslates into a map having a 64-dimensional collocationvector, 2080 quadratic terms and 45760 cubic terms. The sizeof the coefficient matrices in (18) are 64 x 2080 and 64 x45760, for quadratic and cubic terms, respectively. Thereduced scalar map on the center manifold takes the form ofequation (23) with γµ +−= 1c where γ is a versal

deformation parameter and 0511.0,3310.0 32 −=−= aa .The post-bifurcation limit cycle amplitude is derived using thesecond iterate of equation (23) as

3( 2) (1 2 ) ( ) 2 ( ( ))zc zc zcm k m k m kγ δ+ = − − (27)

where 22 3a aδ = + and terms higher than cubic have been

dropped. The fixed points of equation (27) (from which theamplitude of the periodic orbit can be recovered) are obtainedby setting )()2( kmkm zczc =+ as

22 3

( )zcm ka a

γ= ± −

+(28)



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where 1+= cµγ . Now the bifurcation parameter 2/ mlc(which was temporarily omitted to allow for the numericalcomputations) is re-introduced (compare with [33] for theODE case) by computing the gradient cg of γ with respect to

2/ mlc . This is found to be =cg 0.5195 and hence

0583.0=δ which implies that the bifurcation is super-critical [32]. Using equation (28), with )/( 2mlcg c ∆=γ ,the amplitude of the limit cycle is computed and comparedwith the one obtained from the direct numerical simulation.Figure 2 plots the displacement and velocity variables forequation (25) for half the doubled period after steady state isreached computed via DDE23 and Center Manifold Algorithm.The center manifold calculations confirm the amplitude andthe doubled period.

CONCLUSIONSA method to compute center manifold reductions of delay

differential equations with periodic coefficients has beenproposed and illustrated. The method uses classicalperturbation analysis assuming that the nonlinearities aremultiplied by the perturbation parameter. An approximatenonlinear map is constructed using Chebyshev spectralcollocation which takes the solution in each interval of lengthequal to the principal period to the solution in the nextinterval. The method can be extended to the analysis of Hopfbifurcation and to order reduction problems associated withDDEs with periodic coefficients. The scope of the systemsconsidered can also be widened by considering DDEs with arational relationship between the delay and the principalperiods. The use of versal deformation theory allows abifurcation analysis in terms of a parameter as shown in theexample, although the nonlinear map is numericallyconstructed (using MATLAB) to allow for computationaltractability.

ACKNOWLEDGMENTS Financial support provided by the National ScienceFoundation grant, CMS-0114500 is gratefully acknowledged.

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1.Hassard, B. and Wan, Y. H.:Bifurcation formulae derivedfrom center manifold, Journal of Mathematical Analysis andApplications, 63, 297-312 (1978).2.Hassard, B.D., Kazarinoff, N.D., and Wan, Y.-H.: Theoryand Applications of Hopf Bifurcation, London MathematicalSociety Lecture Notes Series, 41, Cambridge University Press,Cambridge (1981).

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Figure 1. A single inverted pendulum subjected to a periodicretarded follower force

Figure 2. Displacement (radians) and velocity (radians/s) forthe system in Fig. 1 for half of the doubled period with DDE23

(+) and Center Manifold Algorithm (continuous)



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