791 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
Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows
An equilibrium of a planar, piecewise-, continuous system of
differential equations that crosses a curve of discontinuity of the Jacobian of
its vector field can undergo a number of discontinuous or border-crossing
bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to
on one side of the discontinuity and
on the other, with ,
and the quantity is nonzero, then a
periodic orbit is created or destroyed as the equilibrium crosses the
discontinuity. This bifurcation is analogous to the classical Andronov-Hopf
bifurcation, and is supercritical if and subcritical if .Comment: laTex, 18 pages, 8 figure
Bifurcation diagram for saddle/source bimodal linear dynamical systems
We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcationPostprint (published version
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