A Heteroclinic Example.

The following system of five equations RuMa:95

$$\begin{array}{rcl} \dot{x} & = & \mu \, x + x\, y - z\, u, \\... ... 4\sigma \\ \dot{v} & = & \zeta u / 4 - \zeta v / 4 \end{array}$$ (25.1)

has been used to describe shearing instabilities in fluid convection. The equations possess a rich structure of local and global bifurcations. Here we shall reproduce a single curve in the $(\sigma,\mu)$-plane of codimension-one heteroclinic orbits connecting a non-trivial equilibrium to the origin for $Q=0$ and $\zeta=4$. The defining problem is contained in equation-file she.f^25.1, and starting data for the orbit at $(\sigma,\mu)=(0.5,0.163875)$ are stored in she.dat, with a truncation interval of PAR(11)=85.07.

We begin by computing towards $\mu=0$ with the option IEQUIB=-2 which means that both equilibria are solved for as part of the continuation process.

@dm she
make first

This yields the output

  BR    PT  TY LAB    PAR(3)        L2-NORM    ...    PAR(1)     
   1     5       2  4.528332E-01  3.726787E-01 ...  1.364973E-01
   1    10       3  3.943370E-01  3.303798E-01 ...  1.044119E-01
   1    15       4  3.358942E-01  2.873213E-01 ...  7.515570E-02
   1    20       5  2.772726E-01  2.433403E-01 ...  4.952636E-02
   1    25       6  2.181955E-01  1.981358E-01 ...  2.845849E-02
   1    30  EP   7  1.581633E-01  1.512340E-01 ...  1.292975E-02

Alternatively, for this problem there exists an analytic expression for the two equilibria. This is specified in the subroutine PVLS of she.f. Re-running with IEQUIB=-1

make second

we obtain the output

  BR    PT  TY LAB    PAR(3)        L2-NORM    ...    PAR(1)     
   1     5       2  4.432015E-01  3.657716E-01 ...  1.310559E-01
   1    10       3  3.723085E-01  3.142439E-01 ...  9.300982E-02
   1    15       4  3.008842E-01  2.611556E-01 ...  5.933966E-02
   1    20       5  2.286652E-01  2.062194E-01 ...  3.179939E-02
   1    25       6  1.555409E-01  1.491652E-01 ...  1.239897E-02
   1    30  EP   7  8.107462E-02  9.143108E-02 ...  2.386616E-03

This output is similar to that above, but note that it is obtained slightly more efficiently because the extra parameters PAR(12-21) representing the coordinates of the equilibria are no longer part of the continuation problem. Also note that AUTO has chosen to take slightly larger steps along the family. Finally, we can continue in the opposite direction along the family from the original starting point (again with IEQUIB=-1).

make third

Figure 25.1: Projections into $(x,y,z)$-space of the family of heteroclinic orbits.

  BR    PT  TY LAB    PAR(3)        L2-NORM    ...    PAR(1)     
   1     5       8  4.997590E-01  4.060153E-01 ...  1.637322E-01
   1    10       9  5.705299E-01  4.551872E-01 ...  2.065264E-01
   1    15      10  6.416439E-01  5.031844E-01 ...  2.507829E-01
   1    20      11  7.133301E-01  5.500668E-01 ...  2.959336E-01
   1    25      12  7.857688E-01  5.958712E-01 ...  3.415492E-01
   1    30      13  8.590970E-01  6.406182E-01 ...  3.872997E-01
   1    35  EP  14  9.334159E-01  6.843173E-01 ...  4.329270E-01

The results of both computations are presented in Figure 25.1, from which we see that the orbit shrinks to zero as PAR(1)=$\mu \to 0$.