8 research outputs found

### Symmetries in the Lorenz-96 model

The Lorenz-96 model is widely used as a test model for various applications,
such as data assimilation methods. This symmetric model has the forcing
$F\in\mathbb{R}$ and the dimension $n\in\mathbb{N}$ as parameters and is
$\mathbb{Z}_n$ equivariant. In this paper, we unravel its dynamics for $F<0$
using equivariant bifurcation theory. Symmetry gives rise to invariant
subspaces, that play an important role in this model. We exploit them in order
to generalise results from a low dimension to all multiples of that dimension.
We discuss symmetry for periodic orbits as well.
Our analysis leads to proofs of the existence of pitchfork bifurcations for
$F<0$ in specific dimensions $n$: In all even dimensions, the equilibrium
$(F,\ldots,F)$ exhibits a supercritical pitchfork bifurcation. In dimensions
$n=4k$, $k\in\mathbb{N}$, a second supercritical pitchfork bifurcation occurs
simultaneously for both equilibria originating from the previous one.
Furthermore, numerical observations reveal that in dimension $n=2^qp$, where
$q\in\mathbb{N}\cup\{0\}$ and $p$ is odd, there is a finite cascade of exactly
$q$ subsequent pitchfork bifurcations, whose bifurcation values are independent
of $n$. This structure is discussed and interpreted in light of the symmetries
of the model.Comment: 31 pages, 9 figures and 3 table

### On Max-Semistable Laws and Extremes for Dynamical Systems

Suppose [Formula: see text] is a measure preserving dynamical system and [Formula: see text] a measurable observable. Let [Formula: see text] denote the time series of observations on the system, and consider the maxima process [Formula: see text]. Under linear scaling of [Formula: see text] , its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process

### Wave propagation in the Lorenz-96 model

In this paper we study the spatiotemporal properties of waves in the
Lorenz-96 model and their dependence on the dimension parameter <i>n</i>
and the forcing parameter <i>F</i>. For <i>F</i>â€‰>â€‰0 the first bifurcation is
either aÂ supercritical Hopf or aÂ double-Hopf bifurcation and the
periodic attractor born at these bifurcations represents aÂ traveling
wave. Its spatial wave number increases linearly with <i>n</i>, but its
period tends to aÂ finite limit as <i>n</i>â€‰â†’â€‰âˆž. For <i>F</i>â€‰<â€‰0 and odd
<i>n</i>, the first bifurcation is again aÂ supercritical Hopf bifurcation,
but in this case the period of the traveling wave also grows
linearly with <i>n</i>. For <i>F</i>â€‰<â€‰0 and even <i>n</i>, however, aÂ Hopf
bifurcation is preceded by either one or two pitchfork bifurcations,
where the number of the latter bifurcations depends on whether <i>n</i>
has remainder 2 or 0 upon division by 4. This bifurcation sequence
leads to stationary waves and their spatiotemporal properties also
depend on the remainder after dividing <i>n</i> by 4. Finally, we explain
how the double-Hopf bifurcation can generate two or more stable
waves with different spatiotemporal properties that coexist for the
same parameter values <i>n</i> and <i>F</i>

### Extreme value laws in dynamical systems under physical observables

Extreme value theory for chaotic dynamical systems is a rapidly expanding
area of research. Given a system and a real function (observable) defined on
its phase space, extreme value theory studies the limit probabilistic laws
obeyed by large values attained by the observable along orbits of the system.
Based on this theory, the so-called block maximum method is often used in
applications for statistical prediction of large value occurrences. In this
method, one performs inference for the parameters of the Generalised Extreme
Value (GEV) distribution, using maxima over blocks of regularly sampled
observations along an orbit of the system. The observables studied so far in
the theory are expressed as functions of the distance with respect to a point,
which is assumed to be a density point of the system's invariant measure.
However, this is not the structure of the observables typically encountered in
physical applications, such as windspeed or vorticity in atmospheric models. In
this paper we consider extreme value limit laws for observables which are not
functions of the distance from a density point of the dynamical system. In such
cases, the limit laws are no longer determined by the functional form of the
observable and the dimension of the invariant measure: they also depend on the
specific geometry of the underlying attractor and of the observable's level
sets. We present a collection of analytical and numerical results, starting
with a toral hyperbolic automorphism as a simple template to illustrate the
main ideas. We then formulate our main results for a uniformly hyperbolic
system, the solenoid map. We also discuss non-uniformly hyperbolic examples of
maps (H\'enon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models).
Our purpose is to outline the main ideas and to highlight several serious
problems found in the numerical estimation of the limit laws

### Chaotic dynamics from a pseudo-linear system

Investigating the possibility of applying techniques from linear systems theory to the setting of non-linear systems has been the focus of many papers. The pseudo-linear (PL) form representation of non-linear dynamical systems has led to the concept of non-linear eigenvalues (NEValues) and non-linear eigenvectors (NEVectors). When the NEVectors do not depend on the state vector of the system, then the NEValues determine the global qualitative behaviour of a non-linear system throughout the state space. The aim of this paper is to use this fact to construct a non-linear dynamical system of which the trajectories of the system show continual stretching and folding. We first prove that the system is globally bounded. Next we analyse the system numerically by studying bifurcations of equilibria and periodic orbits. Chaos arises due to a period doubling cascade of periodic attractors. Chaotic attractors are presumably of HÃ©non-like type, which means that they are the closure of the unstable manifold of a saddle periodic orbit. We also show how PL forms can be used to control the chaotic system and to synchronize two identical chaotic systems

### Global bifurcation analysis of Topp system

In this paper, we study the 3-dimensional Topp model for the dynamics of diabetes. First, we reduce the model to a planar quartic system. In particular, studying global bifurcations, we prove that such a system can have at most two limit cycles. Next, we study the dynamics of the full 3-dimensional model. We show that for suitable parameter values an equilibrium bifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests that near this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arise through period doubling cascades of limit cycles