75 research outputs found
Computability in planar dynamical systems
In this paper we explore the problem of computing attractors and
their respective basins of attraction for continuous-time planar dynamical
systems. We consider C1 systems and show that stability is in general
necessary (but may not be sufficient) to attain computability. In particular,
we show that (a) the problem of determining the number of attractors
in a given compact set is in general undecidable, even for analytic systems
and (b) the attractors are semi-computable for stable systems.
We also show that the basins of attraction are semi-computable if and
only if the system is stable
Computability of differential equations
In this chapter, we provide a survey of results concerning the computability and computational complexity of differential equations. In particular, we study the conditions which ensure computability of the solution to an initial value problem for an ordinary differential equation (ODE) and analyze the computational complexity of a computable solution. We also present computability results concerning the asymptotic behaviors of ODEs as well as several classically important partial differential equations.info:eu-repo/semantics/acceptedVersio
Computability of ordinary differential equations
In this paper we provide a brief review of several results about the
computability of initial-value problems (IVPs) defined with ordinary differential
equations (ODEs). We will consider a variety of settings and analyze
how the computability of the IVP will be affected. Computational
complexity results will also be presented, as well as computable versions
of some classical theorems about the asymptotic behavior of ODEs.info:eu-repo/semantics/publishedVersio
Spiralling dynamics near heteroclinic networks
There are few explicit examples in the literature of vector fields exhibiting
complex dynamics that may be proved analytically. We construct explicitly a
{two parameter family of vector fields} on the three-dimensional sphere
\EU^3, whose flow has a spiralling attractor containing the following: two
hyperbolic equilibria, heteroclinic trajectories connecting them {transversely}
and a non-trivial hyperbolic, invariant and transitive set. The spiralling set
unfolds a heteroclinic network between two symmetric saddle-foci and contains a
sequence of topological horseshoes semiconjugate to full shifts over an
alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The
vector field is the restriction to \EU^3 of a polynomial vector field in
\RR^4. In this article, we also identify global bifurcations that induce
chaotic dynamics of different types.Comment: change in one figur
From limit cycles to strange attractors
We define a quantitative notion of shear for limit cycles of flows. We prove
that strange attractors and SRB measures emerge when systems exhibiting limit
cycles with sufficient shear are subjected to periodic pulsatile drives. The
strange attractors possess a number of precisely-defined dynamical properties
that together imply chaos that is both sustained in time and physically
observable.Comment: 27 page
Limitations of perturbative techniques in the analysis of rhythms and oscillations
Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience
The Construction of Arbitrary Stable Dynamics in Non-Linear Neural Networks
In this paper, two methods for constructing systems of ordinary differential equations realizing any fixed finite set of equilibria in any fixed finite dimension are introduced; no spurious equilibria are possible for either method. By using the first method, one can construct a system with the fewest number of equilibria, given a fixed set of attractors.
Using a strict Lyapunov function for each of these differential equations, a large class of systems with the same set of equilibria is constructed. A method of fitting these nonlinear systems to trajectories is proposed. In addition, a general method which will produce an arbitrary number of periodic orbits of shapes of arbitrary complexity is also discussed.
A more general second method is given to construct a differential equation which converges to a fixed given finite set of equilibria. This technique is much more general in that it allows this set of equilibria to have any of a large class of indices which are consistent with the Morse Inequalities. It is clear that this class is not universal, because there is a large class of additional vector fields with convergent dynamics which cannot be constructed by the above method.
The easiest way to see this is to enumerate the set of Morse indices which can be obtained by the above method and compare this class with the class of Morse indices of arbitrary differential equations with convergent dynamics. The former set of indices are a proper subclass of the latter, therefore, the above construction cannot be universal. In general, it is a difficult open problem to construct a specific example of a differential equation with a given fixed set of equilibria, permissible Morse indices, and permissible connections between stable and unstable manifolds.
A strict Lyapunov function is given for this second case as well. This strict Lyapunov function as above enables construction of a large class of examples consistent with these more complicated dynamics and indices. The determination of all the basins of attraction in the general case for these systems is also difficult and open.Air Force Office of Scientific Research (F49620-86-C-0037
Safety criteria for aperiodic dynamical systems
The
use of
dynamical
system models
is
commonplace
in
many areas of science and
engineering.
One is
often
interested in
whether
the
attracting solutions
in these
models are
robust
to perturbations of
the
equations of motion.
This
question
is
extremely
important
in
situations where
it is
undesirable
to have
a
large
response
to
perturbations
for
reasons
of safety.
An
especially
interesting
case occurs when the
perturbations are aperiodic and
their
exact
form is
unknown.
Unfortunately,
there is
a
lack
of
theory in the literature that
deals
with
this
situation.
It
would
be
extremely useful to have
a practical
technique that
provides
an upper
bound
on the size of the
response
for
an arbitrary perturbation of given
size.
Estimates
of
this form
would allow the
simple
determination
of safety criteria
that
guarantee
the response
falls
within some pre-specified safety
limits. An
excellent area
of application
for this technique
would
be
engineering systems.
Here
one
is frequently
faced
with
the
problem of obtaining safety criteria
for
systems
that in
operational use are
subject
to unknown, aperiodic perturbations.
In this thesis I
show
that
such safety criteria are easy to obtain
by
using
the
concept
of persistence
of
hyperbolicity. This
persistence result
is
well
known in the theory
of
dynamical systems.
The formulation I
give
is functional
analytic
in
nature and
this has
the
advantage
that it is
easy
to
generalise and
is
especially suited to the
problem of
unknown,
aperiodic perturbations.
The
proof
I
give of
the
persistence
theorem
provides
a
technique
for
obtaining
the
safety estimates we want and
the
main part of
this thesis is
an
investigation into how this
can
be
practically
done.
The
usefulness of
the technique is illustrated through two
example systems,
both
of
which are
forced
oscillators.
Firstly, I
consider
the
case where
the
unforced oscillator
has
an asymptotically stable equilibrium.
A
good application of this is the
problem of
ship stability.
The
model
is
called
the
escape equation and
has been
argued to
capture
the relevant
dynamics
of a ship at sea.
The
problem is to find
practical criteria
that
guarantee
the
ship
does not capsize or go
through large
motions when there are external
influences like
wind and waves.
I
show
how
to
provide good criteria which ensure a safe
response when
the
external
forcing is
an arbitrary,
bounded function
of
time. I
also
consider
in
some
detail the
phased-locked loop. This is
a periodically forced
oscillator
which
has
an attracting periodic solution that is
synchronised
(or
phase-locked) with
the
external
forcing. It is interesting to
consider the
effect of small aperiodic variations
in the
external
forcing. For hyperbolic
solutions
I
show that the
phase-locking persists and
I
give
a method
by
which one can
find
an upperbound
on
the
maximum size of
the
response
Local Orthogonal Rectification: A New Tool for Geometric Phase Space Analysis
Local orthogonal rectification (LOR) provides a natural and useful geometric frame for analyzing dynamics relative to manifolds embedded in flows. LOR can be applied to any embedded base manifold in a system of ODEs of arbitrary dimension to establish a corresponding system of LOR equations for analyzing dynamics within the LOR frame. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. Additionally, we illustrate the utility of LOR by showing a wide range of application domains.
In the plane, we use the LOR approach to derive a novel definition for rivers, long-recognized but poorly understood trajectories that locally attract other orbits yet need not be related to invariant manifolds or other familiar phase space structures, and to identify rivers within several example systems.
In higher dimensions, we apply LOR to identify periodic orbits and study the transient dynamics nearby. In the LOR method, %in for any ,
the standard approach of finding periodic orbits by solving for fixed points of a Poincar\'{e} return map is replaced by the solution of a boundary value problem with fixed endpoints, and the computation provides information about the stability of the identified orbit. We detail the general method and derive theory to show that once a periodic orbit has been identified with LOR, the LOR coordinate system allows us to characterize the stability of the periodic orbit, to continue the orbit with respect to system parameters, to identify invariant manifolds attendant to the periodic orbit, and to compute the asymptotic phase associated with points in a neighborhood of the periodic orbit in a novel way.
Finally, we generalize the definition of rivers beyond planar systems, and demonstrate a fundamental connection between canard solutions in two-timescale systems and generalized rivers. We will again use a blow-up transformation on the LOR equations, which provides a useful decomposition for studying trajectories' behavior relative to the embedded base curve
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