3,051 research outputs found
Canards in stiction: on solutions of a friction oscillator by regularization
We study the solutions of a friction oscillator subject to stiction. This
discontinuous model is non-Filippov, and the concept of Filippov solution
cannot be used. Furthermore some Carath\'eodory solutions are unphysical.
Therefore we introduce the concept of stiction solutions: these are the
Carath\'eodory solutions that are physically relevant, i.e. the ones that
follow the stiction law. However, we find that some of the stiction solutions
are forward non-unique in subregions of the slip onset. We call these solutions
singular, in contrast to the regular stiction solutions that are forward
unique. In order to further the understanding of the non-unique dynamics, we
introduce a regularization of the model. This gives a singularly perturbed
problem that captures the main features of the original discontinuous problem.
We identify a repelling slow manifold that separates the forward slipping to
forward sticking solutions, leading to a high sensitivity to the initial
conditions. On this slow manifold we find canard trajectories, that have the
physical interpretation of delaying the slip onset. We show with numerics that
the regularized problem has a family of periodic orbits interacting with the
canards. We observe that this family has a saddle stability and that it
connects, in the rigid body limit, the two regular, slip-stick branches of the
discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure
Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems
This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.
Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems
Systems that are not smooth can undergo bifurcations that are forbidden in
smooth systems. We review some of the phenomena that can occur for
piecewise-smooth, continuous maps and flows when a fixed point or an
equilibrium collides with a surface on which the system is not smooth. Much of
our understanding of these cases relies on a reduction to piecewise linearity
near the border-collision. We also review a number of codimension-two
bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Astrodynamics-optimization theory and guidance theory Research achievements review series no. 15-16
Review of research achievements in astrodynamics, optimization theory, and guidance theor
Variational electrodynamics of Atoms
We generalize Wheeler-Feynman electrodynamics by the minimization of a finite
action functional defined for variational trajectories that are required to
merge continuously into given past and future boundary segments. We prove that
the boundary-value problem is well-posed for two classes of boundary data and
show that the well-posed solution in general has velocity discontinuities,
henceforth broken extrema. Along regular segments, broken extrema satisfy the
Euler-Lagrange neutral differential delay equations with state-dependent
deviating arguments. At points where velocities are discontinuous, broken
extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta
are continuous. The electromagnetic fields of the variational trajectories are
derived quantities that can be extended only to a bounded region B of
space-time. For extrema with a finite number of velocity discontinuities,
extended fields are defined for all point in B with the exception of sets of
zero measure. The extended fields satisfy the integral laws of classical
electrodynamics for most surfaces and curves inside B. As an application, we
study globally bounded trajectories with vanishing far-fields for the
hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses
solutions of the neutral differential delay equations along regular segments
and a variational approximation for the collisional segments. Each hydrogenoid
model predicts a discrete set of finitely measured neighbourhoods of orbits
with vanishing far-fields at the correct atomic magnitude and in quantitative
and qualitative agreement with experiment and quantum mechanics, i.e., the
spacings between consecutive discrete angular momenta agree with Planck's
constant within thirty-percent, while orbital frequencies agree with a
corresponding spectroscopic line within a few percent.Comment: Full re-write using same equations and back to original title
(version 18 compiled with the wrong figure 5). A few commas introduced and
all paragraphs broken into smaller ones whenever possibl
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