Hopf bifurcation and period functions for Wright type delay differential equations
HHOPF BIFURCATION AND PERIOD FUNCTIONS FORWRIGHT-TYPE DELAY DIFFERENTIAL EQUATIONS
ISTV ´AN BAL ´AZSMTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University ofSzeged,Aradi v´ertan´uk tere 1, Szeged, H-6720, [email protected] R ¨OSTMathematical Institute, University of Oxford,Woodstock Road, OX2 6GG, Oxford, United KingdomBolyai Institute, University of Szeged,Aradi v´ertan´uk tere 1, Szeged, H-6720, [email protected] 16, 2018
Abstract
We present the simplest criterion that determines the direction of the Hopf bifurcations of thedelay differential equation x (cid:48) ( t ) = − µf ( x ( t − µ passes through the critical values µ k . We give a complete classification of the possible bifurcation sequences. Using this informationand the Cooke-transformation, we obtain local estimates and monotonicity properties of the periodsof the bifurcating limit cycles along the Hopf-branches. Further, we show how our results relate tothe often required property that the nonlinearity has negative Schwarzian derivative. Keywords:
Hopf bifurcation, supercritical, normal form, delay differential equation, period estimates
The appearance of limit cycles around equilibria via Hopf bifurcations is a common phenomenon for delaydifferential equations, when a parameter of the equation is passing through a critical value and a pair ofeigenvalues of the linearized system is crossing the imaginary axis on the complex plane. Depending on thenature of the nonlinearity, the Hopf bifurcation can be either supercritical or subcritical, i.e. the bifurcat-ing periodic solution can be stable or unstable on the center manifold. It is well known how to determinethe direction of the Hopf bifurcation for delay differential equations at least since the book of Hassard,Kazarinoff and Wan [Hassard et al. (1981)]. One can use bilinear forms, center manifold reduction (see[Diekmann et al. (1995), Hassard et al. (1981)]), Lyapunov-Schmidt method [Guo & Wu(2013)] or alterna-tively, the theory of normal forms for functional differential equations [Faria & Magalhaes(1995)]. Basedon these fundamental techniques, the literature of delay differential equations is vast by papers whereHopf bifurcation results are shown to many particular model systems arising from physical, engineering orbiological applications. However, most of those articles provide only the complicated formula of the firstLyapunov coefficient, which is hard to relate to the original model parameters, and in fact if the readerwants to figure out the direction of the bifurcation in particular cases, it requires almost the same effortas repeating the whole calculation of the general formula. Also, due to the elaborative calculations, theliterature of bifurcation theory is not free of minor mistakes or inaccuracies (some of those are discussed,for example, in [Lani-Wayda(2013)] or [R¨ost(2006)]).1 a r X i v : . [ m a t h . D S ] J a n o remedy this situation, our first aim here is to derive the simplest criterion for the direction of thebifurcations for the class of scalar delay differential equations of the special form x (cid:48) ( t ) = − µf ( x ( t − , which will be then trivial to check in any specific situation. Note that the equation z (cid:48) ( s ) = − f ( z ( s − µ ))can be easily rescaled into the previous one by the change of variables s = tµ and x ( t ) = z ( s ), hence in thesequel we assume that the delay is one and µ will be our bifurcation parameter. This class of equationsis frequently studied and includes such notorious examples as Wright’s equation or the Ikeda-equation.We present concrete examples for all possible sequences of subcritical and supercritical Hopf bifurcations.Our calculations are based on the method of [Faria & Magalhaes(1995)].Next, we use the formulae for the directions of the Hopf bifurcations and combine with the methodof Cooke’s transformation to obtain some information on the periods of the bifurcating solutions. Inparticular, we find narrow estimates of the period function along branches, and explore the relationbetween its monotonicity and the directions of the bifurcations. Finally, we explore the connectionbetween our results and the Schwarzian derivative of the nonlinearity, which plays a significant role inmany global stability results, and show that by local bifurcations one can not disprove the conjecture thatlocal asymptotic stability implies global asymptotic stability for Wright-type delay differential equationswith negative Schwarzian. Consider the scalar delay differential equation x (cid:48) ( t ) = − µf ( x ( t − g ( x t , µ ) , (2.1)where µ ∈ R , f is an R → R , C -smooth function with f (0) = 0 so it can be written as f ( ξ ) = ξ + Bξ + Cξ + h.o.t., where B = f (cid:48)(cid:48) (0) / C = f (cid:48)(cid:48)(cid:48) (0) /
6. Note that f (cid:48) (0) = 1 can be assumed without the loss of generality,as we can normalize it via the parameter µ . It is known that the direction of the Hopf bifurcation dependson the terms of the Taylor-series of the nonlinearity up to order three. In this case, the direction of theHopf bifurcation around the zero equilibrium is determined by a relation between the coefficients B and C . To our surprise, despite the method is well known for a much more general class of equations, we couldnot find a derivation of such a simple, readily available criterion for B and C in the literature for (2.1),only for the first Hopf bifurcation in [Stech(1985)] and in Chapter 6 of the recent book of H. L. Smith[Smith(2011)], and for a different class of equations in [Giannakopoulos & Zapp(1999)]. The main resultof this Section is the general condition for the stability of the Hopf bifurcation at any critical parametervalue. Theorem 2.1. a) Equation (2.1) has Hopf bifurcations from the zero equilibrium at the critical param-eter values µ k = π + 2 kπ, k ∈ Z .b) The k th bifurcation is • supercritical if C < k +1) π − k +1) π B ; • subcritical if C > k +1) π − k +1) π B .c) If a Hopf bifurcation of Equation (2.1) is • supercritical, then the bifurcation branch starts to right if µ k > and left if µ k < ; • subcritical, then the bifurcation branch starts to left if µ k > and right if µ k < .Proof. a) The linearization of equation (2.1) is x (cid:48) ( t ) = − µx ( t − . (2.2)2earching for its solutions in form x ( t ) = e λt , we get λe λt = − µe λ ( t − , and the characteristic equation is λ = − µe − λ . We would like to find the Hopf bifurcation points, so we substitute λ = iω , ω ∈ R \ { } and write iω = − µe − iω = − µ cos ω + iµ sin ω. Taking real and imaginary parts, we get the following system of real equations0 = µ cos ω,ω = µ sin ω. From the first of these equations, we get ω = π + nπ , n ∈ Z . Substituting this into the second equationwe have π nπ = µ sin (cid:16) π nπ (cid:17) . We distinguish two cases: • n = 2 k , k ∈ Z , in which case we find π kπ = µ sin (cid:16) π kπ (cid:17) = µ ; • n = 2 l + 1, l ∈ Z , in which case we find π l + 1) π = µ sin (cid:16) π l + 1) π (cid:17) = − µ, which is equivalent to π − (2 l + 2) π = µ. We find that the via k = − ( l + 1) the two cases can be treated together, and for the critical valueswe may just write µ k = π + 2 kπ = k +12 π , k ∈ Z . For each µ k there is a pair of critical eigenvalues ± iω k , where ω k = k +12 π .b) We follow the procedure developed in [Faria & Magalhaes(1995)]. Let L and F be defined by therelation L ( α ) x t + F ( x t , α ) = − ( µ k + α ) f ( x ( t − , where x t is the solution segment defined by x t ( θ ) = x ( t + θ ) for θ ∈ [ − , L ( α ) is a linear operatorfrom C ([ − , , R ) to R , F is an operator from C ([ − , , R ) × R to R with F (0 ,
0) = 0 and D F (0 ,
0) =0. We write L ( α ) = L + αL + α L + O ( α ) , and for ( x , x , x , x ) ∈ R , F ( x e iω k θ + x e − iω k θ + x · x e iω k θ , B (2 , , , x + B (1 , , , x x + B (1 , , , x x + B (0 , , , x x + B (2 , , , x x + . . . Since µ k = ω k = k +12 π , e iω k = i holds. For φ ∈ C ([ − , , R ) we have L ( φ ) = − µ k φ ( −
1) = − k + 12 πφ ( − . Hence, L (1) = − k + 12 π,L ( θe iω k θ ) = − k + 12 π (cid:16) − e − i k +12 π (cid:17) = − i k + 12 π,L ( e iω k θ ) = − k + 12 π (cid:16) e − i k +12 π (cid:17) = 4 k + 12 π, F can be written as F ( x e iω k θ + x e − iω k θ + x · x e iω k θ , − k + 12 π (cid:0) B ( x ( − i ) + x i + x − x ) + C ( x ( − i ) + x i + x − x ) + h.o.t. (cid:1) . Then the B ( a,b,c,d ) coefficients are B (2 , , , = 4 k + 12 πB, B (1 , , , = − (4 k + 1) πB, B (1 , , , = (4 k + 1) πBi,B (0 , , , = (4 k + 1) πBi, B (2 , , , = 32 (4 k + 1) πCi. According to [Faria & Magalhaes(1995)] (see formula (3.18) and Theorem 3.20), the direction of thebifurcation is determined by the sign of K = Re (cid:20) − L ( θe iω k θ ) (cid:18) B (2 , , , − B (1 , , , B (1 , , , L (1) + B (2 , , , B (0 , , , iω k − L ( e iω k ) (cid:19)(cid:21) . We shall use the notation a ∼ b whenever a = qb for some q >
0. Substituting all terms into K , weneed to find the sign of the real part of11 + i k +12 π (cid:32)
32 (4 k + 1) πCi − − (4 k + 1) πB (4 k + 1) πBi − k +12 π + k +12 πB (4 k + 1) πBii (4 k + 1) π − k +12 π (cid:33) = (2 − (4 k + 1) πi )2(1 + (4 k +1) ) (4 k + 1) π (cid:18) Ci − B i + ( − i − B i (cid:19) ∼ (2 − (4 k + 1) πi )(4 k + 1) (cid:18) Ci + 25 B − B i (cid:19) , the latter expression having the real part(4 k + 1) (cid:20) (4 k + 1) π C + 45 B −
115 (4 k + 1) πB (cid:21) ∼ (cid:20) C − k + 1) π − k + 1) π B (cid:21) . c) To calculate in which direction a pair of characteristic roots crosses the imaginary axis at a bifurcationpoint, we differentiate the real part with respect to the parameter. Let us consider a parameterdependent solution of characteristic equation λ = − µe − λ , written as λ ( µ ) = α ( µ ) + iω ( µ ), where α and ω are the real and imaginary parts. Then we have α + iω = − µe − α − iω = − µe − α (cos ω − i sin ω ) . Separating real and imaginary parts, we get α = − µe − α cos ω,ω = µe − α sin ω. Differentiating these equations with respect to µ , we find α (cid:48) = − e − α cos ω − µe − α ( − α (cid:48) ) cos ω − µe − α ( − sin ω ) ω (cid:48) ,ω (cid:48) = e − α sin ω + µe − α ( − α (cid:48) ) sin ω + µe − α cos( ω ) ω (cid:48) . Assume that the root is critical, i.e. α ( µ k ) = 0, and ω ( µ k ) = ω k . As we have seen in part a), in thecritical case cos ω k = 0 and sin ω k = 1. Then, evaluating the derivatives at a µ k , we obtain α (cid:48) = µ k ω (cid:48) ,ω (cid:48) = 1 − µ k α (cid:48) . ω (cid:48) into the first equation, and express α (cid:48) as α (cid:48) ( µ k ) = µ k µ k . Hence, α (cid:48) ( µ k ) and µ k has the same sign. This means that at a Hopf bifurcation, a pair of characteristicroots crosses the imaginary axis from left to right if and only if µ k >
0. Hence the branch of asupercritical Hopf bifurcation starts to the right if and only if µ k >
0, and the subcritical case is theopposite. - - k Figure 1: Plot of H ( k ) = k +1) π − k +1) π . The values are between π − π and π +845 π , and they tend to as k → ±∞ . According to Theorem 1, when for some k , the value of C/B is below H ( k ), the k thbifurcation is supercritical.As we mentioned, for the special case k = 0, this result can be found in [Smith(2011)], page 97, whichwe now state as a corollary. Corollary 2.2.
The Hopf bifurcation at µ = π/ is supercritical if C < π − π B , and it is subcriticalif C > π − π B . With the notation H ( k ) = k +1) π − k +1) π , we can see that the Hopf-bifurcation is supercritical when C < H ( k ) B . The function H ( k ) is plotted in Figure 1, and from the shape of this function we easilyfind the following. Corollary 2.3. If C < π − π B then every Hopf bifurcation is supercritical, if C > π +845 π B then everyHopf bifurcation is subcritical. For convenience, note that π − π ≈ . π +845 π ≈ .
52. Theorem 1 and its corollaries allow us togive a complete classification of possible bifurcation sequences, which are depicted in Figure 2.
The classical Wright-Hutchinson equation (also called delayed logistic equation) y (cid:48) ( t ) = − µy ( t − y ( t )) , µ > , can be transformed into the form x (cid:48) ( t ) = − µ ( e x ( t − − x ( t ) = ln(1 + y ( t )), for solutions y > −
1. This latter equation is of type (2.1)with f ( ξ ) = e ξ − B = , C = . Since C ≈ . < π − π B ≈ . Corollary 3.1.
In Wright’s equation, every Hopf bifurcation is supercritical. /2 5 π /2 9 π /2 13 π /2 - π /2 - π /2 - π /2 - π /2 amplitude µ π /2 5 π /2 9 π /2 13 π /2 - π /2 - π /2 - π /2 - π /2 amplitude µ (a) C < π − π B : All bifurcations are supercritical. (b) π − π B < C < B : There exists n > k th bifurcation is subcritical iff 0 ≤ k ≤ n . π /2 5 π /2 9 π /2 13 π /2 - π /2 - π /2 - π /2 - π /2 amplitude µ π /2 5 π /2 9 π /2 13 π /2 - π /2 - π /2 - π /2 - π /2 amplitude µ (c) C = B : The k th bifurcation is subcritical iff k ≥
0. (d) B < C < π +845 π B : There exists n < − k th bifurcation is subcritical iff n ≤ k ≤ − π /2 5 π /2 9 π /2 13 π /2 - π /2 - π /2 - π /2 - π /2 amplitude µ amplitude µ π /2 5 π /2 9 π /2 13 π /2 - π /2 - π /2 - π /2 - π /2 4321 17 π /2 - π /2 (e) C > π +845 π B : All bifurcations are subcritical. (f) Wright’s equation Figure 2: The possible configurations of bifurcation branches, based on Theorem 1.
The equation y (cid:48) ( t ) = − sin( y ( t − µ ))arisen in the modeling of optical resonator systems. By rescaling, one has the equivalent form x (cid:48) ( t ) = − µ sin( x ( t − , which fits into (2.1) with f ( ξ ) = sin( ξ ), B = 0, C = − . Since C < π − π B = 0, Corollary 2.3applies. Corollary 3.2.
In the Ikeda equation, every Hopf bifurcation is supercritical. (a) parameter a m p li t ude (b) parameter a m p li t ude Figure 3: (a) The bifurcation branches of the Ikeda equation. (b) The bifurcation branches of our totallysubcritical example.
Consider x (cid:48) ( t ) = − µ ( x ( t −
1) + x ( t −
1) + 1 . x ( t − , that is f ( ξ ) = ξ + ξ + 1 . ξ , B = 1, C = 1 .
44. Then22(4 · π − · π < CB < · π − · π , so the bifurcations at µ and µ are subcritical, the others are supercritical.6 .4 A totally subcritical polynomial equation Consider x (cid:48) ( t ) = − µ (cid:18) x ( t −
1) + x ( t −
1) + 2215 x ( t − (cid:19) , that is f ( ξ ) = ξ + ξ + ξ , B = 1, C = . Then CB = > k +1) π − k +1) π , for all nonnegative integer k , soevery Hopf bifurcation is subcritical for positive critical parameter values (and supercritical for negativeparameter values). Throughout this section we consider µ >
0. The following idea is known in the delay differential equationfolklore as the Cooke-transform, which has been used for example in [Mallet-Paret & Nussbaum(1986),Garab & Krisztin(2011)]. If p ( t ) is a periodic solution of equation (2.1) for parameter value µ = µ ∗ > T , then q ( t ) := p (( lT + 1) t ) is also a periodic solution of equation (2.1) for parameter value µ = µ ∗ ( lT + 1) with period TlT +1 , for any l ∈ N . This can be shown by the straightforward calculations q (cid:48) ( t ) = − ( µ ( lT + 1)) f ( p (( lT + 1) t − lT − − ( µ ( lT + 1)) f ( q ( t − q (cid:18) t + TlT + 1 (cid:19) = p (( lT + 1) t + T ) = p (( lT + 1) t ) = q ( t ) . Thus we can define a map C l : ( µ ∗ , T, p ( t )) (cid:55)→ (cid:18) µ ∗ ( lT + 1) , TlT + 1 , p (( lT + 1) t ) (cid:19) , where p ( t ) is a periodic solution of equation (2.1) with parameter value µ ∗ and period T . Proposition 4.1.
Let k, l ∈ N . Then near the bifurcation points, C l maps the kth bifurcation branch tothe (k+l)th bifurcation branch.Proof. Consider the Hopf-branch of periodic solutions near the parameter value µ k , k ∈ N . Let δ k = 1 ifthe k th bifurcation is supercritical, and let δ k = − k th bifurcation is subcritical. Then, for any k there is a local branch of periodic solutions p kη ( t ) corresponding to parameter value µ = µ k + δ k η with η ∈ (0 , η k ) with some η k >
0. The minimal period of p kη is denoted by T kη . Recall from the previoussection that at the critical values µ k = π + 2 kπ = k +12 π , the critical eigenvalue is iω k = i k +12 π , hence T kη → πω k = k +1 =: T k as η →
0. Notice that µ k ( lT k + 1) = 4 k + 12 π (cid:18) l k + 1 + 1 (cid:19) = π k + l ) + 1) = µ k + l and T k lT k + 1 = 1 l + k +14 = 44 l + 4 k + 1 = 2 πω k + l = T k + l . From the uniqueness of local branches (see [Diekmann et al. (1995), Theorem X.2.7]), we find that theCooke-transform maps Hopf bifurcation branches to Hopf bifurcation branches.
Theorem 4.2.
In equation (2.1) , if k ≥ and the k th Hopf bifurcation is supercritical, then we have thefollowing estimate on the period of the Hopf solution near µ k : T kη ≥ k + 1 + ηπ . Proof.
If the k th bifurcation is supercritical, then δ k = 1, and by Corollary 2, all the ( k + l )th bifurcations( l ∈ N ) are supercritical as well. Then, taking into account Proposition 1,( µ k + η )( lT kη + 1) > µ k + l , (4.1)7hat is T kη > (cid:18) µ k + l µ k + η − (cid:19) l − = 4 − ηlπ k + 1 + ηπ . This inequality holds for any l ∈ N , thus letting l → ∞ we finish the proof. Theorem 4.3.
If in equation (2.1) all Hopf bifurcations are subcritical and k ≥ , then we have thefollowing estimate on the period of the Hopf solution near µ k : T kη ≤ k + 1 − ηπ . Proof.
Now for any k, l ∈ N , δ k = δ l = −
1, and by Proposition 1,( µ k − η )( lT kη + 1) < µ k + l , that is T kη < (cid:18) µ k + l µ k − η − (cid:19) l − = 4 + ηlπ k + 1 − ηπ . This inequality holds for any l ∈ N , thus letting l → ∞ we finish the proof. Theorem 4.4. If π − π B ≤ C < B and k ≥ , then define n := max (cid:26) m ∈ N : C > m + 1) π − m + 1) π B (cid:27) . If k < n then near µ k we have the estimates η ( n − k +1) π k + 1 − ηπ < T kη < η ( n − k ) π k + 1 − ηπ . If k = n then we only have the lower estimate: T kη > ηπ k + 1 − ηπ . Proof.
Assume that k ≤ n . Then the k th bifurcation is subcritical, δ k = − l > k + l )th bifurcation is supercritical. Then( µ k − η )( l T kη + 1) > µ k + l , that implies T kη > ηl π k + 1 − ηπ . Now we choose l to be the minimal index which still gives a supercritical bifurcation, that is l := n − k +1.Next, suppose that the ( k + l )th bifurcation is subcritical. This is only possible if k < n . Then( µ k − η )( l T kη + 1) < µ k + l , that implies T kη < ηl π k + 1 − ηπ . Finally, choose l to be the maximal index that still gives subcritical bifurcation, that is l := n − k .In some situation this theorem provides very sharp estimations of the period function, which isillustrated in Figure 4. Corollary 4.5.
If in equation (2.1) the k th Hopf bifurcation is subcritical for some k ≥ , and the periodssatisfy T kη < T k near µ k , then for all l > the ( k + l ) th Hopf bifurcation is also subcritical. .00084.00064.00044.000241.5705 1.5706 1.5707 1.5708 µ = µ - η T η Figure 4: Narrow estimates on the period of the 0th branch of Example 3.3 by Theorem 4.4 (red curves)compared to numerically obtained periods (blue curve).
Proof.
If the k th Hopf bifurcation is subcritical, then( µ k − η )( lT kη + 1) < µ k ( lT k + 1) = µ k + l . This means that the Cooke-transform maps the k th branch to the left side of µ k + l , thus the ( k + l )thbifurcation is also subcritical.In the situation of π − π B < C < B (see Figure 2.b.), we can infer the monotonicity of the periodfunctions at the subcritical bifurcations, as the next corollary shows. Corollary 4.6.
If in equation (2.1) the k th Hopf bifurcation is subcritical for some k ≥ , but the ( k + l ) thHopf bifurcation is supercritical for any l > , then T kη > is monotone increasing for small η .Proof. If the k th Hopf bifurcation is subcritical, but the ( k + l )th is supercritical, then for η < η wehave ( µ k − η )( lT kη + 1) < ( µ k − η )( lT kη + 1) . This is possible only if T kη < T kη . The Schwarzian derivative of a C function f is defined as( Sf )( ξ ) = f (cid:48)(cid:48)(cid:48) ( ξ ) f (cid:48) ( ξ ) − (cid:18) f (cid:48)(cid:48) ( ξ ) f (cid:48) ( ξ ) (cid:19) at points ξ where f (cid:48) ( ξ ) (cid:54) = 0. This quantity plays an important role in many results regarding the globaldynamics of difference equations, which can be extended to delay differential equations in various cases(see [Liz & R¨ost(2010), Liz & R¨ost(2009), Liz & R¨ost(2013), Liz et al. (2003)] and references thereof). Aglobal stability conjecture was formulated in [Liz et al. (2003)], stating that the zero solution of (2.1) isglobally asymptotically stable whenever it is locally asymptotically stable, if Sf < f with Sf <
0, where the Hopfbifurcation of (2.1) is subcritical at µ . This would provide a counterexample. Since both the directionsof the bifurcation and the sign of the Schwarzian are determined by the derivatives of the nonlinearityup to order three, in view of the results of the previous sections, it is most natural to make a comparisonto check whether such a counterexample is possible. Corollary 5.1. If Sf < , then all Hopf bifurcations are supercritical. Furthermore, if f (cid:48)(cid:48) (0) = 0 , thenfor any k , the k th bifurcation is supercritical if and only if Sf (0) < . roof. From the definition, it is easy to evaluate Sf (0) = 6( C − B ), thus Sf < Sf (0) < C < B . By Corollary 2, all Hopf bifurcations are supercritical. In the special case f (cid:48)(cid:48) (0) = 0, we have B = 0 and Sf (0) = 6 C , thus both the sign of the Schwarzian and the direction of the bifurcation aredetermined by the sign of C .We found that it is not possible to construct a counterexample to the conjecture of Liz et al. bymeans of a subcritical Hopf bifurcation. IB was supported by Hungarian Scientific Research Fund OTKA K109782 and EU-funded Hungariangrant EFOP-3.6.1-16-2016-00008. GR was supported by NKFIH FK124016 and Marie Skodowska-CurieGrant No. 748193. The authors thank Maria Vittoria Barbarossa and Jan Sieber for helping withDDE-BifTool.
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