191 research outputs found
Analysis of a Model for Ship Maneuvering
We analyze numerically and theoretically steady states and bifurcations in a model for ship maneuvering provided by MARIN, and in a simplified model that combines rudder and propeller into an abstract ‘thruster’. Steady states in the model correspond to circular motion of the ship and we compute the corresponding radii.
We non-dimensionalize the models and thereby remove a number of parameters,
so that, due to a scaling symmetry, only the rudder (or thruster) angle remains as a free parameter.
Using ‘degree theory’, we show that a slight modification of the model pos-
sesses at least one steady state for each angle and find certain constraints on the possible steady state configuration. We show that straight motion is unstable for the Hamburg test case and use numerical continuation and bifurcation software to compute a number of curves of states together with their stability, and the corresponding radii of the ship motion. In particular, straight forward motion can be stabilised by increasing the rudder size parameter, and the smallest possible radius is ∼ 119 m.
These analyses illustrate methods and tools from dynamical systems theory that can be used to analyse a model without simulation. Compared with simulations, the numerical bifurcation analysis is much less time consuming. We have implemented the model in MATLAB and the bifurcation software AUTO
Measuring the mixing efficiency in a simple model of stirring:some analytical results and a quantitative study via Frequency Map Analysis
We prove the existence of invariant curves for a --periodic Hamiltonian
system which models a fluid stirring in a cylindrical tank, when is small
and the assigned stirring protocol is piecewise constant. Furthermore, using
the Numerical Analysis of the Fundamental Frequency of Laskar, we investigate
numerically the break down of invariant curves as increases and we give a
quantitative estimate of the efficiency of the mixing.Comment: 10 figure
Homoclinic orbits: Since Poincaré till today
The history and the contemporary results in homoclinic orbits are reported
Recommended from our members
Dynamische Systeme
This workshop, organized by Hakan Eliasson (Paris), Helmut Hofer (Princeton) and Jean-Christophe Yoccoz (Paris), continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser– Zehnder meeting” in 1981. The workshop was attended by more than 50 participants from 12 countries. The main theme of the workshop were the new results and developments in the area of classical dynamical systems, in particular in celestial mechanics and Hamiltonian systems. Among the main topics were KAM theory in finite and infinite dimensions, and new developments in Floer homology (Rabinowitz-Floer homology)
Qualitative Analysis of Polycycles in Filippov Systems
In this paper, we are concerned about the qualitative behaviour of planar
Filippov systems around some typical minimal sets, namely, polycycles. In the
smooth context, a polycycle is a simple closed curve composed by a collection
of singularities and regular orbits, inducing a first return map. Here, this
concept is extended to Filippov systems by allowing typical singularities lying
on the switching manifold. Our main goal consists in developing a method to
investigate the unfolding of polycycles in Filippov systems. In addition, we
applied this method to describe bifurcation diagrams of Filippov systems around
certain polycycles
Global bifurcations close to symmetry
Heteroclinic cycles involving two saddle-foci, where the saddle-foci share
both invariant manifolds, occur persistently in some symmetric differential
equations on the 3-dimensional sphere. We analyse the dynamics around this type
of cycle in the case when trajectories near the two equilibria turn in the same
direction around a 1-dimensional connection - the saddle-foci have the same
chirality. When part of the symmetry is broken, the 2-dimensional invariant
manifolds intersect transversely creating a heteroclinic network of Bykov
cycles.
We show that the proximity of symmetry creates heteroclinic tangencies that
coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies -
trajectories that follow the original cycle n times around before they arrive
at the other node. Each n-pulse heteroclinic tangency is accumulated by a
sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes
defined on an infinite set of disjoint strips, where the first return map is
hyperbolic. We also show how, as the system approaches full symmetry, the
suspended horseshoes are destroyed, creating regions with infinitely many
attracting periodic solutions
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