Next: Bifurcations Up: Continuation of torus bifurcation Previous: Continuation of torus bifurcation   Contents

Mathematical definition

A torus bifurcation of limit cycles (Neimark-Sacker, NS) generically corresponds to a bifurcation to an invariant torus, on which the flow contains periodic or quasi-periodic motion. It can be characterized by adding an extra constraint G=0 to (50) where G is the torus test function which has four components from which two are selected. The complete BVP defining a NS point using a minimally extended system is

ì
ï
ï
ï
í
ï
ï
ï
î
dx

dt

- Tf(x,a)
= 0
x(0) - x(1)
= 0
ó
õ
1

0 
áx(t),
×
x
 

old 
(t) ñdt
= 0
G[x,T,a]
= 0
(77)

where

G= æ
ç
ç
è
G11
G12
G21
G22
ö
÷
÷
ø

is defined by requiring

N3 æ
ç
ç
ç
è
v1
v2
G11
G12
G21
G22
ö
÷
÷
÷
ø
= æ
ç
ç
ç
ç
ç
è
0
  0
0
  0
1
  0
0
  1
ö
÷
÷
÷
÷
÷
ø
.
(78)

Here v1 and v2 are functions and G11,G12,G21 and G22 are scalars and

N3 = é
ê
ê
ê
ê
ê
ë
D-Tfx(x(·),a)
    w11
   w12
d0-2kd1+d2
    w21
    w22
Intv01
    0
    0
Intv02
    0
    0
ù
ú
ú
ú
ú
ú
û
(79)

where the bordering functions v01,v02,w11,w12, vectors w21and w22 are chosen so that N3 is nonsingular []. This method (using system (78) and (79)) is implemented in the curve definition file neimarksacker. The discretization is done using orthogonal collocation over the interval [0 2].