17 research outputs found
Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry
This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages.
This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles
Persistence of the heteroclinic loop under periodic perturbation
We consider an autonomous ordinary differential equation that admits a heteroclinic loop. The unperturbed heteroclinic loop consists of two degenerate heteroclinic orbits and . We assume the variational equation along the degenerate heteroclinic orbit has {d_i}\left({{d_i} > 1, i = 1, 2} \right) linearly independent bounded solutions. Moreover, the splitting indices of the unperturbed heteroclinic orbits are and , respectively. In this paper, we study the persistence of the heteroclinic loop under periodic perturbation. Using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we obtained the bifurcation function, which is defined from to . Under some conditions, the perturbed system can have a heteroclinic loop near the unperturbed heteroclinic loop
Applications of dynamical systems with symmetry
This thesis examines the application of symmetric dynamical systems theory to
two areas in applied mathematics: weakly coupled oscillators with symmetry, and
bifurcations in flame front equations.
After a general introduction in the first chapter, chapter 2 develops a theoretical
framework for the study of identical oscillators with arbitrary symmetry group under an
assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The
structure imposed by the symmetry on the phase space for weakly coupled oscillators
with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries
and network symmetries is shown to cause decoupling under certain conditions.
Chapter 3 discusses what this implies for generic dynamical behaviour of coupled
oscillator systems, and concentrates on application to small numbers of oscillators (three
or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic
cycles.
Following this, chapter 4 reports on experimental results from electronic oscillator
systems and relates it to results in chapter 3. In a forced oscillator system, breakdown
of regular motion is observed to occur through break up of tori followed by a symmetric
bifurcation of chaotic attractors to fully symmetric chaos.
Chapter 5 discusses reduction of a system of identical coupled oscillators to phase
equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian
oscillators with very weakly coupling. This provides a derivation of example phase
equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing
oscillators in the case of a twin-well potential.
Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6
starts by discussing flame front equations in general, and non-linear models in particular.
The Kuramoto-Sivashinsky equation on a rectangular domain with simple
boundary conditions is found to be an example of a large class of systems whose linear
behaviour gives rise to arbitrarily high order mode interactions.
Chapter 7 presents computation of some of these mode interactions using competerised
Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates
the bifurcation diagrams in two parameters
Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds
A primary spectral submanifold (SSM) is the unique smoothest nonlinear
continuation of a nonresonant spectral subspace of a dynamical system
linearized at a fixed point. Passing from the full nonlinear dynamics to the
flow on an attracting primary SSM provides a mathematically precise reduction
of the full system dynamics to a very low-dimensional, smooth model in
polynomial form. A limitation of this model reduction approach has been,
however, that the spectral subspace yielding the SSM must be spanned by
eigenvectors of the same stability type. A further limitation has been that in
some problems, the nonlinear behavior of interest may be far away from the
smoothest nonlinear continuation of the invariant subspace . Here we remove
both of these limitations by constructing a significantly extended class of
SSMs that also contains invariant manifolds with mixed internal stability types
and of lower smoothness class arising from fractional powers in their
parametrization. We show on examples how fractional and mixed-mode SSMs extend
the power of data-driven SSM reduction to transitions in shear flows, dynamic
buckling of beams and periodically forced nonlinear oscillatory systems. More
generally, our results reveal the general function library that should be used
beyond integer-powered polynomials in fitting nonlinear reduced-order models to
data.Comment: To appear in Chao
Forced frequency locking in S1-equivariant differential equations
The aim of this paper is to present a simple analytic stategy for predicting, or engineering, two frequency locking phenomena for S1-equivariant ordinary differential equations. First we consider the forced frequency locking of a rotating wave solution of the unforced equation with a forcing of "rotating wave type", and we describe the creation of modulated wave solutions which is connected with this locking phenomenon. And second, we consider the forced frequency locking of a modulated wave solution with a forcing of "modulated wave type". Especially, we describe the sets of all control parameters and of all forcings such that frequency locking occures, the dynamic stability and the asymptotic behavior (for the forcing intensity tending to zero) of the locked solutions and the structural stability of all the phenomena. This paper is essentially founded on results from our previous work [41] concerning abstract forced symmetry breaking. The equations considered in the present paper are finite dimensional prototypes of certain infinite dimensional models describing the behavior of continuous wave operated or self-pulsating multisection DFB lasers under continuous or pulsating light injection, respectively
Planar Radial Weakly-Dissipative Diffeomorphisms
We study the effect of a small dissipative radial perturbation acting on a one parameter family of area preserving diffeomorphisms. This is a specific type of dissipative perturbation. The interest is on the global effect of the dissipation on a fixed domain around an elliptic fixed/periodic point of the family, rather than on the effects around a single resonance. We describe the local/global bifurcations observed in the transition from the conservative to a weakly dissipative case: the location of the resonant islands, the changes in the domains of attraction of the foci inside these islands, how the resonances disappear, etc. The possible -limits are determined in each case. This topological description gives rise to three different dynamical regimes according to the size of dissipative perturbation. Moreover, we determine the conservative limit of the probability of capture in a generic resonance from the interpolating flow approximation, hence assuming no homoclinics in the resonance. As a paradigm of weakly dissipative radial maps, we use a dissipative version of the Hénon map
A classical treatment of the quadratic Zeeman effect in atomic hydrogen
The classical Hamiltonian describing the hydrogen atom in the presence of a static magnetic field of arbitrary strength for arbitrary angular momentum is derived. For this Hamiltonian the transition from the regular to the chaotic motion is observed by means of the Poincare mappings. Two different classes of non-planar periodic orbits are traced in both regular and irregular regions. The bifurcations and variation of the periodic motion with the change of the total energy parameter throughout the regular regime and into the chaotic regime are given together with the relevant frequencies. For both classes the stability/instability of the periodic orbits is studied by calculating the linearization matrix in the neighbourhood of the corresponding fixed points of the Poincare mappings. In one class, the class of orbits that approach very close to the nucleus, we have surprisingly found that a set of periodic orbits bifurcate from the same periodic orbit along the field at various values of the energy. These values are determined numerically. A repeated pattern of stability and instability of these orbits exists over decreasing intervals of energy until the escape energy is approached. All these periodic orbits are unstable beyond the ionization limit. On the other hand we have found that the bifurcation of the second class of orbits is, generally, generic. Three sets of the energy separation lines due to three types of periodic motions are given when B = 60 kG with m - 0. Other sets of lines are given for B = 42 kG with m = 0, m = -1 and m = -2. Many of these lines coincide with the spectral lines obtained experimentally by A. Holle et al (1986).The energy spacing 0.64 near the ionisation limit, which has been found recently in the experiments of Holle et al (1986) is due to one of the non-planar orbits. Other new predicted spacings arising from other orbits have been seen in high resolution experiments on atoms in external fields (Main et al 1986).<p