A NOTE ON SMOOTH GONJUGACY OF NODE MANIFOLDS

284

A NOTE ON SMOOTH GONJUGACY OF NODE MANIFOLDS

Fatma M. Kandil

Mathematics Department ,Faculty of Science, King Abdul Aziz University, Jeddah - Saudi Arabia

ABSTRACT

It was shown that there exists a unique

C

r.smooth invariant manifold for a given dynamical system.

Keywords: Smoothes, Manifold, Dynamical system . A.M.S.C: 34A26, 34A25, 34A34, 34C30.

1. INTRODUCTION

Consider a vector field

F R

:

p



R

p

, 2

  

p

which is

C

rsmooth

(

r



2)

on the ball

B

_p



{

z

 

}

with a nodal equilibrium at the origin.

F generates the dynamical system

( , )

( , )

x

Lx

P x y

y

My

Q x y

 



 







(S)

where L and M are diagonalizable matrices of order

n n



and

m m



respectively,

(0, 0)

(0, 0)

0 , ( , )

n m

P



Q



x y



R



R

n



m



p

. The eigenvalues of L and M satisfy









* *

Re

( )

Re

(

)

i i

L

j

M

j

 







 

for all integer pairs

( , ) {1, 2,... } {1, 2,... }

i j



n



m

. Let the eigenvalues of be indexed in increasing order. In this paper we shall show that there exists a unique

C

r smooth invariant manifold for (S) even when the large gap conditions

  

*₁

/

*_n

r

(assumed by [4]) does not hold provided the following nonresonance conditions hold. For

j



1, 2,...

n

1 1

, 2

n n

j i i i

i i

k

k

r

 

 











(1)

where

( ,

k k

_{1 2}

,...

k

_n

)

is an integer vector with

k

_i



{0,1, 2,... }

r

. If the resonance condition

1 n

j i i

i

u

k









of

order

2,..., r

holds for some

j



{1, 2,...,

m

}

, and some integer vector

( ,

k k

_{1 2}

,...

k

_n

)

then there are examples of n-dimensional manifolds which are less smooth than the order of the resonance condition Our assumptions are:

A1)

,

n n m m

L



R



M



R

 are diagonalizable

A2)

* *

1

* *

1

1

m

n

r















A3)

1 1

, 2

1 , j=1,2,...,m

n n

j i i i

i i

k

k

r

 

 









 

A4)

,

(

) ,

2

r

P Q



C

B

_

r



A5)

(0, 0)

0 ,

2,...,

k

x

D Q



k



r

under these assumptions then there exists a unique invariant manifold which is the graph of

y



h x

( )

.

Remark: Note that (A3) implies that there exists an analytic change of variables which transforms (S) into

( )

S



such that Q is r semiflat with respect to x . Also



*_m

/

 

₁*

r

implies



k

_i

  

*_{i j}

,

j



1, 2,...,

m

if

1 n

i i

k

r





285

Let X and Y be finite dimensional Banach spaces. All of the major results stated here holds analogously for a map

:

F X



Y

.

Let (D) denote the dynamical system obtained by linearizing the vector field near

z



0

. Let

( )

S



denote the dynamical system obtained from (S) with Q satisfies (A5). It is known that (S) has many local n dimensional invariant manifolds [see 5] which are graphs of

y



g x

( )

where

g x

( )

may only be

C

1 smooth. The partial normal form of one of these manifold can be constructed by solving the following system of partial differential equations over the set of m vectors of polynomials of n variables of order 2 and degree r provided (A3) holds (see 4),

( )[

{ , [ ]]

( )

( , [ ]]}

(0)

(0)

0

Dh x

Lx

P x h x

Mh x

Q x h x

h

Dh





 











_

(M)

However the system

( )

S



with Q satisfying (A5) has a unique solution over the set of r

C

function which are little

0(

x

r

)

. This modivates the following definition.

Definition 1: The generalized Poincare’ manifold for (S) satisfying (A4) is the graph of the unique r

C

function

( )

_p

( )

( )

g x



h x



h x

where

h x

_p

( )

is the vector polynomial determined from (M) and

y



h x

( )

is the manifold.

2. MAIN RESULTS

In this paper we establish the smoothness of the invariant manfold for (S) which is defined to the graph of

( )

y



h x

for

x



B

_



{

x



R

n

:

x

 

}

for some

 

0

. The manifold function h is

C

ron

B

and

(0)

0 ,

0,1,...,

k

D h



k



r

.

A manifold is positively a invariant with respect to (S) if

y t

( )



h x t

[ ( )]

for

t



0

, where

[ ( ), ( )]

x t

y t

is a positive trajectory of (S) passing through

( , )

x y



B

_ such that

y



h x

( )

. From the variation of constant formula one can show that the manifold is positively invariant if and only if the manifold function

h x

( )

is a fixed point of the integral operator T defined by

0

( )( )

Ms

{ ( , , ), [ ( , , )]}

T h x

e Q

s x h h Q s x h

ds





_



(1)

[ ( , , )]

( , , ),

{ ( , , ), [ ( , , )]}

d

t x h

L t x h

P

t x h h

t x h

dt



  

 



(2)

(0, )

x

0 ,

x

B

_

,

0







 

The domain of T is

 

G

_r

( )





C



_r , where

{

(

) :

}

r r

r

C



_



h



C

B

_

h

0,1,...

max sup

k

( )

1 ,

k

(0)

0 ,

0,1, 2,... }

k _{r x}

D h x

D h

k

r









and

0

*

( )

{

(

) :

k

r

G

_

 

h



C

B

_

h

* 0

( )

sup

_k

,

0,1,...

1 and

r

1}

x

h x

k

r

h

x

 



 







From now on assume _{1, , / 2} 1

2

       . Let

K

_k denote a bound on

,

k k

P

Q

computed over

B

_ for

1,...

k



r

and

M

_kdenotes a constant depending on

K K

₁

,

₂

,...

K

_k. Theorem 1: (i) The solution



( , , )

t x h

of (2) satisfies

* 1 1 ( 2 )

( , , )

t x h

x e

  K t

,

x

B

_

,

t

0

286 (ii) The solution

  

_i

( , ,

t x h

_i

) ,

i



1, 2

satisfies

* 1 1 ( 2 )

2 1

sup |

2

[

2

]

1

[ ]

1 1

,

0

K t

t R

h

h

K e

t



  

   

 





1

,

2 r

h h



C



_ .

Proof: The results follows directly by using the variation of constants formula for the solutions of (2) and Grnwall inequality.

Theorem 2: T is a well defined map from



into

G

_r

( )



.

Proof: (i) For

0

  

1

select

0

   

,

/ 2

and choose

 

0

small enough so that

sup ( )

x

  

h x

 

. Since

sup ( , , )

t B

x

t x h





  



 

for

t



0

(because L is a diagonal matrix) the

composition

Q

{ ( , , ), [ ( , , )]}



t x h h



t x h

is well defined and

C

r for

( , )

t x



R





B

_.

(ii)

0 0

{ , [ ]}

( )

M r _r

p

Q

h

Th x

e



ds



_

_











* *



1 2 1 0

2

m r r K s r

r

e

x

K ds



   



_



_{* *}



1

1 2 1 2

r m r r K x Kr



     .

Since



*_m

/

 

*₁

r

, the integral is well defined for if

 

0

is chosen small enough so that

* * 1 1

2

m

r

K

r

  



.

(iii)



* *



1

1 1

( ) k r k2 2

r m

Th x x     K r  r K   

if   1/ r k _for

* *

1 1

0,1,... ,

2

m

r

k

r

K

r

  





and *1 *

1

2 m r

r

K rK     . Thus the range of T is a subset of

( )

r

G

_



.

Theorem 3: T is a contraction on

G

_r

( )



in the norm

* r

. Proof: Select

h h

₁

,

₂



G

_r

( )



, than

2 1 ₀ 2 2 2 1 1 1

0

{

,

[

]}

{ , [ ]

Ms p

Th

Th

e

Q

h

Q

h

ds







_











*

2 2 2 1 2 1

0

{

,

[

]}

{ ,

[ ]}

m u s

e

Q

h

Q

h





_











1 2 1 1 1 1

{ ,

[ ]}

{ , [ ]}

Q

h

Q

h

ds





 





*

2 2 2 1 2 1

2 1

0 2 1

{ , [ ]} { , [ ]}

{

ms r

r r

Q h Q h

e            



2 1 1 1

1 1 1 [ ] [ ] } r r h h

K    ds

 



2 1 2 2 2 1 1 1

0 2 1

{ , [ ]} { , [ ]} 2 m r r s r r

Q h Q h

e    _    __     _  _   _{ }     



2 1 2 2 2 1 1 1

2 1

{ , [ ]} { , [ ]}

2

r r

r r

Q h Q h

      

 

 

*

1 1 2 1 1 1

( 2 )

1

1

[

]

[

]

r r r K s

r

h

h

K e

  

x

ds



 

 



_

287

* 1 ₁

2 1 2 1

1 0

m

r

r k k

s

r k

e

K

  _{ }

 





_





  







*

1 1 1 2 1 1 1

( 2 )

2 1

1

[

]

[

]

2

r r r K s

r r

K h

h

e

  

x

K

ds





_{ }

_



_





   













* *

1 1 1 2 1 1 1

( 2 )

2 2 1 1

0 ₁

[

]

[

]

3

[

]

[

]

m r r K s r

r r

K h

h

e

x

K h

h

ds



   





 









_

 

 

_















_{* *}



1

1

2

1

(3

1

)

2 1 *

r

m r

r

r K



K

K

h

h

   





Thus T is contraction on

G

_r

( )



in the norm

* r

if

* 1 1

2

m

r

K

r



  



and

3

K

_r



(2

r



1)

K

₁

  

r

*_{1 m}.

From now on replacing h by

h T

₀

,

by

T

0, denote

D

_xk



by



_k

,

D h x

k

( )

by

h x

_k

( )

and

( )

k

D Th x

by

T

k . Let

E

k denote the set of function

h x

_k

( )

which are continuous and

B

_ and K linear

operators on

1 k

n

i

x R

 for each fixed

x



B

. The set

k r k

E



G

 is a Banach space with the generalized infinity norm

0

0

( )

( )r k sup k p

k r k

k

h x h x

x



 

 

 .

The equations of variations of order K are given below, Let id denote the identity map on

R

n

,

id x

( )



x

,

1



L

P

x

P h

y 1



1

   



_



_



(3)

1

(0, , )( )

1

,

n n

x h h

I

R





 





1

1 ,

1

{ , ( )} ...

2

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

q

k

k

i i q

k k i

q q k

n

k k

L Dp D P h D D

k

x h h h R





         _

  

   _

 

 ₍₄₎

Similar we can prove he following theorem .

Theorem 4:

T

k is a contraction on

E

k



G

_r k for fixed

h h

₀

, ,...

₁

h

_k1 in the norm

r k_

E

k



Gr



k

_r k . Theorem 5: There exists a unique manifold (constructed as the graph of

y



h x

( )

) which is invariant with respect to (S). The map h is r times continuously differentiable on

B

_ for some

 

0

and

(0)

0 ,

0,1, 2,...

k

D h



k



r

.

Proof: By Theorem 4, and continuity of

0 1 2

( )[ ,

,...

]

k

k

T

h

h h

h

in

( ,

h h

_{0 1}

,...

h

_k

)

for

k



0,1, 2,...

r

the map

(

T T

0

,

1

...

T

r

)

is a fiber contraction from

0

1 1

( ) ( )

r r

r r k r k r k r k

k k

G  C x E_ G G  C x E_ G

  

   

By induction one can show that if

h

₀



C

r



and

h h

₁

,

₂

,...

h

_k



E

1

 

...

E

k then

(

T T

0

,

1

,...

T

r

)

is a map from

r

C

to

C

k in the sense that

T h

(

₀

)



C

k and

D t h

k

(

₀

)



T

k

(

h

₀

)[ ,

h h

_{1 2}

,...

h

_k

]

. By uniform convergence, Taylor’s theorem, and the attractivity property of the filter contraction theorem [2, 6, 8] it follows that

h

* is in

r

C



_for some

 

0

. One can extend the differentialabity of

h

* to

B

_if one employs the continuous extensions of

( , [ ])

k r _k

x



D Q x h x

for

k



1, 2,...,

r

.

288

( )[

{ , ( )}]

( )

{ , ( )}

(0)

0 ,

2,3,...,

k

Dh x

Lx

P x h x

Mh x

Q x h x

D h

k

r





 







(M)

Systems with a form similar to (M) have solutions which provide linearizing and partially linearizing transformations of dynamical systems. Such transformations can be used to constract local horizontal and vertical invariant foliations of

B

_ under milder conditions than those required for the existence of linearizing transformation.

The first step towards developing such a theory is to generalize concept of an invariant manifold by dropping the spectral separation requirement while retaining the explicit construction as the graph of

y



h x

( )

. Thus we can now make an arbitrary assignment of dependent and independent variables in order to consider cases of spectral overlap. If the eigenvalues

 

_i

(

_j

)

are associated with the independent (dependent)variables and are indexed in order of increasing real part then the following cases are possible

(i)

0

       

₁* *_m *₁ *_n

(ii)

0

    

₁* *₁

u

*_m

 

*_n

(iii)

0

    

*_{1 1}*

u

_m*

 

*_n

(iv)

0

       

₁* *₁ *_n *_m

(v)

0

       

*_{1 1}* *_n *_m

(vi)

0

       

*₁ *_n *₁ *_m

Theorem 6: If

DF

(0)

is

C

rand diagonalizable such that the (A1– A4) hold then there exists a unique invariant manifold for (S) which is the graph of

y



h x

( )

.

Proof: The prove is formally identical to that of Theorem 5. The condition (A1) guarantees that one may consider a system with nonlinearity Q in r semiflat form with respect to x . The bound on the spectral gap guarantees that the maps

T

k

,

k



0,1,...

r

, are contractions in the normal _r k for

h h

₀

, ...

₁

h

_k1 fixed. Together with continuity of

k

T

is

( , ...

h h

_{0 1}

h

_k

)

one concludes that

(

T T T

0

, ...

₁ r

)

is a fiber contraction. By induction it can be shown that T

is a map from

C

r to

C

kin the sense previously discussed. Then Taylor’s theorem, uniform convergence, and the attractivity property of the fiber contraction theorem imply

C

rsmoothness of the manifold.

Remarks: (1) The classical linearizing conjugacy theorem of Sternberg is a corollary of Theorem 6, because case (iii) can be modified to include

* * * *

1 1

...

n m

       

with

n



m

(2) All of the sprevious results can be extended to get

C

 manifolds for

C

 vector fields because the limit of the contraction conditions of Theorems 5 as

r

 

can be met by manifold function h defined on

B

_, for some  0.

(iii) A result similar to Theorem 5 for the case when

DF

(0)

is not diagonaliable should hold but the estimates for constractions and the partial normalization are considerably more intricate. There is also a difficulty with local invariance due to polynomial factors which prevent the compositions

Q

0(

id h o t x

, )



( , )

from being well defined on finite time intervals

[0, ]

T

if L is not diagonalizable.

REFERENCES

1- R. Abraham and J. Robbin: Transversal Mapping and Flow. Benjamin, Boston 1967.

2- C. Bota, About the invariant and minimal sets of a dynamical system on a topological space, The Fifth International Worksop on Analele Universitat. Ii din Timi, Soara September 18-22, 2001, Romania Fascicula-Mathematica

3- J. Hale: Integral nanifolds of perturbed differential system. Ann. Math. 73 (1961), 496-531.

4- M. Hirsch, C. Pugh and M. Shub; Invariant Manifold Lect. Note in Math. No 583, Springer Velag 1976.

5- R. Dela Liave: Some new invariant manifold Theorems and applications to smooth conjugacies of dynamical system, mansicript, 1992. 6- A. Osinga: Two-dimensional invariant manifolds in four-dimensional dynamical Systems, Birstol BS8 1TR, UK December 22, 2003. 7- G. Sell: Smooth Linearization near fixed point. Amer. J. Math. Vol.107, (1989), 1035-1091.