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### Open Access Journals

DCDS-B

If financial markets displayed the informational efficiency postulated in
the efficient markets hypothesis (EMH), arbitrage operations would be
self-extinguishing. The present paper considers arbitrage sequences in
foreign exchange (FX) markets, in which trading platforms and information
are fragmented. In [18,9] it was shown
that sequences of triangular arbitrage operations in FX markets containing
$4$ currencies and trader-arbitrageurs tend to display periodicity or grow
exponentially rather than being self-extinguishing. This paper extends the
analysis to $5$ or higher-order currency worlds. The key findings are that
in a $5$-currency world arbitrage sequences may also follow an exponential
law as well as display periodicity, but that in higher-order currency
worlds a double exponential law may additionally apply. There is an
``inheritance of instability'' in the higher-order currency worlds.
Profitable arbitrage operations are thus endemic rather that displaying
the self-extinguishing properties implied by the EMH.

DCDS-B

In the paper a set of necessary and sufficient conditions for

*v-*sufficiency (equiv.*sv-*sufficiency) of jets of map-germs $f:(\mathbb{R}^{n},0)\to (\mathbb{R}^{m},0)$ is proved which generalize both the Kuiper-Kuo and the Thom conditions in the function case ($m=1$) so as the Kuo conditions in the general map case ($m>1$). Contrary to the Kuo conditions the conditions proved in the paper do not require to verify any inequalities in a so-called horn-neighborhood of the (a'priori unknown) set $f^{-1}(0)$. Instead, the proposed conditions reduce the problem on*v-*sufficiency of jets to evaluating the local Łojasiewicz exponents for some constructively built polynomial functions.
DCDS-B

The problem of construction of Barabanov norms for analysis of
properties of the joint (generalized) spectral radius of
matrix sets has been discussed in a number of publications. In
[18, 21] the method of Barabanov
norms was the key instrument in disproving the Lagarias-Wang
Finiteness Conjecture. The related constructions were
essentially based on the study of the geometrical properties of
the unit balls of some specific Barabanov norms. In this
context the situation when one fails to find among current
publications any detailed analysis of the geometrical
properties of the unit balls of Barabanov norms looks a bit
paradoxical. Partially this is explained by the fact that
Barabanov norms are defined nonconstructively, by an implicit
procedure. So, even in simplest cases it is very difficult to
visualize the shape of their unit balls. The present work may
be treated as the first step to make up this deficiency. In the
paper an iteration procedure is considered that allows to build
numerically Barabanov norms for the irreducible matrix sets and
simultaneously to compute the joint spectral radius of these
sets.

DCDS-B

Discrete-time discrete-state random Markov chains with a tridiagonal
generator are shown to have a random attractor consisting of singleton
subsets, essentially a random path, in the simplex of probability vectors.
The proof uses the Hilbert projection metric and the fact that the linear
cocycle generated by the Markov chain is a uniformly contractive mapping
of the positive cone into itself. The proof does not involve probabilistic
properties of the sample path $\omega$ and is thus equally valid in the
nonautonomous deterministic context of Markov chains with, say,
periodically varying transitions probabilities, in which case the
attractor is a periodic path.

DCDS

We consider discrete time systems
$x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter
$\lambda$, and study their trajectories of large amplitudes. The
expansion of the map $U(\cdot;\lambda)$ at infinity contains a principal
linear term, a bounded positively homogeneous nonlinearity, and a
smaller vanishing part. We study Arnold tongues: the sets of
parameter values for which the large-amplitude periodic trajectories
exist. The Arnold tongues in problems at infinity generically are
thick triangles [4]; here we obtain asymptotic estimates for
the length of the Arnold tongues and for the length of their
triangular part. These estimates allow us to study subfurcation at
infinity. In the related problems on small-amplitude periodic orbits
near an equilibrium, similarly defined Arnold tongues have the form
of narrow beaks. For standard pictures associated with the
Neimark-Sacker bifurcation of smooth discrete time systems at an
equilibrium, the Arnold tongues have asymptotically zero width
except for the strong resonance points. The different shape of the
tongues in the problem at infinity is due to the non-polynomial form
of the principal homogeneous nonlinear term of the map
$U(\cdot;\lambda)$: this form implies non-degeneracy of the nonlinear
terms in the expansion of the map iterations and non-degeneracy of
the corresponding resonance functions.

DCDS-S

We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$,
$x\in\R^{N}$, with a complex parameter $\lambda$.
The map $U(\cdot;\lambda)$ at infinity contains a principal linear
term, a bounded positively homogeneous nonlinearity, and a smaller
part. We describe the sets of parameter values for which the
large-amplitude $n$-periodic trajectories exist for a fixed $n$.
In the related problems on small periodic orbits near zero,
similarly defined parameter sets, known as Arnold tongues, are more narrow.

DCDS

The influence of the driving system on a skew-product flow generated by
a triangular system of differential equations can be perturbed in two ways, directly
by perturbing the vector field of the driving system component itself or indirectly
by perturbing its input variable in the vector field of the coupled component. The
effect of such perturbations on a nonautonomous attractor of the driven component
is investigated here. In particular, it is shown that a perturbed nonautonomous
attractor with nearby components exists in the indirect case if the driven system has
an inflated nonautonomous attractor and that the direct case can be reduced to this
case if the driving system is shadowing.

DCDS

A nonautonomous or cocycle dynamical system
that is driven by an autonomous dynamical system acting on a
compact metric space is assumed to have a uniform pullback
attractor. It is shown that discretization by a one-step numerical
scheme gives rise to a discrete time cocycle dynamical system with
a uniform pullback attractor, the component subsets of which
converge upper semi continuously to their continuous time
counterparts as the maximum time step decreases to zero. The proof
involves a Lyapunov function characterizing the uniform pullback
attractor of the original system.

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