3rd Sperlonga Summer School on
Mechanics and Engineering
Sciences
23-27 September 2013
SPERLONGA
CENTER MANIFOLD AND
NORMAL FORM THEORIES
ANGELO LUONGO
1
THE CENTER MANIFOLD METHOD
 Existence of an invariant manifold
 Linear systems
The state-space X   N of the linear system x (t )  Jx(t ) is direct sum of three
invariant sub-spaces, i.e. X  Xc  Xs  Xu , where:
 Xc is the center subspace, of dimension Nc, spanned by the (generalized)
eigenvectors associated with non-hyperbolic eigenvalues of J;
 Xs is the stable subspace, of dimension Ns, spanned by the (generalized)
eigenvectors associated with hyperbolic eigenvalues of J having negative real
part;
 Xu is the unstable subspace, of dimension Nu, spanned by the (generalized)
eigenvectors associated with non-hyperbolic eigenvalues of J having positive
real part.
2
 Nonlinear systems
We consider the nonlinear system (in local form):
x (t )  F(x(t ), μ)
admitting the critical equilibrium (x  0, μc  0) . We assume that J : Fx (0, 0)
posses Nc >0 critical eigenvalues, Ns >0 stable eigenvalues and Nu =0 unstable
eigenvalues.
The Center Manifold Theorem states that the asymptotic dynamics of the system
around the equilibrium point x  0 , at the critical value of the parameters μ  μ c ,
takes place on a (critical) manifold Mc  X , which has the following properties:
 Mc has dimension Nc;
 Mc is tangent to the critical subspace Xc at x  0 ;
 Mc is attractive, i.e. all the orbits tend to it when t   .
3
The center manifold is therefore an Nc-dimensional surface in the
N=Nc +Ns-dimensional state-space.
 Example:
 To analyze the asymptotic dynamics, it needs:
(a) to find the center manifold Mc ;
(b) to obtain the reduced Nc –dimensional equations governing the motion on
Mc (bifurcation equations).
4
 Dependence of the CM on parameters
Since we are interested not only in the dynamics at μ  μ c , but also at the dynamics
at μ close to μ c , we can use the ‘trick’ to consider μ as additional ‘critical’
variables, by considering the extended dynamical system:

x (t )  F(x(t ), μ(t ))
μ (t )  0

X
Therefore, the critical subspace becomes c : Xc  P with P : {μ} the parameter
space. Hence:
 M c has dimension Nc+M;

 M c is tangent to the critical subspace Xc ;



 M c is attractive, i.e. all the orbits tend to it when t   .
5
 Reduction process
 Equations of motion, expanded:
By expanding x (t )  F(x(t ), μ) (and ignoring the dummy equations μ  0 ) for
small x(t ) and μ close to μc , we have:
x (t )  Jx(t )  f (x(t ), μ),
f  O( x(t ) , μ )
2
2
 Linear transformation of the variables:
After letting x(t ) : (xc (t ), xs (t )) , with xc (t )  Xc critical variables and
xs (t )  Xs stable variables, a proper linear transformation uncouples the linear
part of the equations (t omitted):
 x c   J c
 x    0
 s 
0   xc   fc (xc , x s , μ) 
  x    f (x , x , μ) 
Js  s   s c s

6
 Center manifold: active and passive coordinates
Cartesian equations for the CM:
x s  h(x c , μ)
Tangency requirements:
h (0, 0)  0, h x (0, 0)  0, h μ (0, 0)  0 .
The equation expresses the stable variables xs as (unknown) functions of the
critical variables xc; therefore xc are also said active coordinates, and xs
passive coordinates.
 Time derivative of xs:
The passive character of xs also holds for time-derivatives. By using the
chain rule and the upper partition of the equations of motion:
d
h ( x c , μ )  h x ( x c , μ ) x c
dt
 h x ( x c , μ )( J c x c  f c ( x c , h ( x c ), μ ))
x s 
7
 Equation for the center manifold:
The lower-partition of the equation of motion supplies the equation for
determining Mc :
h x (x c , μ )( J c x c  f c ( x c , h ( x c ), μ ))  J s h( x c )  f s ( x c , h (x c ), μ )
This equation can be solved, e.g., by using power series expansions, (starting
from degree-2 polynomial, due to the required tangency):
h(xc , μ)  α 2 z 2  α 3z 3   ,
z : {x c , μ}
 Bifurcation equations:
The upper-partition governs the dynamics on Mc :
x c  J c x c  fc (x c , h(x c ), μ )
8
 Example 1: a static bifurcation
 Nonlinear, 2-dimensional system:
 x   
   
 y 0
0   x   xy  cx3 

   

2
1  y   bx 
 Critical point, critical and stable coordinates:
c  0, xc  {x}, xs  { y}
 Equation for the Center Manifold:
y  h( x,  )  hx ( x,  )[  x  xh( x,  )  cx3 ]  h( x,  )  bx 2
 Power series expansion:
y  h( x,  )  { 20 x 2  11 x   02  2 }  {30 x3  }  
9
 Zeroing separately the different powers:
x 2 :  20  b  0
x : 11  0
 2 :  02  0

 By solving for α’s coefficients, at the lowest order, we have:
y  bx 2  O( ( x,  ) )
3
 Bifurcation equation:
x   x  (b  c) x3
If b+c<0, a supercritical (stable) pitchfork occurs; if b+c>0, a sub-critical
(unstable) pitchfork occurs; if b+c=0, higher-order terms must be evaluated.
10
 Example 2: a dynamical bifurcation
 3-dimensional nonlinear system:
 x   
  
 y   1
 z   0
  
1

0

0  x  
xz

  
0  y   
yz

  2
2
1
 z   x  kxy  y 
 Equation for CM:
z  1 x 2   2 y 2  3 xy   4  x  5  y  O(3)
 z-equation (passive coordinate), transformed:
2( 2  1 ) xy   3 ( x 2  y 2 )   5  x   4  y
 (1 x 2   2 y 2   3 xy   4  x   5  y )  x 2  kxy  y 2  O(3)
11
 Zeroing separately the different powers in the z-equation:
 x2 :
 2
y :

 xy :
 x :

 y :

3  1  1  0
1  1  k / 5
  1  k / 5
 3   2  1  0
 2
2( 2  1 )  3  k  0  3  k / 5
  0
 4  5  0
 4
5  0
5   4  0
 Bifurcation equations:
 x   x  y  x[(1  k / 5) x 2  k / 5 xy  (1  k / 5) y 2 ]

2
2

y

y
x
y
[(1
k
/
5)
x
k
/
5
xy
(1
k
/
5)
y
]








12
THE NORMAL FORM THEORY
 Scope:
To use a smooth nonlinear coordinate transformation, in order to put the
bifurcation equation in the simplest form.
 Algorithm:
Equations of motion:
x  Jx  f (x)
Transformed Equations:
y  Jy  g ( y )
Near-identity transformation:
x  y  h( y )
where x (possibly) includes the dummy variables µ.
13
o Transformed equation in the h(y) unknown:
By differentiating the nearly-identity transformation:
x  [I  h y ( y )]y
the equation of motion is transformed into:
[I  h y ( y )][ Jy  g ( y )]  J[ y  h ( y )]  f ( y  h ( y ))
or:
h y ( y ) Jy  Jh ( y )  f ( y  h ( y ))  [I  h y ( y )] g ( y )
which is a differential equation for h(y) for any given g(y). A series solution is
often necessary.
14
o By letting:
 f ( y )   f 2 ( y )   f3 ( y ) 

 

 


(
)
(
)
(
)
g
y
g
y
g
y

  2   3  
 h(y )   h (y )   h (y ) 

  2   3 
with the homogeneous polynomial of k-degree:
Mk
Mk
Mk
m 1
m 1
m 1
fk (y )    kmp km (y ), g k (y )    kmp km (y ), h k (y )    kmp km (y )
15
o Zeroing the independent monomials:
Lk γ k  α k  βk
where (if J is diagonal):
L k  diag[  i ],
N
 i :  ( m j  j i ),
j 1
N
m
j 1
j
k
o Resonance:
Since  j  0, 0, , i1 , i2 , ,  i  0 for some i (i.e. Lk is singular);
hence, βk  0 must be taken for compatibility, and resonant terms survive in
the normal form!!!
16
 Example 1: System independent of parameters
o System at a Hopf bifurcation:
3
2
2
3
 x  i 0   x    31 x   32 x x   33 xx   34 x 
 , x,   

   

3
2
2
3 
 x   0 i   x    31 x   32 x x   33 xx   34 x 
o Normal form:
y  i y   31 y 3   32 y 2 y   33 yy 2   34 y 3
o Near-identity transformation:
x  y   31 y 3   32 y 2 y   33 yy 2   34 y 3
17
o Equation for the coefficients:
 2i
 0

 0

 0
0
0
0
0
0 2i
0
0
0    31    31   31 
  
 

0    32    32   32 






0   33
 33  33 
 

   
4i    34    34   34 
o Solution:
 31   33   34  0,  32   32
o Normal form:
y  i y   32 y 2 y
2
y
y is resonant, since 21  2  1 , i.e. 2(i )  (i )  i
The term
18
o Amplitude equation:
Changing the variables according to:
y (t )  A(t ) eit ,
A(t )  
the normal form is transformed into:
[ A (t )  i A(t )]eit  i A(t ) eit   32 A2 (t ) A(t ) e(21)it
or:
A (t )   32 A2 (t ) A(t )
3
This is called Amplitude Modulation Equation (AME). Since A  O( A ) , the
AME describes a slow modulation. Therefore, the change of variable filters the
fast dynamics.
19
 Example 2: Non-diagonalizable system
o Double-zero bifurcation equations:
2
2
 x1  0 1   x1    21 x1   22 x1 x2   23 x2 

 
   

2
2
 x2  0 0   x2    24 x1   25 x1 x2   26 x2 
o Normal Form:
2
2
 y1  0 1   y1    21 y1   22 y1 y2   23 y2 

  
   

2
2
 y2  0 0   y2    24 y1   25 y1 y2   26 y2 
o Near-Identity transformation:
2
2
 x1   y1    21 y1   22 y1 y2   23 y2 
 
      
2
2
 x2   y2    24 y1   25 y1 y2   26 y2 
20
o By zeroing the coefficients of the three monomials in the two transformed
equations:
0
2

0

0
0

0
0
0
1
0
0
0
0
0
0
0
0
0
1 0 0    21    21   21 
  

0 1 0    22    22   22 
0 0 1   23    23   23 

   
0 0 0    24    24   24 
2 0 0    25    25   25 

   
0 1 0    26    26   26 
Since Rank[L 2 ]  4 :
K (L 2 )  span{u 21 , u 22 }, u 21 : (0,1, 0, 0, 0,1)T , u 22  (0, 0,1, 0, 0, 0)T
K (LT2 )  span{v 21 , v 22 }, v 21  (2, 0, 0, 0,1, 0)T , v 22  (0, 0, 0,1, 0, 0)T
21
o Compatibility condition:
2( 21   21 )  ( 25   25 )  0,
 24   24  0
It is possible to take  22   23   26  0 and  21  0 or  25  0 .
o By taking  25  0, 21   21   25 / 2 the NF reads (Takens normal form):
1

2


(
)


y
y
y
0
1
 1 
  1   21
25
1 


2
  
 y  
0
0
y
  2   y2
 2 

 24 1

o By taking  21  0,  25   25  2 21 the NF reads (Bogdanov normal form):
0

 y1   0 1   y1  

  
 

2
y
0
0
y
  2    24 y1  ( 25  2 21 ) y1 y2 
 2 
The Normal Form is not unique!!
22