1,867 research outputs found
Simultaneous Border-Collision and Period-Doubling Bifurcations
We unfold the codimension-two simultaneous occurrence of a border-collision
bifurcation and a period-doubling bifurcation for a general piecewise-smooth,
continuous map. We find that, with sufficient non-degeneracy conditions, a
locus of period-doubling bifurcations emanates non-tangentially from a locus of
border-collision bifurcations. The corresponding period-doubled solution
undergoes a border-collision bifurcation along a curve emanating from the
codimension-two point and tangent to the period-doubling locus here. In the
case that the map is one-dimensional local dynamics are completely classified;
in particular, we give conditions that ensure chaos.Comment: 22 pages; 5 figure
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
Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps
Resonance tongues are mode-locking regions of parameter space in which stable
periodic solutions occur; they commonly occur, for example, near Neimark-Sacker
bifurcations. For piecewise-smooth, continuous maps these tongues typically
have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation
diagrams. We give a symbolic description of a class of "rotational" periodic
solutions that display lens-chain structures for a general -dimensional map.
We then unfold the codimension-two, shrinking point bifurcation, where the
tongues have zero width. A number of codimension-one bifurcation curves emanate
from shrinking points and we determine those that form tongue boundaries.Comment: 27 pages, 6 figure
The Role of Constraints in a Segregation Model: The Symmetric Case
In this paper we study the effects of constraints on the dynamics of an
adaptive segregation model introduced by Bischi and Merlone (2011). The model
is described by a two dimensional piecewise smooth dynamical system in discrete
time. It models the dynamics of entry and exit of two populations into a
system, whose members have a limited tolerance about the presence of
individuals of the other group. The constraints are given by the upper limits
for the number of individuals of a population that are allowed to enter the
system. They represent possible exogenous controls imposed by an authority in
order to regulate the system. Using analytical, geometric and numerical
methods, we investigate the border collision bifurcations generated by these
constraints assuming that the two groups have similar characteristics and have
the same level of tolerance toward the members of the other group. We also
discuss the policy implications of the constraints to avoid segregation
Bifurcations in the Lozi map
We study the presence in the Lozi map of a type of abrupt order-to-order and
order-to-chaos transitions which are mediated by an attractor made of a
continuum of neutrally stable limit cycles, all with the same period.Comment: 17 pages, 12 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
Bifurcations of piecewise smooth ļ¬ows: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
- ā¦