Electronic Circuit of Freire et al.

Consider the following model of a three-variable electronic circuit [#!FrRLuGaPo:93!#]

$$\left \{ \begin{array}{rcl} \dot{x} & = & \left [-(\beta+\nu) x +... ...beta x -(\beta+\gamma)y -z -b_3(y-x)^3, \\ \dot{z} & = & y. \end{array} \right.$$ (24.1)

These autonomous equations are also considered in the AUTO demo tor.

First, we copy the demo into a new directory and compile

@dm cir

The system is contained in the equation-file cir.f and the initial run-time constants are stored in c.cir.1 and h.cir.1. We begin by starting from the data from cir.dat for a saddle-focus homoclinic orbit at $\nu=-0.721309$, $\beta=0.6$, $\gamma=0$, $r=0.6$, $A_3=0.328578$ and $B_3=0.933578$, which was obtained by shooting over the time interval $2T=$PAR(11)$=36.13$. We wish to follow the family in the $(\beta,\nu)$-plane, but first we perform continuation in $(T,\nu)$ to obtain a better approximation to a homoclinic orbit.

make first

yields the output

 BR  PT  TY LAB     PERIOD       L2-NORM    ...   PAR(1)     
  1  21  UZ   2  1.000000E+02  1.286637E-01 ... -7.213093E-01
  1  42  UZ   3  2.000000E+02  9.097899E-02 ... -7.213093E-01
  1  50  EP   4  2.400000E+02  8.305208E-02 ... -7.213093E-01

Note that $\nu=$PAR(1) remains constant during the continuation as the parameter values do not change, only the the length of the interval over which the approximate homoclinic solution is computed. Note from the eigenvalues, stored in d.1 that this is a homoclinic orbit to a saddle-focus with a one-dimensional unstable manifold.

We now restart at LAB=3, corresponding to a time interval $2T=200$, and change the principal continuation parameters to be $(\nu,\beta)$. The new constants defining the continuation are given in c.cir.2 and h.cir.2. We also activate the test functions pertinent to codimension-two singularities which may be encountered along a family of saddle-focus homoclinic orbits, viz. $\psi_2$, $\psi_4$, $\psi_5$, $\psi_9$ and $\psi_{10}$. This must be specified in three ways: by choosing NPSI=5 and appropriate IPSI(I) in h.cir.2, by adding the corresponding parameter labels to the list of continuation parameters ICP(I) in c.cir.2 (recall that these parameter indices are 20 more than the corresponding $\psi$ indices), and finally adding USZR functions defining zeros of these parameters in c.cir.2. Running

make second

results in

BR  PT  TY LAB    PAR(1)     ...    PAR(2)     ...    PAR(25)       PAR(29)    
1   17  UZ   5 -7.256925E-01 ...  4.535645E-01 ... -1.765251E-05 -2.888436E-01
1   75  UZ   6 -1.014704E+00 ...  9.998966E-03 ...  1.664509E+00 -5.035997E-03
1   78  UZ   7 -1.026445E+00 ... -2.330391E-05 ...  1.710804E+00  1.165176E-05
1   81  UZ   8 -1.038012E+00 ... -1.000144E-02 ...  1.756690E+00  4.964621E-03  
1  100  EP   9 -1.164160E+00 ... -1.087732E-01 ...  2.230329E+00  5.042736E-02

with results saved in b.2, s.2, d.2.

Figure 24.1: Solutions of the boundary value problem at labels 6 and 8, either side of the Shil'nikov-Hopf bifurcation

Figure 24.2: Phase portraits of three homoclinic orbits on the family, showing the saddle-focus to saddle transition

Upon inspection of the output, note that label 5, where PAR(25)$\approx 0$, corresponds to a neutrally-divergent saddle-focus, $\psi_5=0$. Label 7, where PAR(29)$\approx 0$ corresponds to a local bifurcation, $\psi_9=0$, which we note from the eigenvalues stored in d.2 corresponds to a Shil'nikov-Hopf bifurcation. Note that PAR(2) is also approximately zero at label 7, which accords with the analytical observation that the origin of (24.1) undergoes a Hopf bifurcation when $\beta=0$. Labels 6 and 8 are the user-defined output points, the solutions at which are plotted in Fig. 24.1. Note that solutions beyond label 7 (e.g., the plotted solution at label 8) do not correspond to homoclinic orbits, but to point-to-cycle heteroclinic orbits (c.f. Section 2.2.1 of ChKuSa:95).

We now continue in the other direction along the family. It turns out that starting from the initial point in the other direction results in missing a codim 2 point which is close to the starting point. Instead we start from the first saved point from the previous computation (label 5 in s.2):

make third

The output

 BR  PT  TY LAB    PAR(1)     ...    PAR(2)        PAR(22)       PAR(24)    
  1   9  UZ  10 -7.204001E-01 ...  5.912315E-01 -1.725669E+00 -3.295862E-05
  1  18  UZ  11 -7.590583E-01 ...  7.428734E-01  3.432139E-05 -2.822988E-01
  1  26  UZ  12 -7.746686E-01 ...  7.746147E-01  5.833163E-01  1.637611E-07
  1  28  EP  13 -7.746628E-01 ...  7.746453E-01  5.908902E-01  1.426214E-04

contains a neutral saddle-focus (a Belyakov transition) at LAB=10 ($\psi_4=0$), a double real leading eigenvalue (saddle-focus to saddle transition) at LAB =11 ($\psi_2=0$) and a neutral saddle at LAB=12 ($\psi_4=0$). Data at several points on the complete family are plotted in Fig. 24.2. If we had continued further (by increasing NMX), the computation would end at a no convergence error TY=MX owing to the homoclinic family approaching a Bogdanov-Takens singularity at small amplitude. To compute further towards the BT point we would first need to continue to a higher value of PAR(11).