Example

In this section we discuss a non-generic situation, i.e. a case with a symmetry and a continuation of BPC points that involves two effective parameters and one artificial parameter.

For this example the following model is used:

\begin{displaymath}
\left\{\begin{array}{rcl}
\dot{x}&=&(-(\beta+\nu)x+\beta y...
...a+\gamma)y-z-b_3 (y-x)^3\\
\dot{z}&=y
\end{array}\right.
\end{displaymath} (87)

which is the same system as in the torus of cycles continuation. It has a trivial solution branch $x=y=z=0$ for all parameter values. Moreover, it has the $Z_2$ - symmetry $x\rightarrow -x$, $y\rightarrow -y$. To compute the branch of BPC points with respect to $\nu$ through the BPC point that we will detect on a limitcycle continued with free parameters $\nu,\beta$, we need to introduce an additional free parameter that breaks the symmetry. There are many choices for this; we choose to introduce a parameter $\epsilon$ and extend the system (87) by simply adding a term $+\epsilon$ to the first right-hand-side. For $\epsilon=0$ this reduces to (87) while for $\epsilon \neq 0$ the symmetry is broken.

We start by computing the trivial branch with fixed parameters $\gamma=-0.6$, $r=0.6$, $a_3=0.328578$, $b_3=0.933578$, $\beta=0.5$,$\epsilon=0$ and free parameter $\nu$ with initially $\nu=-0.9$. On this branch a Hopf point is detected for $\nu=-0.58933644$ and a branch point of equilibria for $\nu=-0.5$.

>> p=[0.5;-0.6;0.6;0.32858;0.93358;-0.9;0];
>> [x0,v0]=init_EP_EP(@torBPC,[0;0;0],p,[6]);
>> opt=contset; opt= contset(opt,'Singularities',1);
>> opt=contset(opt,'MaxNumPoints',50);
>> [x,v,s,h,f]=cont(@equilibrium,x0,[],opt);
first point found
tangent vector to first point found
label = H , x = ( 0.000000 0.000000 0.000000 -0.589336 )
First Lyapunov coefficient = -4.563631e-001
label = BP, x = ( 0.000000 0.000000 0.000000 -0.500000 )

elapsed time  = 0.3 secs
npoints curve = 50

These computations can be done by running the script testtorBPC4.

From the Hopf point we start the computation of a curve of limit cycles, using 25 test intervals and 4 collocation points. This is clearly a branch of symmetric solutions of (87); we detect one LPC and two BPC, see Fig. 28.

>> x1=x(1:3,s(2).index);p(6)= x(end,s(2).index);ap = 6;
>> [x0,v0]=init_H_LC(@torBPC,x1,p,ap,0.0001,25,4);
>> opt=contset; opt= contset(opt,'Singularities',1);
>> opt=contset(opt,'Multipliers',1);
>> opt=contset(opt,'MaxNumPoints',150);
>> opt=contset(opt,'Adapt',5);
>> [xlc,vlc,slc,hlc,flc]=cont(@limitcycle,x0,v0,opt);
first point found
tangent vector to first point found
Limit point cycle (period = 8.426472e+000, parameter = -5.843348e-001)
Normal form coefficient = 1.553595e-001
Branch Point cycle(period = 8.689669e+000, parameter = -5.870290e-001)
Neimark-Sacker (period = 8.743033e+000, parameter = -5.881194e-001)
Neutral saddle

elapsed time  = 30.2 secs
npoints curve = 150
>> plotcycle(xlc,vlc,slc,[1 2]);

These computations can be done by running the script testtorBPC5.

Figure 28: Curve of limit cycles with LPC and branch points in the circuit example.
\includegraphics[scale=0.8]{ex/LCBPC1.eps}
We continue the secondary cycle branch passing through the BPC point. From Fig. 29 it is clear that in the secondary cycle the symmetry is broken.
>> [x1,v1]=init_BPC_LC(@torBPC,xlc,vlc,slc(3),25,4,1e-6);
>> opt=contset(opt,'MaxNumPoints',50);
>> opt=contset(opt,'Backward',1);
>> [xlc1,vlc1,slc1,hlc1,flc1]=cont(@limitcycle,x1,v1,opt);
first point found
tangent vector to first point found
Neimark-Sacker (period = 8.794152e+000, parameter = -5.916502e-001)
Normal form coefficient = -8.661266e-003
Period Doubling (period = 9.266303e+000, parameter = -6.149552e-001)
Normal form coefficient = -6.374237e-003

elapsed time  = 13.6 secs
npoints curve = 50
>> plotcycle(xlc1,vlc1,slc1,[1 2]);

These computations can be done by running the script testtorBPC6.

Figure 29: Asymmetric curve of limit cycles in the circuit example.
\includegraphics[scale=0.8]{ex/LCBPC2.eps}
Using the code for the continuation of generic BPC points with three free parameters $\nu,\beta,\epsilon$ we continue the curve of non-generic BPC points, where $\epsilon$ remains close to zero. The picture in Fig. 30 clearly shows that the symmetry is preserved (the axis labels were added manually).
>> [x1,v1]=init_BPC_BPC(@torBPC,xlc,slc(3),[1 6 7],25,4,ap);
>> opt=contset(opt,'MaxNumPoints',200);
>> [xbpc,vbpc,sbpc,hbpc,fbpc]=cont(@branchpointcycle,x1,v1,opt);
first point found
tangent vector to first point found

elapsed time  = 158.1 secs
npoints curve = 200
>> plotcycle(xbpc,vbpc,sbpc,[1 2]);

These computations can be done by running the script testtorBPC7.

Figure 30: Curve of BPC points in the circuit example.
\includegraphics[scale=0.8]{ex/Figure30.eps}