Ion Anca Veronica. 142KB Jun 04 2011 12:09:03 AM

ON THE BAUTIN BIFURCATION FOR SYSTEMS OF

DELAY DIFFERENTIAL EQUATIONS

by

Anca–Veronica Ion

Abstract. For systems of delay differential equations the Hopf bifurcation was investigated by several authors. The problem we solve here is that of the possibility of emergence of a codimension two bifurcation, namely the Bautin bifurcation, for some such systems.

Keywords: bifurcation theory, Bautin, delay differential equation

1. Introduction

The existence of periodic solutions for evolution equations is of certain interest for both pure and applied mathematicians. Even for bidimensional systems of differential equations the detection of limit cycles by theoretical means is difficult. The bifurcation theory offers a strong tool for finding limit cycles, namely the theory concerning the Hopf bifurcation (when there is a varying parameter)[2], [6]. Several authors studied the Hopf bifurcation for delay differential equations (e.g. [4], [7], [1], [5] ). We are interested to find sufficient conditions for the Bautin bifurcation for a class of such systems.

2. Setting of the problem, theoretical frame

Consider a system of the form

( ) ( ) ( ) ( )

α α

(

( ) ( )

, ,α

)

)

(t A x t B x t r f x t x t r

x = + − + −

•

, (1)

( )

,

[ ]

,0

)

(s s s r

x =φ ∈ − , (2) where x=

(

x₁,..., x_n

)

∈Rn, α =

(

α₁,α₂

)

∈R2,A

( )

α ,B(α) are nxn matrices over R, f =

(

f₁,..., f_n

)

is continuously differentiable on its domain of definition, ⊂ 2n+1

D R . Moreover, f

(

0,0,α

)

=0and the differential of f in the first two vectorial variables, calculated at

(

0,0,α

)

is equal to zero. φis an element of the Banach space B=C

(

[ ]

−r,0,Rn

)

(column vectors). In order to write eq. (1) as a differential equation in a Banach space, the space

[ ]

[

)

{

n n

}

R

R ∃ ∈

→ −

= : ,0 , iscontinuouson -r,0 and lim_→ ( ) B

-0 s

is considered in section 8.2 of [4]. Its elements are ψ =ϕ +X₀σ , with

n

R

∈

∈ σ

ϕ B, (column vector) and

  

= < ≤ − =

, 0 ,

0 ,

0 ) ( 0

s I

s r s

X

n

where 0,I_nare the zero and, respectively, the unity nxn matrix. The norm of ψ is defined as the sum of the norm of ϕ in B and the norm of σ in n

R . The complexifications B ,_C B_0C of B , respectivelyB , are used below. ₀

Considerδ(⋅)- the Dirac function, and the nxn matrix valued function

n

I s

s) ( )

( =δ

∆ . Also consider the bounded linear operator L_α :B→Rn,

∫

−

= 0

) ( ) (

rd s s

L_αϕ η_α ϕ ,

with )η_α(s)=A(α)∆(s)−B(α)∆(s+r . By denoting

[ ]

,0. ),

( )

(s x t s s r

x_t = + ∈ − ,

we have in the spirit of [4], the following relations, equivalent with (1), (2):

( )

0, ( ), ), (

) ( ) 0

( L_α x f x x r α dt

dx

t t t

t = + −

(3)

), ( )

( s

ds dx s dt

dx_{t t}

= (4) .

0 =ϕ

x (5) Define (see also [4]) the linear operator, ₀ ( )

( )

0 ,

~

  

 ₋

+

=ϕ• ϕ ϕ•

ϕ α

α X L

A

[ ]

(

r

)

BC BC

C

A : 1 − ,0,Rn ⊂ →

~

α . Now we can rewrite the above problem as

), ), ( ), 0 ( (

0 ~

α

α x X f x x r

A dt dx

t t t

t = + −

, (6) .

0 =ϕ

x (7) The last term of (6) may be written as

∫

∫

− ∆ − ∆ +

0 0

0f( _rd (s)xt(s), _rd (s r)xt(s), )

X α .

We define

∫

∫

−

− ∆ ∆ +

= 0 0

) ), ( ) ( ,

) ( ) ( ( ) , (

r t

r t

t f d s x s d s r x s

x

F α α . Thus (6) and

(7) take the form

), , (

0 ~

α

α t t

t

x F X x A dt dx

+

ϕ = 0

x , (9) this being the abstract problem in B equivalent to (1),(2). ₀

The eigenvalues of α

~

A are (see [4]) the roots of the equation

( )

( )

(

)

0

detλI −Aα −e−λrBα =

. We assume the following hypothesis, that we denote H1.

H1. An open set U exists in the parameter plane such that for every α∈U,

there is a pair of complex conjugated simple eigenvalues

( ) ( )

α µ α ω

( )

α

λ_1,2 = ±i , with the property that there is a α₀∈U such that

( )

0

( )

0 0

2 ,

1 α ω α ω

λ =±i =±i , with ω₀ >0 and for every α∈U, all other

eigenvalues have strictly negative real parts, uniformly bounded from above by

a negative number.

By a simple eigenvalue we mean an eigenvalue having the algebraic multiplicity equal to 1.

We remark that H1 implies the existence of a neighborhood of α₀such that each eigenvalue different from λ_1,2

( )

α has real part strictly less than µ

( )

α . The eigenvectors corresponding to λ_i

( )

α , i=1,2, are elements ofB -the _C complexification of B , namely

( )( )

s e ( ) i

( )( )

0, s

[ ]

r,0

s i

i ∈ −

= ϕ α

α

ϕ λ α _,

where ϕ_i

( )( )

α 0 is a solution of

(

λ_i

( )

α I −A

( )

α −e−λi( )α rB

( )

α

)

ϕ =0_{. Obviously,}

( )

₁( )

2 α ϕ α

ϕ = .

Denote by _{{λ ( )α}}

2 , 1

M the linear subspace of B , spanned by _C {ϕ₁

( ) ( )

α ,ϕ₂ α } and Φ

( )

α the matrix having as columns the vectors

( ) ( )

α ϕ α

ϕ1 , 2 . Let {ψ1

( ) ( )

α ,ψ2 α } be two eigenvectors for the adjoint problem ([3], [4]), corresponding to the eigenvalues −λ_1,2

( )

α of the infinitesimal generator of the adjoint problem. They are elements of B*_C- the complexification of B* =C

(

[ ]

0,r,Rn

)

-row vectors, and we assume that they are selected such that, Ψ

( )

α being the matrix having as rows the vectors

( ) ( )

α ψ α

ψ1 , 2 , the relation (Ψ

( ) ( )

α ,Φα )= I2 holds, where

( ) ( )

0 0 ( ) ( ) ( ) , B , B . )

,

( *_{C C}

0

0

∈ ∈

− −

=

∫∫

− ψ ξ θ η θ ϕ ξ ξ ψ ϕ

ϕ ψ ϕ

ψ θ α

r

d d

In [4] a projection π _{{λ ( )α}}

2 , 1

0C B

: →M , is defined by

(

ϕ α

) ( ) ( )

α

[

(

α ϕ

) ( )

α

]

π +X₀ =Φ Ψ , +Ψ 0 . With this projection the space B is _0C decomposed as B _{{λ ( )α}} Kerπ

2 , 1

0C =M ⊕ . Since the solution xtof (8) and (9)

belongs to C1([−r,0],Rn), it is decomposed as )

( ) ( )

( u t y t

x_t =Φ α + , (10) with u(t)=

(

Ψ(α),xt

)

and y(t)=(I−π)xt, where ( ) ( ( ), ( ))

_

t z t z t

u = -column

vector,z(t)∈C. Let us define

  

 =

) ( 0

0 ) ( ) (

1 1

α λ α λ α

B .

The projection of eq. (8) on _{{λ ( )α}}

2 , 1

M is

( )

α u=Φ

( ) ( )

α Bα u+Φ

( ) ( )

α Ψ α (0)F(Φ(α)u(t)+ y(t),α)

Φ • ,

and since Φ(α) is invertible, this is equivalent to

( )

α u

( )

α (0)F( (α)u(t) y(t),α)

B

u• = +Ψ Φ + . (11) By projecting the initial condition we find

(

(α),ϕ

)

) 0

( = Ψ

u .

3. Existence of the invariant manifold and the restricted equation

If α∈U \

{ }

α₀ , and Reλ_1,2

( )

α >0 then, since these are the only two eigenvalues with positive real part (and they are simple), there is a local invariant manifold for the problem, namely the local unstable manifold, tangent to the space _{{λ ( )α}}

2 , 1

M , [3], [7].

For α =α₀, since Reλ_1,2

( )

α₀ =0, there is a local invariant manifold for the problem, namely the local center manifold, tangent to the space _{{ ( )}}

0 2 , 1 α

λ

M ,

[7].

Hence, for every α∈U with µ

( )

α ≥0, there is a neighborhood

( )

α

graph of a C1function. That is, the local invariant manifold may be expressed as

( )

α

{

ϕ wα

( )

ϕ ϕ {λ ( )α} V

( )

α

}

W_loc = + ∈ ∩

2 , 1

; M ,

where w_{α {λ ( )α}}→Kerπ 2

, 1

:M is a C1 function, w_α

( )

0 =0, and it has zero differential at 0. Since ϕ _{{λ ( )α}}

2 , 1

M

∈ , we have ϕ

( )

α =zϕ₁

( )

α +zϕ₁

( )

α , with

C

∈

z . This relation induces a dependence of z,z to w_α

( )

ϕ that justifies the notation w_α

( )

ϕ =w_α

( )

z,z .

Equation (11) implies

( )

α

(

ϕ α ϕ α α

)

ψ α

λ( ) ( ) (0) ( ) ( ) ( ) ( ) α( ( ), ( )),

)

(t ₁ z t ₁ F z t ₁ z t ₁ w z t z t

z = + + +

•

, (12)

(

ψ (α),ϕ

)

) 0

( = ₁

z . (13) Let S_α(t)φ be the solution of eq. (1) corresponding to the initial condition φ, at the moment t .

If φ∈Wloc(α), then

)) ( ), ( ( ) ( ) ( ) ( ) ( )

(t z t ₁ z t ₁ w z t z t

S_α φ = ϕ α + ϕ α + _α . (14) By using again the function f, (12) becomes

( )

(0) ( ( ) (0), ( ) ( ), ) )

( ) ( )

(t λ₁ α z t ψ₁ α f S_α t φ S_α t φ r α

z• = + − ,

(15) or,

), , ) ( ), ( ( ) ( ) ( )

(t λ₁ α z t g z t z t α

z• = + (16) by denoting

( )

(0) ( ( ) (0), ( ) ( ), ) )

, ) ( ), (

(z t z t α ψ₁ α f S_α t φ S_α t φ r α

g = − . (17)

4. The equations for the invariant manifold

The following proposition is a natural consequence of the invariance of

( )

α

loc

W . A similar result is given in [7], on the center manifold. We give the proof for the sake of completeness.

Proposition 1. Let φ∈Wloc

( )

α be the initial value for the problem (1). Then the

( )( )

( )( )

[ ]

,0 ,

), ))( ( ), ( ( ) ), ( ), ( ( ) ), ( ), ( ( ) ))( ( ), (

( _{1 1}

r s s t z t z w s s t z t z g s t z t z g s t z t z w t − ∈ ∂∂ = = + + ∂∂ α α α ϕ α α ϕ α (18)

( )( )

( )( )

) ), ( ) ( ), 0 ( ) ( ( ) ))( ( ), ( ( ) ( ) 0 ))( ( ), ( ( ) ( 0 ) ), ( ), ( ( 0 ) ), ( ), ( ( ) 0 ))( ( ), (

( _{1 2}

α φ φ α α α ϕ α α ϕ α α α α α α r t S t S f r t z t z w B t z t z w A t z t z g t z t z g t z t z w t − + − + = = + + ∂∂ (19)

with z(t) solution of the Cauchy problem (16),(13) and g defined by (17).

Proof Since φ∈W_loc

( )

α and W_loc

( )

α is invariant, S_α(t)φ∈W_loc(α). Let us denote, for t ≥0 and s∈

[ ]

−r,0 , )S_α(t)φ(s)=x(φ)(t+s . Obviously ) ( ) ( ) ( ) ( s t s x s t t x + ∂ ∂ = + ∂ ∂ φ φ _.

This and (14) imply

( )

( )

( )

( )

( )

( )

( ) ( )

( )

( ) )) ( ), ( ( ) ( ) ( ) ( ) ( )) ( ), ( ( 1 1 1 1 s t z s t z s t z t z w s s t z s t z s t z t z w t α ϕ α ϕ α ϕ α ϕ α α • • • • + + ∂∂ = = + +

∂∂ ₍₂₀₎

(here ₁

( )

( ) ₁

( )

(s)

ds d s ϕ α

α

ϕ• = ) and thus

( )

( )

w z t z t

( )

s

s s t z s t z s t z t z w

t ( ( ), ( )) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ( ), ( ))

_ 1 1 1 1 _ α α λ α ϕ α λ α ϕ α = _∂∂      ₋ +     ₋ + ∂∂ • •

. (21) With (16) we obtain (18).

On another side, since S_α(t)φis a solution of equation (1), we have

and, by taking s=0, we obtain (19).

This proposition allows the determination of the coefficients of the series of powers in z and

_

z of the functionw_α. Indeed, let us write

( )

, ! ! 1 ) ), ( ) ( ), 0 ( ) ( ( 2 k j k j

jk z z

F k j r t S t S f

∑

≥ + = − α α φ φ α

α (22)

( )

( )

j k

k j

jk z z

g k j t z t z g

∑

≥ + =

2 ! !

1 )

, ), (

( α α ,

( )

j k

k j

jk z z

w k j z z w

∑

≥ + = 2 , ! ! 1 ) , , (θ θ α α (23) where g_jk(α)=ψ₁(0)F_jk(α).

By replacing (22) and (23) in (18) and by matching the obtained series, we get first order linear differential equations for w_jk.

Thus, equation (18) implies

( )

( )

( )( )

( )

( )( )

( )

, . ! ! 1 ! ! 1 ! ! 1 , ! ! 1 1 1 2 1 2 1 2 2     ₊ +     + +     = • − • − ≥ + ≥ + ≥ + ≥ +

∑

∑

∑

∑

z z kz z z jz s w k j s z z g k j s z z g k j z z s w ds d k j k j k j k j jk k j k j jk k j k j jk k j k j jk α α ϕ α α ϕ α α (24) In this equality z• and

•

z will be replaced with the right hand side of (16) to obtain

( )

( )

( )( )

( )

( )( )

( )

(

)

( )

( )

( )

. ! ! 1 ! ! 1 , ! ! 1 ) ( ) ( , ! ! 1 ! ! 1 ! ! 1 , ! ! 1 2 _ 1 2 1 2 1 1 2 1 2 1 2 2     _      +       + + + +     + +     =

∑

∑

∑

∑

∑

∑

∑

≥ + − ≥ + − ≥ + ≥ + ≥ + ≥ + ≥ + m l m l lm k j m l m l lm k j k j jk k j k j jk k j k j jk k j k j jk k j k j jk z z g m l z kz z z g m l z jz s w k j z z k j s w k j s z z g k j s z z g k j z z s w ds d k j α α α α λ α λ α α ϕ α α ϕ α α (25) By matching the same order terms we obtain first order differential equations for wjk(.,α).

( )

(

)

( )

( )

( )

( )

( )( )

( )

( )( )

( )

(

)

( )

. ! ! 1 , ! ! 1 ) ( , 0 ! ! 1 ) ( 0 ! ! 1 0 ! ! 1 ! ! 1 ! ! 1 , 0 ! ! 1 ) ( ) ( , 0 ! ! 1 2 2 2 1 2 1 2 2 _ 1 2 1 2 1 1 2 k j k j jk k j k j jk k j k j jk k j k j jk k j k j jk m l m l lm k j m l m l lm k j k j jk k j k j jk z z F k j z z r w k j B z z w k j A z z g k j z z g k j z z g m l z kz z z g m l z jz w k j z z k j w k j

∑

∑

∑

∑

∑

∑

∑

∑

∑

≥ + ≥ + ≥ + ≥ + ≥ + ≥ + − ≥ + − ≥ + ≥ + + − + = =     +     + +     _      +       + + + α α α α α α ϕ α α ϕ α α α α α λ α λ α (26)

The relations obtained by equating the terms with similar powers of z, _

z in (26) are used as conditions to determine the constants that appear in the general form of w_jk obtained above.

In this theoretical form the calculus is very lengthy and we do not make it explicitly here.

We firstly remark that the coefficients of the second order terms in z and _

z in the expansion of fi(S_α(t)φ(0),S_α(t)φ(−r),α),i=1,...,n, are independent on those of w ,_α they depend only on the coefficients of the Taylor series of

(

x,y,α

)

fi . The similar assertion holds for the coefficients of g(z(t),z(t),α).

Hence g₂₀

( ) ( ) ( )

α ,g₁₁ α ,g₀₂ α are known, given fi and ψ1

( )

α . The following algorithm to determine wjk

( )

α must be used.

- w₂₀

( ) ( ) ( )

α ,w₁₁ α ,w₀₂ α are determined from the equations obtained by identifying the terms containing z2,zz,z2respectively, in (25), with initial conditions obtained by the same method from (26); they depend on g₂₀

( ) ( ) ( )

α ,g₁₁ α ,g₀₂ α .

- w₂₀

( ) ( ) ( )

α ,w₁₁ α ,w₀₂ α are used to compute g_jk

( )

α , j+k=3, (from (14), (17), (23) ) .

- w_jk

( )

α , j+k=3, are determined from the equations (25) and

- w_jk

( )

α , j+k≤3 are used to compute g_jk

( )

α , j+k=4 (from (14), (17), (23) ).

- w_jk

( )

α , j+k=4, are determined from the equations (25) and

conditions (26); they depend on g_jk

( )

α , j+k ≤4. - wjk

( )

α , j+k ≤4 are used to compute gjk

( )

α , j+k=5.

We do not need so many terms to have an accurate form of the invariant manifold, but we need them in order to discuss the behaviour of the solution z

of (16) that determines the solution of (1) on the invariant manifold.

5. The Bautin bifurcation

After applying the above algorithm, equation (16) has the form

( )

( )

6

5 2 1

! !

1 )

( ) ( )

( g z z O z

k j t

z t

z j k

k j

jk +

+

=

∑

≤ + ≤ •

α α

λ . (27)

Now, let l₁(α),l₂(α)be the first and the second Lyapunov coefficient respectively, associated to (27), defined in [6]. They are functions of g_kl(α). More specific, l₁(α)is function of g_kl(α) with 2≤k +l ≤3, and l₂(α) is function of g_kl(α) with 2≤k+l ≤5.

Let us define, with [6],

( )

( )

α ωµ α

ν1 = , ν2 =l1

( )

α , )ν =(ν1,ν2 . Let us consider the following hypothesis, denoted H2.

H2. l₁

( )

α₀ =0, l₂

( )

α₀ ≠0, and the map

(

α₁,α₂

)

→(ν₁,ν₂) is regular at α₀. The value in α₀of the first Lyapunov coefficient is

(

( ) ( ) ( )

)

Re 2

1 )

( _{20 0 11 0 0 21 0}

2 0 0

1 α α ω α

ω

α ig g g

l = + .

The value in α₀of the second Lyapunov coefficient is much more complicated and we do not reproduce it here (see [6]).

Th. 8.2. of [6] asserts that, if H2 is satisfied for eq. (16), then eq. (16) may be transformed, by smooth invertible changes of the complex function z and time reparametrizations, into

(

)

( )

4

( )

6

2 2 2

1 i z zz L zz O z

where L₂

( )

ν =l₂

( )

α

( )

ν .

We suppose that H2 is satisfied for our problem.

Problem (28) is then (see [6]) locally, near z=0, topologically equivalent to the following complex normal form of the Bautin bifurcation

(

₁ i

)

z ₂zz2 szz4,

z• = β + +β + (29) where β₁ =ν₁,β₂ = L₂

( )

ν ν₂, )s=signl₂(α_0. .

Assume the third important fact denoted by H3. H3. l₂

( )

α

₀ >0 (hence s=1).

Consider the polar form of (29), in the hypothesis H3

(

)

   

=

+ +

=

• •

, 1

,

4 2 2 1

η

ρ ρ β β ρ ρ

(30) with (ρ,η) the polar co-ordinates.

Due to H2, there is a C1bijection T between a neighbourhood U₁ of α₀, in the )

,

(α₁ α₂ plane and a neighbourhood W of (0,0) in the (β₁,β₂) plane.

We study the behaviour of the solutions only for β₁ ≥0 (β₁ and µ

( )

α have the same sign and we proved the existence of the invariant manifold only for µ

( )

α ≥0).

For thoseβ∈Wwith 0β₁ ≥0,β₂ ≥ , (30) has an repulsive focus in 0, and so does (29). Then, since the dynamical system generated by (29) is topologically equivalent to that generated by (16), this one has an unstable equilibrium in 0 (a repulsive focus or node, since they cannot be distinguished by topological equivalence).

In order to transport these conclusions to the solution of eq. (1), we remind relation (14):

)) ( ), ( , ( ) )( ( ) ( ) )( ( ) ( ) ( ) (

_

1

1 z t w z t z t

t z t

S_α φ θ = ϕ α θ + ϕ α θ + _α θ .

In the intersection of the quadrant β₁ ≥0,β₂ <0 and Wthere is a zone, situated between the axisβ₂ =0and the curve Γ=

{

(

β₁,β₂

)

;β₂ =−2 β₁

}

where eq. β₁+β₂ρ2 +ρ4 =0 has no positive solutions and thus there are no limit cycles for (30). The local phase portrait for (1) on the invariant manifold is as described above (unstable equilibrium point), for the corresponding (through −1

T ) zone in the α plane, let us denote it by V₂. Now let us put 2

1 V

V

V = ∪ .

In the zone situated between the curve Γ=

{

(

β₁,β₂

)

;β₂ =−2 β₁

}

and the axis β₁ =0 eq. β₁ +β₂ρ2 +ρ4 =0 has two positive solutions,

( ) ( )

β ρ β

ρ1 , 2 . Thus two concentric closed orbits for (30), hence for (29), exist. Since, for the corresponding zone of the αplane (let us denote it by *

V ), the problems (29) and (16) are topologically equivalent, eq. (16) will also have two periodic solutions. Relation (14) shows that, in this case, on the unstable manifold there exist two limit cycles for (1).

As we cross Γ, leaving the zone in the β plane described above, the two limit cycles collide and disappear.

6. Conclusions

The facts discussed above lead to the following result.

Theorem. If H1, H2, H3 are satisfied for eq. (1), then at α₀a Bautin type

bifurcation takes place. There is a neighbourhood U of ₁ α₀in the α plane

having a subset V (with α₀∈V ) with the property that for every α∈V , 0 is

an unstable equilibrium point (focus or node) for the problem (1) restricted to the bidimensional invariant manifold defined above.

There is also a subset V* of U , (having ₁ α₀as a limit point) with the

property that for every α∈V*, the restriction of problem (1) to the unstable

manifold has two concentric limit cycles. In this case also 0 is an unstable

equilibrium point.

References:

[1] R. Culshaw, S. Ruan, G. Webb, A mathematical model of cell to cell spread

of HIV that includes a time delay, J. Math. Biology, 46(2003), 425-444;

[2] Georgescu, A., M. Moroianu, I. Oprea, Bifurcation theory. Principles and

applications, Series of Applied and Industrial Mathematics of University of

Piteşti, 1999 (in Romanian);

[3] J. Hale, Functional Differential Equations, Applied Mathematical Sciences, 3, Springer, New York, 1971;

[4] J. Hale, L. Magalhães, W. Oliva, Dynamics in Infinite Dimensions,

Springer, New York, 2002;

[5] J. C. Ji, Stability and Hopf bifurcation of a magnetic bearing systems with

time delays, J. of Sound and Vibration, 259(4) (2003), 845-856;

[6] Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences,112, Springer, New York, 1995;

[7] G. Mircea, M. Neamţu, D. Opriş, Dynamical systems from economy,

mechanics, biology, described by delay equations,Mirton, Timişoara, 2003 (in

Romanian).

Author: