Lyapunov theory is not the only available tool to
analyze the general behaviour of dynamical systems.
Other methodologies like the direct simulation of so-
lutions with many different initial conditions are also
useful. However, this is a highly costly computational
method that provides only limited information about
the very general behaviour of the system, unless esti-
mates are available, e.g. when shadowing solutions.
Other, more sophisticated methods like the com-
putation of invariant manifolds, forming the bound-
aries of basins of attraction of attractors (Krauskopf
et al., 2005) are still unpractical because they re-
quire additional analysis of the parts with gradient-
like ﬂow.
Other methods divide the phase space into cells
to compute the dynamics between them, see for ex-
ample (Osipenko, 2007). Such methods are known
as cell mapping approach (Hsu, 1987) or set oriented
methods (Dellnitz and Junge, 2002) and they are also
capable of obtaining complete Lyapunov functions.
We will consider a general autonomous ODE of
the form (1). The ﬁrst proof of the existence of a
complete Lyapunov function for dynamical systems
was given in (Auslander, 1964; Conley, 1978b; Con-
ley, 1978a). These proofs holds for a compact met-
ric space and considers the corresponding attractor-
repeller pairs while constructing a function which is
1 on the repeller, 0 on the attractor and decreasing
in between. Then these functions are summed up
over all attractor-repeller pairs. Later, Hurley gen-
eralized these ideas to more general spaces (Hurley,
1992; Hurley, 1995; Hurley, 1998).
Several other computational approaches to con-
struct complete Lyapunov functions have been pub-
lished, for example (Kalies et al., 2005; Ban and
Kalies, 2006; Goullet et al., 2015), where the phase
space was subdivided into cells and a discrete-time
system was given by the time-T map. Then, a multi-
valued map was introduced using the computer pack-
age GAIO (Dellnitz et al., 2001) to compute the dy-
namics between them. Using graph algorithms (Ban
and Kalies, 2006), an approximate complete Lya-
punov function was computed. This approach, how-
ever, is not efﬁcient because it requires a high number
of cells even for low dimensions.
Our methodology, which is faster and works well
in higher dimensions, is inspired by the construction
of classical Lyapunov functions. In (Bj
¨
ornsson et al.,
2014a), the method of (Ban and Kalies, 2006) is com-
pared to mesh-free collocation (see below) for one
particular example. While the mesh-free collocation
method is very efﬁcient on the gradient-like part, in
that particular case, the approach of (Ban and Kalies,
2006) works properly only on the chain-recurrent set
but requires a much ﬁner grid.
Another method construct a complete Lyapunov
as a continuous piecewise afﬁne (CPA) function,
afﬁne on a ﬁxed simplicial complex, see (Bj
¨
ornsson
et al., 2015). However, they require a superset of the
chain-recurrent set as an input. In this paper, the pro-
posed method does not require any information about
the system under consideration except for the right-
hand side function f of system (1).
Let us give an overview of this paper: in Section 2
we discuss mesh-free collocation to approximate so-
lutions of a general linear PDE as well as previous,
related algorithms. In Section 3 we present our algo-
rithm to compute a complete Lyapunov function, ap-
ply the method to an example, and discuss the results
in detail before we conclude.
2 PRELIMINARIES
2.1 Mesh-free Collocation
Numerous numerical construction methods have been
proposed to obtain classical Lyapunov functions,
e.g. (Johansson, 1999; Johansen, 2000; Marin
´
osson,
2002; Giesl, 2007; Hafstein, 2007; Bj
¨
ornsson et al.,
2014b; Kamyar and Peet, 2015; Anderson and Pa-
pachristodoulou, 2015; Doban, 2016; Doban and
Lazar, 2016), see also the review (Giesl and Hafstein,
2015). In this paper, like previously in (Arg
´
aez et al.,
2017a; Arg
´
aez et al., 2018b), our algorithm will be
based on the Radial Basis Function (RBF) method
with mesh-free collocation. Under this approach, the
solution of a linear PDE is approximated and the or-
bital derivative is speciﬁed to approximate the solu-
tion of a linear PDE.
Mesh-free collocation, in particular using Ra-
dial Basis Functions, solves generalized interpolation
problems efﬁciently. In our case, we interpolate the
data given by a partial differential operator applied to
the function at given collocation points. Examples for
Radial Basis Functions include the Gaussians, multi-
quadrics, inverse multiquadrics and Wendland’s com-
pactly supported basis functions. An interpolation
example, as well as a C++ code to compute Wend-
land’s basis functions of arbitrary order, can be found
in (Arg
´
aez et al., 2017b).
We assume that the target function belongs to
a Hilbert space H of continuous functions (often a
Sobolev space). We assume that the Hilbert space
H has a reproducing kernel ϕ : R
n
× R
n
→ R, given
by a convenient Radial Basis Function Φ through
ϕ(x,y) := Φ(x − y), where Φ(x) = ψ
0
(kxk) is a ra-
dial function.
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