Bifurcation of Periodic Delay Differential Equations at Points of 1:4 Resonance
aa r X i v : . [ m a t h . D S ] J a n Bifurcation of Periodic Delay DifferentialEquations at Points of 1:4 Resonance ∗ Gergely R¨ost † Abstract
The time-periodic scalar delay differential equation ˙ x ( t ) = γf ( t, x ( t − z ( t ) = − γr ( t ) g ( x ( t − Keywords: bifurcation of maps, periodic delay equation, Floquet multi-pliers, spectral projection, center manifold, projection method, 1:4 resonance
AMS 2000
The generic bifurcation of planar discrete dynamical systems is the Neimark-Sacker bifurcation, where a complex conjugate pair of multipliers crosses theunit circle at critical values of the parameter and an invariant curve bifurcates ∗ Supported by the Hungarian Foundation for Scientific Research, grant T 049516. † Bolyai Institute, Univ. Szeged, Hungary, H-6720 Szeged, Aradi v´ertan´uk tere 1. x ( t ) = γf ( t, x ( t − , (1)where γ is a real parameter, f : R × R → R is a C -smooth function satisfying f ( t + 1 , ξ ) = f ( t, ξ )and f ( t,
0) = 0for all t, ξ ∈ R . Such equations arise very naturally in several applications,i.e. in population dynamics. A nice overview of related models can be foundin [19]. The periodicity is due to the periodic fluctuation of the environment.Denote the Banach space of continuous real and complex valued functionson the interval [-1,0] by C and C C , respectively, with the norm || φ || = sup − ≤ t ≤ | φ ( t ) | . Every φ ∈ C determines a unique continuous function x φ : [ − , ∞ ) → R ,which is differentiable on (0 , ∞ ), satisfies (1) for all t > x φ ( t ) = φ ( t )for all t ∈ [ − , x φ is called the solution of (1) with theinitial value φ . The time-one map F : C → C is defined by the relations F ( φ ) = x φ , x t ( s ) = x ( t + s ) , s ∈ [ − , . F γ emphasizes the dependence of the time-one map on theparameter. The spectrum σ ( U ) of the monodromy operator U (the derivativeof the time-one map F at 0) determines the behavior of solutions close tothe equilibrium 0. The monodromy operator is a linear continuous map andwith the relation U ( ψ ) = U (Re ψ ) + iU (Im ψ ) considered as an operator C C → C C and given by U ( ψ ) = y ψ , where y ψ is the solution of the linearvariational equation ˙ y ( t ) = γf ξ ( t, y ( t − , (2)where y ψ | [ − , ≡ ψ . The operator U is compact, therefore all the non-zero points of the spectrum are isolated points and eigenvalues of finitemultiplicity with finite dimensional range of the associated eigenprojection P µ : C C → C C , where µ ∈ σ ( U ) , µ = 0. These eigenvalues are called Floquetmultipliers. The spectral theory and other properties of different types ofdelay differential equations were extensively studied in [5] and [12].In [18], the equation˙ x ( t ) = γ (cid:0) a ( t ) x ( t ) + f ( t, x ( t − (cid:1) (3)was studied. Varying γ , Floquet multipliers cross the unit circle and bi-furcation of an invariant curve occurs, supposing that the critical Floquetmultipliers are not third or fourth roots of unity. For equation (1), the cri-tical eigenvalues are i and − i , that is a strong resonance, and the results of[18] are not valid anymore. In the case of strong resonance, in general onecan not expect the appearance of the invariant curve (see [1] or [2]). In [21],a condition was given which guarantees the appearance of the invariant curvefor 2-dimensional maps even at points of resonance. Independently, a similarresult was presented in [17]. Roughly speaking, if there are no bifurcatingfour-periodic points, the invariant curve occurs. The stability of the four-periodic points was treated in [14]. To apply this to our infinite-dimensionalsystem, we use center manifold reduction.The classical process of computing the dynamical system restricted to thecenter manifold using bilinear forms for delay differential equations (see [12]for the theory and [13] for applications) can not be applied directly to peri-odic equations. Faria ([6] and [7]) presented the method of normal forms forperiodic functional differential equations with autonomous linear part. Weestablished a spectral projection method in [18] for periodic scalar equations.3he spectral projection is represented by a Riesz-Dunford integral. The re-solvent of the monodromy operator of a periodic delay differential equationand corresponding spectral projections were calculated in the paper of Fras-son and Verduyn Lunel ([9, Section 6.2.]) in a more general setting. Certaincomputations done in [18] for equation (3), can be used for equation (1), sim-ply taking a ( t ) ≡
0. We remark that these arguments work only if the periodand the delay are the same. If the delay is not a multiple of the period, thenwe can not compute the Floquet multipliers by the characteristic equation.Some information can be obtained on the Floquet multipliers in a similarproblem in [20], there the period is three and the delay is one. The mostdifficult case, when the delay is incommensurable with the period, there areno results in this direction.The paper is organized as follows. In
Section 2 we summarize some previ-ous results, follow by the general theory ([5],[12]) or obtained in [18].
Section3 is devoted to the bifurcation analysis of strong resonance. We give anexplicit condition in terms of f and its partial derivatives to ensure the bi-furcation of an invariant curve or four-periodic points, and determine thedirection of the appearance and the stability properties. We apply our re-sults to equations with periodic coefficient in Section 4 , showing that theresonance does not cause any ”anomalies” for this class of equations. In
Section 5 we illustrate the results on the example of the celebrated Wrightequation with periodic coefficient.
A non-zero point µ of the spectrum of the monodromy operator U is calleda Floquet multiplier of equation (2) and any λ for which µ = e λ is calleda Floquet exponent of equation (2). By the Floquet theory ([12, p. 237]), µ = e λ is a Floquet multiplier of equation (2) if and only if there is a nonzerosolution of equation (2) of the form y ( t ) = p ( t ) e λt , where p ( t + 1) = p ( t ).Substituting this solution into equation (2), one can easily deduce that theFloquet exponents are the zeros of the characteristic function h ( λ ) = λ − γβe − λ , (4)where β = Z − f ξ ( t, dt.
4e assume that β = 0. The eigenfunctions have the form χ µ ( t ) : [ − , ∋ t e γe − λ R t − f ξ ( s, ds ∈ C . For any root of the characteristic equation h ( λ ) = 0, the corresponding χ µ ( t ) defines a Floquet solution of equation (2), hence the Floquet exponentscoincide with the roots of the characteristic function .Let ∆( z ) = z − e γβz . The equation ∆( z ) = 0 is equivalent to the characteristic equation. Anycomplex number µ = e λ is a root of ∆( z ) if and only if λ is a Floquetexponent. Applying Theorem 3 . . of [12, p. 247] to equation (1), one findsthat the Floquet multipliers consist of the roots of ∆( z ) and the algebraicmultiplicity of an eigenvalue µ equals to the order of µ as a zero of ∆( z ).When this number is 1, we call µ a simple eigenvalue. According to the Riesz-Schauder Theorem, if U : C C → C C is a compact operator with a simpleeigenvalue µ , then there are two closed subspaces E µ and Q µ such that E µ isone-dimensional, E µ ⊕ Q µ = C C , furthermore the relations U ( E µ ) ⊂ E µ and U ( Q µ ) ⊂ Q µ , σ ( U | E µ ) = { µ } and σ ( U | Q µ ) = σ ( U ) \{ µ } hold. The spectralprojection P µ onto E µ along Q µ can be represented by the Riesz-Dunfordintegral P µ = 12 πi Z Γ µ ( zI − U ) − dz = Res z = µ ( zI − U ) − , where Γ µ is a small circle around µ such that µ is the only singularity of( zI − U ) − inside Γ µ .For simplicity, let b ( t ) = γf ξ ( t,
0) and B ( t ) = R t − b ( s ) ds . With thisnotation the linearized equation takes the form˙ y ( t ) = b ( t ) y ( t − ,β = γ R − b ( t ) dt . By the variation-of-constants formula for ordinary differen-tial equations we find the following representation of the time-one map F ( φ )( t ) = φ (0) + Z t − γf ( s, φ ( s )) ds, t ∈ [ − , , (5)which implies for the monodromy operator U ( φ )( t ) = φ (0) + Z t − b ( s ) φ ( s ) ds, t ∈ [ − , . (6)5e need the derivatives of the operator F up to order three, evaluated at 0.Let V = D F (0) and W = D F (0). V and W are n -linear operators with n = 2 and n = 3, respectively. By the representation (5), one has V ( φ , φ )( t ) = Z t − γf ξξ ( s, φ ( s ) φ ( s ) ds, t ∈ [ − , , and W ( φ , φ , φ )( t ) = Z t − γf ξξξ ( s, φ ( s ) φ ( s ) φ ( s ) ds, t ∈ [ − , . The following statements are special cases of Lemma 4 and Theorem 2 of[18], setting a ( t ) ≡ Proposition 1 ([18]) The resolvent of the monodromy operator can be ex-pressed as ( zI − U ) − ( ψ )( t ) = e R t − b ( u ) z du H ( t ) , t ∈ [ − , , (7) where H ( t ) = (cid:16) z ψ (0) + e R − b ( u ) z du R − z e − R s − b ( u ) z du b ( s ) ψ ( s ) ds (cid:17) ·· (cid:18)(cid:16) z − e R − b ( u ) z du (cid:17) − + z e − R t − b ( u ) z du ψ ( t ) + R t − z e − R s − b ( u ) z du b ( s ) ψ ( s ) ds (cid:19) . The spectral projection operator, corresponding to a simple eigenvalue µ , hasthe representation P µ ( ψ ) = χ µ R µ ( ψ ) , where R µ ( ψ ) = (cid:18) µ + γβ (cid:19)(cid:18) ψ (0) + Z − b ( s ) ψ ( s ) χ µ ( s ) ds (cid:19) . (8)Notice that R µ ( χ µ ) = 1. Consider the decomposition C = T c ⊕ T su , where T c = Re E µ ⊕ Im E µ is the critical 2-dimensional realified centereigenspace corresponding to µ and spanned by { Re χ µ , Im χ µ } , moreover T su = Re Q µ ⊕ Im Q µ is the 2-codimensional realified stable-unstable sub-space corresponding to the other part of σ ( U ). The idea of the projection6ethod is that we introduce new variables x, y and use them as coordinateson these subspaces. Suppose we have a map˜ x = A ( x ) + g ( x, y ) , ˜ y = A ( y ) + h ( x, y ) , where A and A are linear maps on the corresponding subspaces and g (0 ,
0) = 0 , Dg (0 ,
0) = 0 ,h (0 ,
0) = 0 , Dh (0 ,
0) = 0 . For y = M ( x ) we have ˜ x = A ( x ) + g ( x, M ( x ))˜ y = A ( M ( x )) + h ( x, M ( x )) . If M ( x ) denotes the center manifold then by the invariance ˜ y = M (˜ x ), andthus M ( A ( x ) + g ( x, M ( x ))) = A ( M ( x )) + h ( x, M ( x )) . (9)The coefficients of the Taylor-expansion of M ( x ) can be calculated by thisformula. For details and examples we refer to [16] and [23]. The computationsin the infinite dimensional case can be found in [18]. Represent the Taylor-expansion of F in the form F ( φ ) = U ( φ ) + 12 V ( φ, φ ) + 16 W ( φ, φ, φ ) + O ( || φ || ) . Let Z ( φ ) = F ( φ ) − U ( φ ) be the nonlinear part of F . Now decompose φ ∈ C as φ = zχ µ + ¯ z ¯ χ µ + ψ, where z = R µ ( φ ) ∈ C , zχ µ + ¯ z ¯ χ µ ∈ T c and ψ ∈ T su . The complex variable z is a coordinate on the 2-dimensional real eigenspace T c and the function ψ isa variable in T su . The subspaces T c and T su are invariant under U . For anyreal φ , φ ∈ T su if and only if P µ ( φ ) = 0. U ( χ µ ) = µχ µ implies U (¯ χ µ ) = ¯ µ ¯ χ µ , R µ = R ¯ µ . 7 roposition 2 ([18]) The restricted map can be written as ˜ z = µz + 12 ρ z + ρ z ¯ z + 12 ρ ¯ z + 12 ρ z ¯ z + 16 ρ ¯ z + ..., (10) where ρ = R µ ( V ( χ µ , χ µ )) (11) ρ = R µ ( V ( χ µ , ¯ χ µ )) ρ = R µ ( V (¯ χ µ , ¯ χ µ )) ρ = R µ ( W ( χ µ , χ µ , ¯ χ µ )) + 2 R µ ( V ( χ µ , (1 − U ) − V ( χ µ , ¯ χ µ ))) ++ R µ ( V (¯ χ µ , ( µ − U ) − V ( χ µ , χ µ ))) ++ µ (1 − µ )1 − µ R µ ( V ( χ µ , χ µ )) R µ ( V ( χ µ , ¯ χ µ )) −− − µ | R µ ( V ( χ µ , ¯ χ µ )) | − µµ − | R µ ( V (¯ χ µ , ¯ χ µ )) | ,ρ = R µ ( W (¯ χ µ , ¯ χ µ , ¯ χ µ )) + 3 R µ (cid:16) V (¯ χ µ , ( µ − I − U ) − ·· (cid:0) V (¯ χ µ , ¯ χ µ − R µ ( V (¯ χ µ , ¯ χ µ )) χ µ − R ¯ µ ( V (¯ χ µ , ¯ χ µ ))¯ χ µ (cid:1)(cid:17) . The coefficients ρ , ρ , ρ , ρ are computed in [18, Section 5 ]. In thenon-resonant case ρ is not needed, but can be obtained completely analo-gously, hence the computation is omitted here. Two conditions are formulated in the Neimark-Sacker bifurcation theorem:the transversality condition, and the non-resonance condition, viz. ∂µ ( γ ) ∂γ | γ j =0 and µ j = 1 , µ j = 1, where γ j is a critical parameter value and µ j is acorresponding critical multiplier. The following two lemmas show that thetransversality condition is always fulfilled for equation (1), while µ j = 1.This situation is a 1:4 strong resonance.8 emma 1 The critical values of (2) are γ j = − π + 2 jπβ , j ∈ Z , and the corresponding critical Floquet multipliers are µ j = e λ j = i and ¯ µ j = e ¯ λ j = − i. These Floquet-multipliers are simple eigenvalues and the criticaleigenfunctions are χ ± i ( t ) : [ − , ∋ t e ∓ iB ( t ) ∈ C . Proof
One can check easily that i and − i can not be a double rootof ∆( z ), thus if i or − i is a Floquet-multiplier, then it is always a sim-ple eigenvalue. Suppose that λ = iθ is a critical Floquet-exponent, thenby the real part of (4) we have cos( θ ) = 0, hence θ = π + 2 kπ or θ = − π + 2 kπ , where k ∈ Z . Taking into account the imaginary part of (4),both options lead to the statements of the lemma by simple calculations.Introduce the notation B = B (0) = γβ . Lemma 2 ∂µ ( γ ) ∂γ | γ j = β λ ( γ j ) = β B (1 + iB ) Proof
By the characteristic equation and the Implicit Function Theorem µ ( γ ) = e γβµ ( γ ) is defined in a neighborhood of γ j . Differentiating with respectto γ gives µ ′ ( γ ) = e γβµ ( γ ) (cid:16) βµ ( γ ) − βγµ ′ ( γ ) µ ( γ ) (cid:17) = β − λ ( γ ) µ ′ ( γ ) . This yields µ ′ ( γ ) = β λ ( γ ) . Setting γ = γ j one has λ = − iγ j β = − iB andthe lemma is proved.The next proposition is the Poincar´e normal form map for 1:4 resonance. Proposition 3 ([16, p. 436])
Suppose that we have a map g = g ( γ ) : C C ,depending on the parameter γ ∈ R , and g has the form g ( z ) = µz + ρ z + ρ z ¯ z + ρ z + ρ z (12)+ ρ z ¯ z + ρ z ¯ z + ρ z + O ( | z | ) , here µ = µ ( γ ) and ρ kl = ρ kl ( γ ) depends on the parameter smoothly and µ ( γ j ) = i for some critical value γ = γ j . Then by a coordinate transformationdepending smoothly on the parameter, in the critical case the transformed maptakes the form ˜ g ( w ) = iw + c w ¯ w + c ¯ w + O ( | w | ) , where c = 1 + 3 i ρ ρ + 1 − i ρ ¯ ρ + − − i ρ ¯ ρ + ρ and c = i − ρ ρ + − i − ρ ¯ ρ + ρ . Note that similar, but different formulas are presented for c in [14, Chap-ter IV] and [21]. These formulas are miscalculated and false. One can checkdirectly by a straightforward, but rather elaborative computation that theformula of [16, p. 436], presented in Proposition 3 is the correct one. How-ever, in the literature the wrong formula of [14] is spreading, see for examplethe recent papers [10] and [22], where applications of the resonant normalform to mechanical systems are presented. Since the applied formula is notcorrect, the obtained results may not be correct as well.Define a = c i , a = c i and d = ∂ | µ ( γ ) | ∂γ | γ = γ j . Proposition 4 (Resonant bifurcation theorem,[14],[21])
Suppose thatwe have a map g ( z ) : C C of the form (12), depending smoothly on theparameter γ , satisfying d = 0 and µ ( γ j ) = i .If | Im ( a d ) | > | a d | , then a unique invariant curve bifurcates (and no pe-riodic points of order ) from the equilibrium as the parameter γ passesthrough γ j . The cases Re a < and Re a > are called supercritical andsubcritical bifurcations. In the supercritical case a stable invariant curve ap-pears for γ > γ j , while in the subcritical case an unstable invariant curvedisappears when γ increases through γ j .If | Im ( a d ) | < | a d | , then two families of periodic points of order bifurcate(and no invariant curve). Moreover, if | a | > | a | , the two families bifurcateon the same side and at least one of them is unstable. If | a | < | a | , then thetwo families bifurcate on opposite sides and both of them are unstable. emma 3 For the restricted map of the time-one map corresponding to equa-tion (1) we have a = 3 − i ρ ρ − i | ρ | − − i | ρ | − i ρ == − i h R i ( W ( χ i , χ i , ¯ χ i )) + 2 R i ( V ( χ i , (1 − U ) − V ( χ i , ¯ χ i ))) ++ R i ( V (¯ χ i , ( i − U ) − V ( χ i , χ i ))) i , and a = − i ρ ρ + 1 + i ρ ρ − i ρ == − i h R i ( W (¯ χ i , ¯ χ i , ¯ χ i )) + 3 R i (cid:16) V (¯ χ i , ( i − I − U ) − (cid:0) V (¯ χ i , ¯ χ i ) (cid:1)(cid:17)i . Proof
Apply Proposition 2 and Proposition 3 with µ = i . We obtain thelemma by a simple calculation.Let us define δ = | Im ( a ) − B Re ( a ) | − | a |√ B . (13)Remark that δ depends on the parameter. Some additional computationyields | Im ( a d ) | > | a d | ⇔ | Im ( a (1 − iB ) | > | a (1 − iB ) | , that is δ > F . See [15] for the existence and [8] for the smoothnessresult. Summarizing all the previous lemmas and propositions of Section 2 and
Section 3 , combining with the center manifold theorem and the reductionprinciple (for details see [3],[16] and [23]), we obtain our main theorem.
Theorem 1
The family of time-one maps F γ , corresponding to equation (1),has at the critical value γ = γ j the fixed point φ = 0 with exactly two simpleFloquet-multipliers µ j = i and ¯ µ j = − i on the unit circle. This is a 1:4 strong esonance. The transversality condition is fulfilled. There is a neighborhoodof in which a unique invariant curve (and no 4-periodic points) bifurcatesfrom , providing that δ > . The direction of the bifurcation is determinedby the sign of
Re ( a ) . If δ < , then two families of 4-periodic points (andno invariant curve) bifurcate from the equilibrium in a neighborhood of .Furthermore, if | a | > | a | , the two families bifurcate on the same side andat least one of them is unstable. If | a | < | a | , then the two families bifurcateon the opposite side and both of them are unstable. The conditions given in the theorem can be checked for any given equa-tion, we can compute γ j , a , a and B explicitly by terms of f ( t, ξ ) and itspartial derivatives. In this section we consider the equation˙ z ( t ) = − γr ( t ) g ( z ( t − , (14)where γ is a real parameter, r : R → R is a continuous function satisfying r ( t + 1) = r ( t ) for all t ∈ R , g ( ξ ) is a C -smooth function satisfying g (0) = 0.Without loss of generality we may suppose that g ( ξ ) = ξ + S ξ + T ξ + O ( ξ ) , where S, T ∈ R . With our previous notations we have f ( t, ξ ) = − r ( t ) g ( ξ ) ,f ξ ( t,
0) = − r ( t ) ,f ξξ ( t,
0) = − Sr ( t ) ,f ξξξ ( t,
0) = − T r ( t ) , and b ( t ) = − γr ( t ) . We show that this equation behaves at the bifurcation points as a nonreso-nant equation: an invariant curve bifurcates and no 4-periodic points fromthe equilibrium 0. The following lemma is used many times during the de-tailed computations. 12 emma 4
Let B ( t ) = R t − b ( s ) ds . Then Z t − e B ( s ) b ( s ) ds = e B ( t ) − , Z t − e B ( s ) b ( s ) B ( s ) ds = e B ( t ) B ( t ) − e B ( t ) + 1 . Proof
The first identity is obvious, the second can be deduced from thefirst by a partial integration.
Theorem 2
For any family of time-one maps corresponding to equation(14), if T = S then a unique invariant curve bifurcates from the equilib-rium as the parameter γ passes through γ j . The bifurcation is supercriticalif T < S (cid:0) B +25 B (cid:1) and subcritical if T > S (cid:0) B +25 B (cid:1) . Proof
Let us fix γ = γ j to be a critical parameter value. Using Lemma4, we have V ( χ i , χ i )( t ) = Z t − Sb ( s ) e − iB ( s ) ds = S − i ( e − iB ( t ) −
1) = iS e − iB ( t ) − ,V ( χ i , ¯ χ i )( t ) = SB ( t ) ,V (¯ χ i , ¯ χ i )( t ) = Z t − Sb ( s ) e iB ( s ) ds = S i ( e iB ( t ) −
1) = − iS e iB ( t ) − ,W ( χ i , χ i , ¯ χ i ) = Z t − T b ( s ) e − iB ( s ) ds = iT ( e − iB ( t ) − , and W (¯ χ i , ¯ χ i , ¯ χ i ) = Z t − T b ( s ) e iB ( s ) ds = − iT e iB ( t ) − . Notice that B = − π + 2 jπ , hence e iB = cos B + i sin B = − i , and e imB =( − i ) m for all m ∈ Z . Taking into account this fact, one obtains13 i ( e miB ( t ) ) = (cid:18) i + B (cid:19) (cid:18) e miB + Z − b ( s ) e ( m +1) iB ( s ) ds (cid:19) = (15)= (cid:18) i + B (cid:19) (cid:18) e miB + 1( m + 1) i ( e ( m +1) iB − (cid:19) == (cid:18) i + B (cid:19) (cid:18) ( − i ) m − i ( − i ) m +1 − m + 1 (cid:19) == m ( − i ) m + i ( i + B )( m + 1)for any m = −
1. Observe that R i ( e iB ( t ) ) = ii + B = R i (1) . If m = −
1, we getthe eigenfunction e − iB ( t ) , and as in general, R i ( e − iB ( t ) ) = R i ( χ i ( t )) == (cid:18) i + B (cid:19)(cid:18) e − iB + Z − b ( s ) ds (cid:19) = i + Bi + B = 1 . Now let us evaluate the resolvent by Proposition 1 and Lemma 4:(1 − U ) − V ( χ i , ¯ χ i ) = e B ( t ) (cid:16) ( SB + e B Z − e − B ( s ) b ( s ) SB ( s ) ds )(1 − e B ) − + e − B ( t ) SB ( t ) + Z t − e − B ( s ) b ( s ) SB ( s ) ds (cid:17) = Se B ( t ) (cid:16)(cid:0) B + e B ( − e − B B − e − B + 1) (cid:1) (1 − e B ) − + e − B ( t ) B ( t ) − e − B ( t ) B ( t ) − e − B ( t ) + 1 (cid:17) = Se B ( t ) ( − − e − B ( t ) + 1) = − S. Referring to Lemma 3 and ( i − U ) − = ( i − − U ) − = ( − − U ) − , we stillneed ( − − U ) − ( e miB ( t ) ) = e − B ( t ) H ( t ) , (16)where 14 ( t ) = ( − − U ) − ( e miB ( t ) ) = e − B ( t ) (cid:16)(cid:0) − e miB + e − B R − b ( s ) e ( mi +1) B ( s ) ds (cid:1) ·· (cid:0) − − e − B (cid:1) − − e (1+ mi ) B ( t ) + R t − b ( s ) e (1+ mi ) B ( s ) ds (cid:17) == e − B ( t ) (cid:16)(cid:0) − e miB + e − B e ( mi +1) B − mi +1 (cid:1)(cid:0) − − e − B (cid:1) − − e (1+ mi ) B ( t ) + e ( mi +1) B ( t ) − mi +1 (cid:17) == e − B ( t ) ( − i ) m mi − e − B )( mi +1) − e miB ( t ) mimi +1 . Particularly,( − − U ) − ( e iB ( t ) −
1) == e − B ( t ) ( − i ) i − e − B )(2 i +1) − e iB ( t ) 2 i i +1 − e − B ( t ) − e − B == − e iB ( t ) 2 i i +1 . Similarly, one finds( − − U ) − ( e − iB ( t ) −
1) = e − iB ( t ) i − i + 1 . (17)Now we are ready to compute the coefficients of the normal form given inLemma 3, namely a = − i R i h W (¯ χ i , ¯ χ i , ¯ χ i ) + 3 V (¯ χ i , ( − − U ) − (cid:0) V (¯ χ i , ¯ χ i ) (cid:1)i = (18)= − i R i h − iT e iB ( t ) − − iS V (¯ χ i , ( − − U ) − (cid:0) e iB ( t ) − (cid:1)i = − T R i h ( e iB ( t ) − i − S R i h V (¯ χ i , − e iB ( t ) i i + 1 i = 0 , where we used the linearity of R i and the identity1 S R i ( V (¯ χ i , e iB ( t ) )) = 1 T R i ( W (¯ χ i , ¯ χ i ¯ χ i ) = 13 i R i ( e iB ( t ) −
1) = 0 .
15e use (15) and (16) to conclude a = − i R i h W ( χ i , χ i , ¯ χ i ) + 2 V ( χ i , − S ) + (19)+ V (¯ χ i , ( − − U ) − V ( χ i , χ i )) i = − i R i h T i ( e − iB ( t ) −
1) + 2( − i ) S ( e − iB ( t ) − i − i R i h V (cid:0) ¯ χ i , ( − − U ) − (cid:16) iS e − iB ( t ) − (cid:1)(cid:17)i = T − S R i [ e − iB ( t ) −
1] + S R i h V (cid:0) ¯ χ i , e − iB ( t ) i − i + 1 i = ( T − S ) B i + B ) + Si − i R i [ V (¯ χ i , e − iB ( t ) )]= ( T − S ) B i + B ) + Si − i R i [ S − i ( e − iB ( t ) − T − S ) B i + B ) − S B (2 − i )( i + B ) = B i + B ) (cid:0) T − S
11 + 2 i (cid:1) . Applying (18),(19) and i + B = B − i B to (13), we find2(1 + B )Re ( a ) = T B − BS B + 25 , (20)2(1 + B )Im ( a ) = − T B + BS − B . The sign of Re ( a ) determines the direction of the bifurcation, as formulatedin the theorem, which is the same as the sign of T − S (cid:0) B +25 B (cid:1) . Substitutingthe previous two formulas into (13), we deduce δ = 12(1 + B ) | B | · | − T − T B + S − B + 2 B + 11 B ) | = | B | · | T − S | . Since B = 0, the condition T = S guarantees that δ > An example
The classical form of the celebrated Wright-Hutchinson equation (or delayedlogistic equation) is ˙ y ( t ) = − αy ( t − y ( t )) . The change of variable z ( t ) = ln(1 + y ( t )) transforms Wright’s equation into˙ z ( t ) = − α ( e z ( t − − . Since the pioneer works of Wright ([24]), a huge amount of papers concernedwith the dynamical properties of this equation and its generalizations. Herewe consider this equation with a periodic coefficient:˙ z ( t ) = − αr ( t )( e z ( t − − , (21)where α > r ( t ) is a continuous function satisfying r ( t + 1) = r ( t ) for all t ∈ R . Without loss of generality we may suppose R − r ( s ) ds = 1. We have g ( ξ ) = ξ + ξ + ξ + O ( ξ ) , that is S = 1 and T = 1. The next theorem isa direct application of Theorem 2 and Lemma 1. Theorem 3
The family of time-one maps corresponding to equation (21),undergoes a supercritical bifurcation and a unique invariant curve bifurcatesfrom the equilibrium as the parameter α passes through π . Remark that taking r ( t ) ≡ α = π a periodic solution emerges from the equilibrium by a supercritical Hopfbifurcation. This is consistent with Theorem 3. Acknowledgement .The author would like to thank the invaluable work of the referees.17 eferences [1] V. I. Arnold,
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