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In dynamical systems theory, an orbit corresponding to a solution u(t) is
called homoclinic to the equilibrium point u0 of the dynamical system if u(t) ® u0 as t ® ±¥.
There are two types of homoclinic orbits with codimension 1, namely
homoclinic-to-hyperbolic-saddle ( HHS), if u0 is a saddle (saddle-focus or bi-focus), and
homoclinic-to-saddle-node ( HSN), if u0 is a saddle-node (i.e., exhibits a limit point
bifurcation). We recall that
auto has a toolbox for homoclinic continuation, named HomCont [22], [23].
During the continuation, it is necessary to keep track of several eigenspaces
of the equilibrium in each step. To do this in an efficient way, MatCont
incorporates the continuation of invariant subspaces [24]
into the defining system. For some details on the implementation of the homoclinic
continuation we refer to [25].
12.1.1 Homoclinic-to-Hyperbolic-Saddle Orbits
To continue HHS orbits in two free parameters, we use an extended defining system
that consists of several parts.
First, the infinite time interval is truncated, so that instead of [-¥,+¥]
we use [-T,+T], which is scaled to [0,1] and divided into mesh-intervals.
The mesh is nonuniform and adaptive. Each mesh interval is further subdivided by
equidistant fine mesh points. Also, each mesh interval contains a number of collocation
points. (This discretization is the same as that in auto for boundary value problems.)
The equation
|
×
x
|
(t) - 2 T f(x(t),a) = 0, |
| (90) |
must be satisfied in each collocation point.
The second part is the equilibrium condition
Third, there is a so-called phase condition needed for the homoclinic solution, similar to
periodic solutions
|
ó õ
|
1
0
|
|
×
|
*
|
(t)[x(t) - |
~
x
|
(t)]dt = 0. |
| (92) |
Here [x\tilde](t) is some initial guess for the solution,
typically obtained from the previous continuation step.
We note that in the literature another phase condition
is also used, see, for example [26]. However, in the present implementation we employ
the condition (92).
Fourth, there are the homoclinic-specific constraints to the solution.
For these we need access to the stable and unstable eigenspaces of the system in the
equilibrium point after each step. It is not efficient to recompute the spaces from
scratch in each continuation-step. Instead, we use the algorithm for continuing
invariant subspaces, as described in [24]. This method adds two small-sized
vectors (YS and YU) to the system variables, from which the necessary eigenspaces
(stable and unstable, respectively) can easily be computed in each step.
If Q(0) is an orthogonal matrix whose first m columns form a basis for the invariant
subspace under consideration in the previous step, and A=fx(x0,a) is the Jacobian
at the new
equilibrium point, then we first compute the so-called Ricatti-blocks, Tij, by the formula
|
é ê
ê ë
|
|
ù ú
ú û
|
= Q(0)T A Q(0). |
| (93) |
If n is the number of state variables, then T11 is of size m ×m and T22
is (n-m) ×(n-m). This is done for the stable and unstable eigenspaces separately.
Now YS and YU are obtained from the Ricatti equations
|
T22U YU - YU T11U + T21U - YU T12U YU = 0, |
|
T22S YS - YS T11S + T21S - YS T12S YS = 0. |
|
|
|
| (94) |
Now we can formulate constraints on the behavior of the solution close to the
equilibrium x0. The initial vector of the orbit, (x(0) - x0), is placed in the unstable
eigenspace of the system in the equilibrium. We express that by the requirement that it is
orthogonal to the orthogonal complement of the unstable eigenspace. Using YU, we can
compute the orthogonal complement of the unstable eigenspace. If QU(0) is the
orthogonal matrix from the previous step, related to the unstable invariant subspace,
then a basis for the orthogonal complement in the new step QU^(s) is
QU^(s) = QU(0) |
é ê
ê ë
|
|
|
ù ú
ú û
|
. |
|
Note that QU^(s) is not orthogonal. The full orthogonal matrix Q1U needed for the
next step, is computed separately after each step. The equations to be added
to the system are (after analogous preparatory computations for the stable eigenspace)
Finally, the distances between x(0) (resp., x(1)) and x0 must be small enough,
so that
|
|| x(0) - x0 || - e0 = 0, |
|
|| x(1) - x0 || - e1 = 0. |
|
|
|
| (96) |
A system consisting of all equations (90), (91),
(92), (94), (95) and (96), is overdetermined.
The basic defining system for the continuation of a HHS orbit in two free parameters
consists of (90), (91), (94), (95), and
(96) with fixed e0,1, so that the phase condition (92) is
not used. The variables in this system are stored in one vector. It contains
the values of x(t) in the fine mesh points including x(0) and x(1),
the truncation time T, two free system parameters, the coordinates
of the saddle x0, and the elements of the matrices YS and YU.
Alternatively, the phase condition (92) can be added if T is kept fixed
but e0 and e1 are allowed to vary.
It is also possible to fix T and e0, say, and allow e1 to vary,
again with no phase condition. Other combinations are also possible, in particular, when
the homotopy method [24] is used to compute a starting homoclinic solution.
12.1.2 Homoclinic-to-Saddle-Node Orbits
For a homoclinic orbit to a saddle-node equilibrium, the extended defining system undergoes some
small changes. Now (x(0) - x0) has to be placed in the center-unstable subspace. Analogously,
(x(1) - x0) must be in the center-stable subspace.
This again is implemented by requiring that the vector is orthogonal to the orthogonal
complement of the corresponding space. So the equations (95) themselves do not really
change; the changes happen in the computation of the matrices Q.
The defining system now has one equation less than in the HHS case (ns + nu < n, with ns
the dimension of the stable, and nu of the unstable eigenspace);
the number of equations is restored however, by adding the constraint that the equilibrium
must be a saddle-node. For this we use the bordering technique, as described in section 4.2.1
of [27].