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# Detective bifurcations

Detective bifurcations, the double Hopf-case:
- a self-excited non lynear system with non-symmetric stiffness and damping;
- linear stability of the trivial equilibrium: exact analysis;
- linear stability of the trivial equilibrium: perturbation analysis;
- nonlinear, multiple-scale bifurcation analysis. Vedi di più

Esame di Dynamical systems and bifurcation theory docente Prof. A. Luongo

Anteprima

### ESTRATTO DOCUMENTO

perturbation analysis

Linear stability of the trivial equilibrium:

The stability analysis of the trivial equilibrium is repeated, as an example, via

evaluation of the eigenvalue sensitivities.

• Rescaling: µ ν σ ε µ ν σ

( , , ) ( , , )

• Characteristic equation:

2 2 2 2 2

λ ω ε µλ λ ω σ νλ

+ − + + − =

( ) [ ( ) )] 0

• Series expansion

ε λ ω ω

→ → ± ±

0 ,

i i

When , then . A must

fractional power series expansion

be used: 1/ 2 3/ 2

λ λ ε λ ελ ε λ

= + + + +

0 1 2 3 6

• Perturbation equations:

0 2 2

ε λ ω

+ =

: ( ) 0

0

1/ 2 2 2

ε λ ω λ λ

+ =

: ( )4 0

0 0 1

1 2 2 2 2 2 2 2

ε λ ω λ λ λ λ ω λ ω λ µ σ λ ν

+ = − + + + + −

: ( )4 2 (3 ) ( )

0 0 2 1 0 0 0 0

3/ 2 2 2 3 2 2 2 2

ε λ ω λ λ λ λ λ λ λ ω λ ω λ µ λν

+ = − + + + −

: ( )4 4 4 (3 ) (3 )

0 0 3 0 1 1 2 0 0 1 1

• Generating solution: λ ω

= i

0

λ ω

= −

i

(the solutions generated by is obtained by complex conjugation).

0

1/ 2

ε

• -order: trivially satisfied 7

ε

• solvability at -order: 1

2

λ σ ων

= − +

( )

i

1 2

ω

4

3/ 2

ε

• solvability at -order: σ 1

λ λ λ µ

= − +

( )

i

1 2 1 3

ω

8 4

Two cases arise: σ ν

and

(a) in which do not vanish simultaneously

generic perturbation,

λ ≠

( 0) ;

1 3/ 2

ε

λ

σ ν =

( 0)

and

(b) in which vanish . -solvability

singular perturbation 1

is trivially satisfied! 8

generic perturbation: σ

1 1

(1,2) (1,2)

λ σ ων λ µ

= ± − + = − +

, ,

i i

1 2 3

ω ω

2 8 4

ε

from which, after reabsorbing : σ

1 1

(1,2)

λ ω σ ων µ

= ± − + + −

i i i 3

ω ω

2 4 8

σ ν

= =

0, 0

not valid close to .

singular perturbation:

An ordering violation occurs, since the leading term vanishes. An

integer power expansion would be necessary. Not an efficient

procedure! 9

Reconstitution method:

A uniformly valid expression is built up, recombining all the

in a whole

solvability conditions: 1/ 2 3/ 2

λ λ λ ε λ ελ ε λ

∆ = − = + + +

0 1 2 3

2 2 3/ 2

λ ελ ε λ λ

∆ = + +

2

1 1 2 σ

1 1

3/ 2

ε σ ων ε λ µ

= − + + − + =

( ) 2 ( )

i i

1

2 3

ω ω

4 8 4

σ

1 1

ε σ ων λ µ

− + + ∆ − +

[ ( ) 2 ( )]

i i

2 3

ω ω

4 8 4

ε

After reabsorbing , a is obtained:

reconstituted sensitivity equation

σ

1 1

2

λ µ λ σ ων

∆ + − ∆ + − =

( ) ( ) 0

i i

3 2

ω ω

2 4 4 10

• Asymptotic expression for the critical manifold

λ λ λ β

≡ ∆ = ∆ =

Re( ) Re( ) 0 i

On the critical manifold . In order that :

σ σ

2

β β ν ωµβ

+ − = = −

0, 2

3 2

ω ω

4 4

β

By eliminating : σ

ν µ σ σ µ σ µσ

= ± + + = ± +

( 1 ) O( )

4

ω

16 2

ε

O( )

to within an error of order , not

which recovers the exact result

accounted for in the analysis. 11

Nonlinear, multiple-scale bifurcation analysis

We investigate the dynamics of the nonlinear system around the bifurcation

point.

• Rescaling:

By introducing the rescaling: 1/ 2

ε

µ ν σ ε µ ν σ →

→ ( , ) ( , )

x y x y

( , , ) ( , , ) ,

the equations read:  

 

     

2 3

2 µ

− − − − − −

ω ( )( ) ( ) 0

x x x b y x y x c y x

1  

ε

+ + =

 

     

 

2

2 3

ω

     

0

y y σ ν

 

0 − + − − + −

 

( )( ) ( )

x x b y x y x c y x 12

• Fractional series expansions:

       

 

ε

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

x t t x t t x t t x t t

( ; )

x t 0 0 1 1 0 1 2 0 1 3 0 1

1/ 2 3/ 2

ε ε ε

= + + + +

       

 

ε

 

( ; ) ( , , ) ( , , ) ( , , ) ( , , )

y t y t t y t t y t t y t t

       

0 0 1 1 0 1 2 0 1 3 0 1

/2

k

ε

= =

( 0,1, )

t k

where . By applying the chain rule:

k

d 1/ 2 3/ 2

ε ε ε

= + + + +

d d d d

0 1 2 3

d t

2

d 2 1/ 2 2 3/ 2

ε ε ε

= + + + + + +

d 2 d d (d 2 d d ) 2 (d d d d )

0 0 1 1 0 2 0 3 1 2

2

d t 13

• Perturbation equations:

 2 2

ω

+ + =

 d 0

x x y

0 0 0 0

0

ε 

: 2 2

 ω

+ =

d 0

y y

0 0 0

 2 2

ω

 + + = −

d 2 d d

x x y x

0 1 1 1 0 1 0

1/ 2

ε 

: 2 2

 ω

+ = −

d 2 d d

y y y

0 1 1 0 1 0

 2 2 2

ω µ

+ + = − + − +

d (d 2 d d ) 2 d d d

x x y x x x

0 2 2 2 1 0 2 0 0 1 1 0 0

 2 2 3

+ + − − + −

 d ( ) (d d ) ( )

b x x b y x y x c y x

1 0 0 0 0 0 0 0 0 0 0 0 0

ε 

: 2 2 2 σ ν

ω

+ = − + − − +

d (d 2 d d ) 2 d d d

y x x

y y y

 0 1 1 0 0 0

0 2 2 1 0 2 0

 2 2 3

+ + − − − − −

d ( ) (d d ) ( )

b y y b y x y x c y x

 2 0 0 0 0 0 0 0 0 0 0 0 0

(continue)

14

 2 2 2

ω

+ + = − + − + −

d 2(d d d d ) (d 2 d d ) 2 d d

x x y x x x

0 3 3 3 0 3 1 2 0 1 0 2 1 0 1 2

 µ

+ +

(d d )

x x

 1 0 0 1

 2

+ + +

[ (d d ) 2 d ]

b x x x x x x

 1 0 1 0 0 1 0 1 0 0

 2

+ − − + −

{( ) [(d ( ) d ( )]

b y x y x y x

0 0 0 1 0 0 0 1 1

 + − − −

2( )( ) d ( )]

y x y x y x

 0 0 1 1 0 0 0

 2

+ − −

 3 ( ) ( )

c y x y x

0 0 1 1

3/ 2

ε 

: 2 2 2

ω

+ = − + − + −

d 2(d d d d ) (d 2 d d ) 2 d d

y y y

y y

 0 3 3 0 3 1 2 0 1 0 2 1 0 1 2

 ν

+ +

(d d )

x x

 1 0 0 1

 2

+ + +

[ (d d ) 2 d ]

b y y y y y y

2 0 1 0 0 1 0 1 0 0

 2

− − − + −

 {( ) [(d ( ) d ( )]

b y x y x y x

0 0 0 1 0 0 0 1 1

 + − − −

2( )( ) d ( )]

y x y x y x

 0 0 1 1 0 0 0

 2

− − −

3 ( ) ( )

c y x y x

 0 0 1 1

15

• Generating solution:

The generating equation admits the general solution:

 i

ω ω

i t i t

= + +

( , , , ) e ( , , , ) e .

x A t t t B t t t t c c

0 0.

0 1 2 3 1 2 3 0

ω

 2

 ω

i t

= +

( , , , ) e . .

y B t t t c c

 0

0 1 2 3 = 0

## B

. To eliminate secular terms, must be taken; therefore:

with ∈

A

( , )

B  ω

i t

 =

( , , , ) e

x A t t t 0

0 1 2 3

 = 0

y 0 16

• Higher-order equations:

They are, at any order, of the following type:

 ω

ik t

2 2

ω

+ + = +

d e . .

x x y f c c

0

0 j j j jk

 =

1,3,

k

 ∑ ω

ik t

2 2

ω

+ = +

 d e . .

y y g c c

0

0 j j jk

 =

1,3,

k

∈ t

( , )

with constant on the -scale.

f g 0

jk jk

• Higher-order solutions:

Solutions are harmonic and polynomial-harmonic. By ignoring these latter

(secular terms), we let: ∑ ω

ik t

= +

ˆ ˆ

( , ) ( , ) e . .

x y x y c c

0

j j jk jk

k

from which an algebraic problem follows: 17

 2 2

ω − + =

ˆ ˆ

(1 )

k x y f

 jk jk jk

 2 2

ω − =

ˆ

 (1 )

k y g

jk jk

1

k

if ( forcing terms) the equation are and

non-resonant non-singular

therefore they admit an unique solution:

 f g

jk jk

= −

ˆ

x

 jk 2 2 2 2 4

ω ω

 − −

(1 ) (1 )

k k

 g

 jk

=

ˆ

y

 jk 2 2

ω

 (1 )

k

= 1

k

If ( forcing terms), the equations are and

resonant singular,

therefore call for a compatibility (or condition:

solvability)

= 0

g 1

j

If this holds, they admit infinite solutions: 18

 =

ˆ

x C

1

j

 ∀ ∈

=

ˆ

 y f

1 1

j j

ω =

exp( )

C i t 0

## C

However, since +c.c. repeats the generating solution,

0

is taken:  =

ˆ 0

x 1

j

 =

ˆ

 y f

1 1

j j 19

1/ 2

ε

• -order:

equations:  ω

i t

2 2

ω ω

 + + = − +

d 2 d e . .

x x y i A c c

0

0 1 1 1 1

 2 2

 ω

+ =

d 0

y y

0 1 1

solvability condition: automatically satisfied

solution:  = 0

x

1

 ω

i t

 ω

= − +

2 d e . .

y i A c c

0

1 1

d A

while remains undetermined at this order.

1 20

• ε-order:

equations:  ω ω

3

i t i t

2 2

ω

+ + = + +

 d e e . .

x x y f f c c

0 0

0 2 2 2 21 23

 ω ω

3

i t i t

2 2

 ω

+ = + +

d e e . .

y y g g c c

0 0

0 2 2 21 23

where:  

 

     

ω

ωµ ω

− − + 1

3 ( )

c i b b

f 2

i i

0 1

21 2 2

= + − −  

  d d

  A A A A A

    2 1

2

σ ων ω

− + + ω

   

3 0

g i c i b

     

4

21 0

   

ω

− − +

( )

f c i b b

23 0 1 3

=

    A

ω

+

g c i b

   

23 0

solvability condition:

= 0

g

By requiring , it follows:

21 1

2 2

σ ων ω

= − + + +

d [( ) (3 ) ]

A i A c i b A A

1 0

2

ω

4 21

PAGINE

37

PESO

1.02 MB

AUTORE

PUBBLICATO

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### DESCRIZIONE DISPENSA

Detective bifurcations, the double Hopf-case:
- a self-excited non lynear system with non-symmetric stiffness and damping;
- linear stability of the trivial equilibrium: exact analysis;
- linear stability of the trivial equilibrium: perturbation analysis;
- nonlinear, multiple-scale bifurcation analysis.

DETTAGLI
Corso di laurea: Corso di laurea magistrale in ingegneria matematica
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 Dynamical systems and bifurcation theory 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.

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