Starting from a general point and setting
and
,
we find by time integration a stable equilibrium at
.
By equilibrium continuation with free, we find two limit points (LP), at
for
with normal form coefficient
and at
for
with normal form coefficient
(note the
reflection).
By continuation of the limit points with free, MATCONT detects a
cusp point CP at
for
and
.
Also detected are two
Zero-Hopf points ZH for
at
and
, but these are in fact Neutral Saddles. Further, two Bogdanov-Takens points BT are found for
at
and
. The normal form
coefficients are
.
A bifurcation diagram of (97) is shown in Figure 32.
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We will now start an orbit homoclinic to saddle from a limit cycle with large period.
First select the BT point at
, and then compute
a curve of Hopf points H passing through it (Type
Curve
Hopf), along which one encounters a Generalized
Hopf bifurcation GH where
and
Stop the continuation before the GH point, e.g. for
where
and
From this Hopf point start the continuation of limit cycles (this is the default Curve type) with
and amplitude equal to
We then obtain limit cycles for slowly decreasing values of At
the parameter
does
not decrease further (at least as seen for this number of digits) but the period increases at each step by approximately the amount of MAXSTEPSIZE.
Stop the continuation when the period reaches (or a nearby value).
Now select the last LC of this curve (Select Initial point), and declare it to be a homoclinic orbit by clicking on Type
Curve
Homoclinic to Saddle.
For a homoclinic continuation you need 2 free parameters (here and
) and 1 or 2 homoclinic parameters. Pick
and
. With these settings, you can start the homoclinic curve (Compute
Backward).
See Figure 33.
For speed purposes, you can increase some Tolerances in the Continuer window, or set Adapt to 0.
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One can also monitor the eigenvalues of the equilibrium during continuation, by displaying them in the Numerical window. This is a useful feature, because it gives indications on what further bifurcations might be expected. For example, a non-central homoclinic-to-saddle-node reveals itself by the fact that one eigenvalue approaches zero.