Analysis
of Degenerate Solutions to ODEs with Variational Equations of High Order
Victor Varin
Keldysh
Institute of Applied Mathematics, Miusskaja square. 4, 125047, Moscow, Russia,
varin@keldish.ru
Abstract.
We give a definition of variational equations of
high order that does not involve infinite dimentional functional
spaces and Frechet derivatives. We use instead formal differentiation on a
finite dimentional analytical manifold. This technique is used for an analysis
of degenerate solutions on families of periodic solutions. We demonstrate that
for any
degenerate solution there exists a non-degenerate system of boundary value
problems that allows to compute this solution
with the same accuracy as a generic non-degernerate solution on the family.
Keywords: Degeneracy, variational equations, analytical family.
PACS: 02.30.Hq, 02.30.Oz,
02.30.Xx, 95.10.Ce
The neighborhoods
of degenerate solutions in ODEs, i.e. local degeneracies, are usually studied
with the method of normal forms [1]. To reduce a system of ODEs to its normal
form, we need to know the degenerate solution
(a stationary point, a known periodic solution, etc.) in advance. Then, using
the normal form of the equations, we can describe solutions in the neighborhood.
But there are fairly typical situations in multi-parameter problems when
the degenerate solution is not known, but its existence is obvious. For
example, a numerical continuation of a family of solutions by a parameter
encounters some problems. It can be a bifurcation with another family, or some
topological peculiarity in the family. Such degeneracies are non-local, and,
until recently [2], they were computed only with the help of interpolation. We
propose a method of computation of such degeneracies with the help of
variational equations of higher order. The method is based on treating the
solutions to an ODE for all possible values of parameters as some surface or
characteristic manifold, included in a finite dimensional Euclidean space. Then
each solution to the ODE is a function on the manifold; so the partial
derivatives of any order of a solution to the ODE with respect to the local
coordinates on the characteristic manifold satisfy variational equations, which
are obtained recursively by formal differentiation of the ODE with respect to
initial data and parameters. This allows to express the degeneracy in terms of
geometry of the characteristic manifold and obtain equations for its
computation with the same accuracy as an ordinary solution.
We illustrate this
approach on a pendulum type equation that describes oscillations of a satellite
on an elliptic orbit, i.e. the Beletsky
equation:
(1)
where δ is the double angle between the radius-vector of the
mass-center of the satellite; the independent variable ν is the true
anomaly of the position of the satellite on the orbit, e is
the eccentricity of the orbit; and μ is the inertial parameter. This equation has infinitely many
two-parameter families of periodic solutions with various types of degeneracies
on them [2].
We define an
analytical manifold of all possible solutions to the equation (1) as the
following set:
(2)
where δ(0), δ'(0) are initial values of a solution with the
fixed parameters e and μ, and
δ(π), δ'(π)
are computed at ν=π. The first four values in the definition
of χ are local coordinates on the manifold.
Each point on the manifold χ corresponds
to a unique solution to the equation (1), and the equation itself is a
differential identity on the manifold. So we can differentiate this identity
with respect to the local coordinates and obtain variational equations of an arbitrary order and in every
combination of initial values and parameters.
Let us denote the
variations of δ by the letter y,
for example:
(3)
Initial values for are
and for
are
. For all other variations initial values are zeros [2].
Degenerate
solutions are expressed in a remarkably simple form in terms of variations. The
following figure shows some of these solutions at and close to the intersection
of three manifolds of periodic solutions to the Beletsky equation. The curve is the intersection
of the families of symmetric and
asymmetric periodic solutions. This intersection is given by the equation
The same equation
holds for the curve
, where the manifold of the integrable case μ=0
intersects with the manifold of asymmetric solutions. So the point
is where three 2-dimentional manifolds intersect. It is given
by the equation
[2].
Figure 1. Some degeneracies in the families of symmetric and asymmetric periodic solutions.
Here we can not
derive formulas for all the degenerate solutions shown in Fig. 1. Instead, we
only describe some of them and refer the reader to the paper [2], where
rigorous proofs are given. But we will show how to use this technique on the
example of the point corresponding to the
Whitney cusp on the family of symmetric
periodic solutions.
Extremal points on
characteristics are very important, although they are not usually considered as
degeneracies. The point on the curve
, where the eccentricity attains its extremal value, is given by the equation
. The point
, where the inertial parameter attains its extremal value, is given by the same
equation, but with the substitution of e instead
of μ. This is a typical situation
with variational equations, when one does not need to derive equations for each
degeneracy anew. In fact, the latter
equation holds for every similar degenerate solution in similar pendulum type
equations, i.e. with two parameters and the same symmetries.
The curve γ
in Fig.1 corresponds to the ridge of a symmetric cusp in the family of
asymmetric periodic solutions. It is given by the equation Tr=2, where Tr is
the trace of the monodromy matrix. The
point is where the ridge is
attached to the curve . It given by the equation
. This equation also holds
for the point
, where a similar symmetric cusp originates. Remaining points
in Fig. 1, together with their numerical values, see in [2].
Consider the first
and the second variations of the equation (1) with respect to the initial
value :
(4)
Now we introduce
the following system of boundary value problems for the equations (1), (4):
(5)
where k is an iteger.
Theorem. [2] A solution
to the system (5) for the unknowns , e, and μ corresponds to the vertex of a
Whitney cusp in the family of symmetric generalized periodic solutions to the
Beletsky equation with the rotation number k.
In particular, the point satisfies the system
(5) for k=0. Note that the vertex of
a Whitney cusp is a degeneracy of co-dimension 2, and hence it is an isolated
point on the manifold χ. But using this theorem, we can compute these
points in various families of periodic solutions tracing them by the parameter k. The following figure shows two
analytical curves that satisfy the system (5) for continuous values of k. Bullets in Fig. 2 show the places
where k is integer, and so they are
the origins of Whitney cusps. Projection of one of such cusp is shown in Fig. 2
for the rotation number k=1.
Figure 2. Analitical curves and the vertices of Whitney cusps for integer k.
So variational
equations allow to compute non-generic solutions in families of periodic solutions, where ordinary numerical
methods fail. To obtain variational equations, we only need to use formal
differentiation with respect to initial values and parameters of the problem.
These equations form triangular systems of ODEs that can be numerically solved
with the same accuracy as the original ODEs. The solutions in the neighborhood
of a degenerate solution are represented in a small parameter series whose
terms are expressed through variational equations as well.
An application of
variational equations for the analysis of degenerate limit cycles see in [3].
In [4], degenerate solutions were computed in the restricted 3-body problem.
This work was
supported by RFBR grant 11-01-00023.
1. A. D. Bruno, Local Methods in Nonlinear Differential Equations , Berlin: Springer-Verlag,
1989.
2. V. P. Varin, “Degeneracies
of Periodic Solutions to the Beletsky Equation” in Regular and Chaotic
Dynamics, 5, N 3, 2000, pp. 313-328.
3. V. P. Varin, “Poincaré Map for Some
Polynomial Systems of Differential Equations” in Sbornik: Mathematics, 195, N 7,
2004, pp. 917-934.
4. A. D. Bruno and V. P. Varin, “Closed Families of Periodic
Solutions of a Restricted Three-Body Problem” in Solar
System Research, 43, N 3, 2009, pp. 253-276.