3,087 research outputs found
The dynamics of a low-order coupled ocean-atmosphere model
A system of five ordinary differential equations is studied which combines
the Lorenz-84 model for the atmosphere and a box model for the ocean. The
behaviour of this system is studied as a function of the coupling parameters.
For most parameter values, the dynamics of the atmosphere model is dominant.
For a range of parameter values, competing attractors exist. The Kaplan-Yorke
dimension and the correlation dimension of the chaotic attractor are
numerically calculated and compared to the values found in the uncoupled Lorenz
model. In the transition from periodic behaviour to chaos intermittency is
observed. The intermittent behaviour occurs near a Neimark-Sacker bifurcation
at which a periodic solution loses its stability. The length of the periodic
intervals is governed by the time scale of the ocean component. Thus, in this
regime the ocean model has a considerable influence on the dynamics of the
coupled system.Comment: 20 pages, 15 figures, uses AmsTex, Amssymb and epsfig package.
Submitted to the Journal of Nonlinear Scienc
The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictible zone
We have applied harmonic expansion to derive an analytical solution for the
Lorenz-Haken equations. This method is used to describe the regular and
periodic self-pulsing regime of the single mode homogeneously broadened laser.
These periodic solutions emerge when the ratio of the population decay rates is
smaller than 0.11. We have also demonstrated the tendency of the Lorenz-Haken
dissipative system to behave periodic for a characteristic pumping rate "2CP"
[4], close to the second laser threshold "2C2th" (threshold of instability).
When the pumping parameter "2C" increases, the laser undergoes a
period-doubling sequence. This cascade of period doubling leads towards chaos.
We study this type of solutions and indicate the zone of the control parameters
for which the system undergoes irregular pulsing solutions. We had previously
applied this analytical procedure to derive the amplitude of the first, third
and the fifth order harmonics for the laser-field expansion [4, 14]. In this
work, we extend this method in the aim of obtaining the higher harmonics. We
show that this iterative method is indeed limited to the fifth order, and that
above, the obtained analytical solution diverges from the numerical direct
resolution of the equations.Comment: 20 pages, 4 figures, 1 anne
Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically.Comment: submitted to EP
The role of data in model building and prediction: a survey through examples
The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play-and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components
Cluster-based reduced-order modelling of a mixing layer
We propose a novel cluster-based reduced-order modelling (CROM) strategy of
unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's
group (Burkardt et al. 2006) and and transition matrix models introduced in
fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a
potential alternative to POD models and generalises the Ulam-Galerkin method
classically used in dynamical systems to determine a finite-rank approximation
of the Perron-Frobenius operator. The proposed strategy processes a
time-resolved sequence of flow snapshots in two steps. First, the snapshot data
are clustered into a small number of representative states, called centroids,
in the state space. These centroids partition the state space in complementary
non-overlapping regions (centroidal Voronoi cells). Departing from the standard
algorithm, the probabilities of the clusters are determined, and the states are
sorted by analysis of the transition matrix. Secondly, the transitions between
the states are dynamically modelled using a Markov process. Physical mechanisms
are then distilled by a refined analysis of the Markov process, e.g. using
finite-time Lyapunov exponent and entropic methods. This CROM framework is
applied to the Lorenz attractor (as illustrative example), to velocity fields
of the spatially evolving incompressible mixing layer and the three-dimensional
turbulent wake of a bluff body. For these examples, CROM is shown to identify
non-trivial quasi-attractors and transition processes in an unsupervised
manner. CROM has numerous potential applications for the systematic
identification of physical mechanisms of complex dynamics, for comparison of
flow evolution models, for the identification of precursors to desirable and
undesirable events, and for flow control applications exploiting nonlinear
actuation dynamics.Comment: 48 pages, 30 figures. Revised version with additional material.
Accepted for publication in Journal of Fluid Mechanic
Hidden attractors in fundamental problems and engineering models
Recently a concept of self-excited and hidden attractors was suggested: an
attractor is called a self-excited attractor if its basin of attraction
overlaps with neighborhood of an equilibrium, otherwise it is called a hidden
attractor. For example, hidden attractors are attractors in systems with no
equilibria or with only one stable equilibrium (a special case of
multistability and coexistence of attractors). While coexisting self-excited
attractors can be found using the standard computational procedure, there is no
standard way of predicting the existence or coexistence of hidden attractors in
a system. In this plenary survey lecture the concept of self-excited and hidden
attractors is discussed, and various corresponding examples of self-excited and
hidden attractors are considered
On the nature of the torus in the complex Lorenz equations.
The complex Lorenz equations are a nonlinear fifth-order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus. In this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter) and between zero and nonzero r2(complexity parameter); in so doing, we are able to gain insight on the effect of the rotational invariance of the system, and on how extra weak dispersion (r2 ≠0) affects the chaotic behavior of the real Lorenz system (which describes a weakly dissipative, dispersive instability)
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