Che materia stai cercando?

Anteprima

ESTRATTO DOCUMENTO

Zeroing separately the different powers in the z-equation:

 2 α α

+ − = 

: 1 0

x α = −

1 / 5

k

3 1

 1

2 α α α

− + − = = +

 : 1 0

y 1 / 5

k

3 2 2

 

α α α α

 

− + − = =

: 2( ) 0 / 5

xy k k

2 1 3 3

 

µ α α α

+ = =

: 0 0

x

 

4 5 4

 

α

µ α α =

− = 0

: 0

y 

 5

5 4

Bifurcation equations:

 2 2

µ

 = − + − + + +

[(1 / 5) / 5 (1 / 5) ]

x x y x k x k xy k y

 2 2

 µ

= − + − + + +

[(1 / 5) / 5 (1 / 5) ]

y y x y k x k xy k y

≠ =

z

0 0

k k

Note: if , the inertia contributes to the motion; if , contributes

z

statically. 11

6.THE NORMAL FORM THEORY

• Scope:

To use a smooth in order to put the

nonlinear coordinate transformation,

bifurcation equation in the simplest form.

• Algorithm: = +

x Jx f x

( )

Equations of motion: = +

y Jy g y

( )

Transformed Equations: = +

x y h y

( )

Near-identity transformation:

x µ.

where (possibly) includes the dummy variables 12

h(y)

Transformed equation in the unknown:

o By differentiating the nearly-identity transformation:

= +

x I h y y

[ ( )]

y

the equation of motion is transformed into:

+ + = + + +

I h y Jy g y J y h y f y h y

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

y

or: − = + − +

h y Jy Jh y f y h y I h y g y

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

y y

h(y) g(y).

which is a differential equation for for any given A series solution is

often necessary. 13

Series solution:

o  

 

  f y

f y ( )

( )

f y

( ) 3

2  

 

  = + +

g y g y g y

( ) ( ) ( )

 

 

  2 3

     

h y

( ) h y

( ) h y

  ( )

   

2 3

Chain of equations:

o − = −

h y Jy Jh y f y g y

( ) ( ) ( ) ( )

y

2, 2 2 2

− = + − −

h y Jy Jh y f y f y h y h y g y g y

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

y y y

3, 3 3 2, 2 2, 2 3

i.e.: ˆ

= − =

h y f y g y

( ) ( ) ( ), 2,3,

k

k k k 14

Homogeneous polynomial of k-degree:

o M M M

k k k

∑ ∑ ∑

ˆ α β γ

= = =

f y p y g y p y h y p y

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

k km km k km km k km km

= = =

1 1 1

m m m

Zeroing the independent monomials:

o = −

L γ α β

k k k k

J

where (if is diagonal): N N

∑ ∑

λ λ

= Λ Λ = − =

L diag[ ], : ( ),

m m k

k i i j j i j

= =

1 1

j j

Resonance:

o Λ

λ ω ω =

= ± ± 0

0, 0, , , ,

i i L

Since , for some (i.e. is singular);

i k

i

1 2

j ≠ 0

β

hence, must be taken for compatibility, and resonant terms survive in

k

the normal form!!! 15

• Example 1: System independent of parameters

System a Hopf bifurcation:

at

o  

  3 2 2 3

    α α α α

ω + + +

x x x x xx x

0

i x 31 32 33 34

  α

= + ∈

 

, ,

x

 

   

ω

− 3 2 2 3

   

0 i x α α α α

 

x + + +

x x x xx x

 

31 32 33 34

Normal form:

o 3 2 2 3

ω β β β β

= + + + +

y i y y y y yy y

31 32 33 34

Near-identity transformation:

o 3 2 2 3

γ γ γ γ

= + + + +

x y y y y yy y

31 32 33 34 16

Equation for the coefficients:

o      

  γ α β

ω

2 0 0 0

i 31 31 31

     

  γ α β

0 0 0 0      

  32 32 32

= −

     

 

ω γ α β

0 0 2 0

i 33 33 33

     

       

ω γ α β

 

0 0 0 4

i      

34 34 34

Solution:

o β β β β α

= = = =

0,

31 33 34 32 32

Normal form:

o 2

ω α

= +

y i y y y

32

2 λ λ λ ω ω

+ =

y y ω

2 ( ) ( )

i i

+ = i

2

The term is since , i.e.

resonant, 1 2 1 17

• Example 2: System dependent on a parameter

System to a Hopf bifurcation:

close

o  

  3 2 2 3

    α α α α

µ ω + + +

+

x x x x xx x

0

i x 31 32 33 34

  α

= + ∈

 

, ,

x

 

   

µ ω

− 3 2 2 3

   

0 i x α α α α

 

x + + +

x x x xx x

 

31 32 33 34

or, extending the state-space:  

3 2 2 3

  α α α α

+ + +

     

x x x xx x

ω µ

x 0 0

i x x 31 32 33 34

 

     

 

3 2 2 3

ω µ α α α α

 

= − + + + + +

0 0

x i x x x x x xx x

     

  31 32 33 34

 

   

   

µ µ

0 0 0 0

      0

   

O(2) O(3) 18

First equation:

o 3 2 2 3

ω µ α α α α

= + + + + +

( ) ( )

x i x x x x x xx x

31 32 33 34

Normal form (reduced number of monomials tried):

o 3 2 2 3

ω β β β β

= + + + +

y i y y y y yy y

31 32 33 34

2 2

β µ µ β β β

+ + + + +

( ) ( ) ignored terms

y y yy y

20 35 36 37

Near-identity transformation (reduced number of monomials tried):

o 3 2 2 3

γ γ γ γ

= + + + +

x y y y y yy y

31 32 33 34

2 2

γ µ µ γ γ γ

+ + + + +

( ) ( ) ignored terms

y y yy y

20 35 36 37 19

By operating as in Example 1, at O(2) it follows:

o ⇒

γ β β γ

= − = =

0 1 1, 0

20 20 20 20

At O(3), in addition to the equations of Example 1, one has:

o    

 

  γ β

α

ω 3

0 0

i 35 35

21

   

 

  i

ω γ α β

− = −

0 0 3

i    

 

  36 22 36

ω

3  

   

 

ω α

γ β

0 0 3

i

   

   

23

37 37

Since no further resonances appear,

β β β

= = = 0 .

35 36 37

Normal form:

o 2

µ ω α

= + +

( )

y i y y y

32 20

Amplitude equation:

o Changing the variables according to: ω

i t

= ∈

( ) ( ) e , ( )

y t A t A t

the normal form is transformed into:

ω ω ω

2 (2 1)

i t i t i t

ω µ ω α

+ = + +

[ ( ) ( )]e ( ) ( ) e ( ) ( ) e

A t i A t i A t A t A t

32

or: 2

µ α

= +

( ) ( ) ( ) ( )

A t A t A t A t

32

This is called (AME).

Amplitude Modulation Equation

3

µ

= O( , )

A A A

Since , the AME describes a slow modulation. Therefore,

the change of variable filters the fast dynamics. 21


PAGINE

25

PESO

160.96 KB

AUTORE

Atreyu

PUBBLICATO

+1 anno fa


DESCRIZIONE DISPENSA

The Center Manifold Method
- Existence of an invariant manifold
- Dependence of the CM on parameters
- Reduction process
- Example 1: a static bifurcation
- Example 2: a dynamical bifurcation
The Normal Form Theory
- Scope: to use a smooth nonlinear coordinate transformation, in order to put the bifurcation equation in the simplest form.
- Algorithm
- Example 1: System independent of parameters
- Example 2: System dependent on a parameter
- Example 3: Non-diagonalizable system.


DETTAGLI
Corso di laurea: Corso di laurea magistrale in ingegneria civile
SSD:
Università: L'Aquila - Univaq
A.A.: 2011-2012

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Sistemi dinamici e teoria della biforcazione e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università L'Aquila - Univaq o del prof Luongo Angelo.

Acquista con carta o conto PayPal

Scarica il file tutte le volte che vuoi

Paga con un conto PayPal per usufruire della garanzia Soddisfatto o rimborsato

Recensioni
Ti è piaciuto questo appunto? Valutalo!

Altri appunti di Sistemi dinamici e teoria della biforcazione

Biforcazione - Teoria
Dispensa
Biforcazione
Dispensa