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# Resonant double-Hopf bifurcations

Resonant double-Hopf bifurcations of 1:1 and 1:3 type:
- internal resonance occurs when the critical frequencies rationally linearly dependent;
- example of Rayleigh-Duffing coupled oscillators in 1:1 and 1:3 internal resonance;
- the r = 1 case;
- the r = 3 case;
- steady-state solutions and fixed points of RAME.

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

Anteprima

### ESTRATTO DOCUMENTO

Since phases appear as a linear combination, we introduce a phase-

combination: γ θ θ

= −

: 1 2

γ θ θ

= −

and recombine the phase-equations according . We obtain:

1 2

γ

{ , , }

a a

RAME in the state-variables :

three 1 2

 1 3 3

2 3 2 2

µ ω ω γ

= − + − +

( ) [2 cos 2 ]

a a b b a b a a

 1 1 0 1 1 1 0 1 1 2

2 8 8

 9 3

 2 2 2 3

ω γ ω γ

+ +

cos cos

b a a b a

 0 1 1 2 0 1 2

8 8

 1 3 3

 2 3 2 2

ν ω ω γ

= − + − +

( ) [2 cos 2 ]

a a b b a b a a

 2 2 0 2 1 2 0 1 1 2

2 8 8

 3 9

 2 3 2 2

ω γ ω γ

+ +

cos cos

b a b a a

0 1 1 0 1 1 2

 8 8

 3 3

c c

 2 3 2 3

γ σ ω γ ω γ

= − + + + −

( sin 2 ) ( sin 2 )

a a a a b a a b a a

0 1 1 2

1 2 1 2 0 1 1 2

ω ω

 8 8

1 1

 3 3 3

 2 4 2 2 2 2 4

ω γ ω γ ω γ

− − −

sin sin sin

b a b a a b a

0 1 1 0 1 1 2 0 1 2

 8 4 8 9

two phase-equations:

 3 3 3 3

c

3 2 2 2 3 2 2

θ ω γ ω γ ω γ

= + + −

sin sin sin 2

a a b a a b a b a a

 1 1 1 0 1 1 2 0 1 2 0 1 1 2

ω

 8 8 8 8

1

 3 3 3 3

c

 3 2 3 2 2 2 2

θ σ ω γ ω γ ω γ

= + + + −

sin sin sin 2

a a a b a b a a b a a

 2 2 2 2 0 1 1 0 1 1 2 0 1 1 2

ω

8 8 8 8

1

Once the RAME have been solved, the phase-equations can be integrated

by quadrature.

Note: while the RAME of a system are pure-amplitude

non-resonant

equations, those of a system are mixed-amplitude-phase

resonant

equations. 10

The r=3 case

In a similar way, the complex AME are found to be:

 1 3 9

c 2 2 2 2 2

µ ω ω ω

= + − + − −

d [ ( ) ] 27

A A i b b A A b A A A b A A

 1 1 1 1 0 1 1 1 0 1 1 2 2 0 1 1 2

ω

 2 2 2

1

 1 1 1

c

 2 2 2 2 3

ν σ ω ω ω

= + + − + − −

d ( ) [ 27( ) ] 3

A i A i b b A A b A A A b A

 1 2 2 2 0 1 2 2 0 1 1 2 1 0 1 1

ω

2 2 6

1

ω ω

= 3

in which has been substituted.

2 1 11

After parameter reabsorbing, and use of the polar representation, we obtain

four real bifurcation equations:

 1 3 27 9

2 3 2 2 2 2

µ ω ω ω θ θ

= − + − − −

( ) cos(3 )

a a b b a b a a b a a

 1 1 0 1 1 1 0 1 1 2 0 1 1 2 1 2

2 8 4 8

 1 27 3 1

 2 3 2 2 2 3

ν ω ω ω θ θ

= − + − − −

( ) cos(3 )

a a b b a b a a b a

 2 2 0 2 1 2 0 1 1 2 0 1 1 1 2

2 8 4 24

 3 9

c

 3 2 2

θ ω θ θ

= + −

sin(3 )]

a a b a a

1 1 1 0 1 1 2 1 2

ω

 8 8

1

 1 1

c

 3 2 3 θ

θ σ ω θ

= + − −

sin(3 )]

a a a b a

2 2 2 2 0 1 1 1 2

 ω

8 24

1

They suggest the following definition for the phase-combination:

γ θ θ

= −

: 3 1 2 12

The RAME are:

 1 3 27 9

2 3 2 2 2 2

µ ω ω ω γ

= − + − −

( ) cos

a a b b a b a a b a a

 1 1 0 1 1 1 0 1 1 2 0 1 1 2

2 8 4 8

 1 27 3 1

2 3 2 2 2 3

ν ω ω ω γ

 = − + − + +

( ) ( ) cos

a a b b a b a a b b a

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

2 8 4 24

 9 1 1 27

c c

3 3 2 4 2 2 2

γ σ ω γ ω γ

= − + − + +

sin sin

a a a a a a a a b a b a a

 1 2 1 2 1 2 1 2 0 1 1 0 1 1 2

ω ω

8 8 24 8

 1 1

The phase-equations are:

 3 9

c

3 2 2

θ ω γ

= + sin

a a b a a

 1 1 1 0 1 1 2

ω

 8 8

1

 1 1

c

 3 2 3

θ σ ω γ

= + − sin

a a a b a

 2 2 2 2 0 1 1

ω

8 24

1 13

• Response (r =1,3 cases) γ

( ), ( ), ( )

a t a t t

After integration, the RAME furnish ; successively, the phase

1 2

θ θ

( ), ( )

t t . The response read:

equations give 1 2  = Φ +

( ) cos( ( )) . . .

x a t t h o t

1 1

 = Φ +

( ) cos( ( )) . . .

y a t t h o t

 2 2

where:

ω θ ω θ

Φ = + Φ = +

( ) : ( ), ( ) : ( )

t t t t t t

1 1 1 2 2 2

are total phases. 14

Steady-state solutions and fixed points of RAME

• The RAME, are of the following type:

 γ

= ( , , )

a F a a

1 1 1 2

 γ

 =

( , , )

a F a a

2 2 1 2

 γ γ

= ( , , )

a a G a a

 1 2 1 2

and phase-equations are of the type:

θ γ

=

 ( , , )

a H a a

1 1 1 1 2

 θ γ

= ( , , )

a H a a

2 2 2 1 2 =

z F z

( )

Note: The RAME can be put in the standard form , with

γ ≠ ≠

= 0, 0

a a

z : ( , , )

a a , if and only if (complete solutions).

1 2

1 2 = =

0, and/or 0

a a

Note: in incomplete solutions ( ), the phases of the

1 2

zero-amplitudes remains undetermined; however, they are inessential. 15

γ =

• ( , , ) const

a a

The fixed points of RAME are solutions of:

1 2

s s s  γ =

( , , ) 0

F a a

1 1 2

s s s

 γ

 =

( , , ) 0

F a a

2 1 2

s s s

 γ =

( , , ) 0

G a a

 1 2

s s s

Consequently, the associated phases (if determined) are linearly varying

functions: 0 0

θ κ θ θ κ θ

= + = +

( ) , ( )

t t t t

1 1 1 2 2 2

s s s s s s

κ =

κ

( , ) const

with the frequency corrections.

1 2

s s 16

• For a complete solution, we prove that (non-trivial)

the fixed points of the

(for incomplete solution,

RAME are periodic motions for the original system

this is a trivial matter). Indeed:

a constant phase-difference:

0 0

γ θ θ κ θ κ θ

= − = + − +

: const

= =

r ( ) ( )

r t t 1,3

r

1 2

s s 1 1 2 2

s s s s s

entails a relation between frequency corrections and initial phases:

0 0

κ κ θ θ γ

− = − =

0,

r r

1 2 1 2

s s s s s

ω ω

= r

consequently, since , the total phases read:

2 1 0

ω θ ω κ θ

Φ = + = + +

( ) : ( ) ( )

t t t t

1 1 1 1 1 1

s s s

0 0

ω θ ω κ θ ω κ θ

Φ = + = + + = + +

( ) : ( ) ( ) [ ( ) ]

t t t t r t

2 2 2 2 2 2 1 1 2

s s s s s

i.e. the nonlinear frequencies are in the same integer ratio r as the

k

ω

linear frequencies :

k

ω κ ω κ

Ω = + Ω = + = Ω

: , : r

1 1 1 2 2 2 1

s s s s 17

The steady response, therefore, is periodic, and it reads:

 0

θ

= Ω + +

 ( ) cos( ) . . .

x a t t h o t

1 1 1

s s

 0

 θ γ

= Ω + − +

( ) cos[ ( ) ] . . .

y a t r t h o t

2 1 1

s s s

γ

Note: the phase difference is given by the solution; however, an initial

s

θ 0

phase, e.g. remains undetermined, since the limit cycle can be

1s

traveled starting from any of its points. 18

Finding the fixed points of RAME

• In the case, the RAME admit:

r=1

(T) the trivial solution:

γ µ ν σ

= = ∀ ∀

0, , ( , , )

a a

1 2

## T T T

γ

with the phase-difference being undetermined.

(P) a number of (or complete) periodic solutions:

bimodal

µ ν σ µ ν σ γ γ µ ν σ

= = =

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

a a a a

1 1 2 2

## P P P P P P

θ θ

with associated, determined phases and .

1P 2P 19

• In the case the RAME admit:

r=3

(T) the trivial solution:

γ µ ν σ

= = ∀ ∀

0, , ( , , )

a a

1 2

## T T T

(M) a (incomplete) periodic solution:

mono-modal

ν γ

= = ∀

0, ( ),

a a a

1 2 2

with: θ θ σ ν θ

= ∀

( , ),

2 2 1

## M M M

(P) one or more (complete) periodic solutions:

bimodal

µ ν σ µ ν σ γ γ µ ν σ

= = =

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

a a a a

1 1 2 2

## P P P P P P

θ θ

with associated phases and .

1P 2P 20

Stability of steady solutions

It needs to distinguish: (s=P): since all quantities are

The steady-solution is complete

=

z F z

( )

determined, and the RAME can be put in the normal form ,

γ

=

z : ( , , )

a a

with , stability is governed by the variational equation:

1 2 δ δ

=

z J z

## J

A zero eigenvalue of denotes a branching of a new periodic

## P

solution; a pair of purely imaginary eigenvalues denotes a branching of

a solution (i.e. a periodically modulated periodic

quasi-periodic

motion). γ

(s=T,M): since is undetermined,

The steady-solution is incomplete

s

and the RAME are in standard form, use of the (not reduced) AME

not

must be made. Examples are given below. 21

• Stability of the trivial solution (r=1,3 cases)

= = 0

## A A

The variation of the AME, based on , reads:

1 2

 1

δ µδ

=

## A A

 1 1

2

 1

δ ν σ δ

= +

( )

A i A

 2 2

2

whose solution is: 1 1

ˆ ˆ

δ δ µ δ δ ν σ

= = +

exp( ), exp[( ) ]

A A t A A i t

1 1 2 2

2 2

ˆ ˆ

δ δ

,

A A constants. The trivial solution is therefore stable when

with 1 2

µ ν

< <

0, 0 . 22

• Stability of the mono-modal solution (r=3 case)

The variation of the AME, based on:

1 0

κ θ

= = = +

0, : exp[ ( )]

A A A a i t

1 2 2 2 2 2

## M M M M M M

2

assumes the following (uncoupled) form:

 1

2

δ δ

= +

( )

 1 1 2 2 1

## M

4

 1 1

 2 2 0

δ δ κ θ δ

= + + +

( ) exp[2 ( )]

A C C a A C a i t A

 2 1 2 2 2 3 2 2 2 2

## M M M M

4 4

∈ = +

, : ∈

R C R iI

where are coefficients.

j j j j ≠

κ

Note: due to the presence of the frequency correction , ;

const

## A

2M 2 M

consequently, the second variational equation depends on time. 23

a change of variable is

To render the second equation autonomous,

performed: δ δ α β

= +

exp[ ( )]

A B i t

2 2

β

with to be determined. By requiring the coefficients are

α, θ

= = 0

β

κ

α

independent of time, it follows: ; moreover is taken

2 M 2 M

for simplicity.

In the new variables, the variational equations read:

 1

2

δ δ

= +

( )

 1 1 2 2 1

## M

4

 1 1

 2 2

δ κ δ δ

= + − +

( )

B C C a i B C a B

 2 1 2 2 2 2 3 2 2

## M M M

4 4

Since the equations are linear, a Cartesian representation is better

suited: 24

δ δ

= + = +

,

A p iq B p iq

1 1 1 2 2 2

leading to four real variational equations:

   

 

p p

0

1 1

11

   

 

q q

0 J

   

 

1 1

22

=

   

 

J J

p p

33 34

2 2

   

 

J J

 

q q

   

43 44

2 2

where: R 2

2

= = +

:

J J R a

11 22 1 2 M

4

2 2

a a κ

2 2

## M M

= + + = − + − + +

( ) , ( )

## J R R R J I I I

33 1 2 3 34 1 2 3 2 M

4 4 ,

2 2

a a

κ

2 2

= + + − = + −

( ) , ( )

## J I I I J R R R

33 1 2 3 2 34 1 2 3

## M

4 4

The eigenvalues (four real, or two real and two complex conjugate),

govern the stability of the M-solution. 25

PAGINE

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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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