3,012 research outputs found
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
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
Statics and dynamics of elastic manifolds in media with long-range correlated disorder
We study the statics and dynamics of an elastic manifold in a disordered
medium with quenched defects correlated as r^{-a} for large separation r. We
derive the functional renormalization-group equations to one-loop order, which
allow us to describe the universal properties of the system in equilibrium and
at the depinning transition. Using a double epsilon=4-d and delta=4-a
expansion, we compute the fixed points characterizing different universality
classes and analyze their regions of stability. The long-range
disorder-correlator remains analytic but generates short-range disorder whose
correlator exhibits the usual cusp. The critical exponents and universal
amplitudes are computed to first order in epsilon and delta at the fixed
points. At depinning, a velocity-versus-force exponent beta larger than unity
can occur. We discuss possible realizations using extended defects.Comment: 16 pages, 11 figures, revtex
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