3,605 research outputs found
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
Global dynamics of a harmonically excited oscillator with a play : Numerical studies
This work was supported by the National Secretariat of Science, Technology and Innovation of Ecuador (SENESCYT); the Escuela Superior Politécnica del Litoral of Ecuador (ESPOL); the National Natural Science Foundation of China (11272268, 11572263) and Scholarship of China. A.S.E. Chong and Y. Yue acknowledge the hospitality of the Centre of Applied Dynamics Research at the University of Aberdeen.Peer reviewedPostprin
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
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.
Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System
We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a
canard near which both small amplitude and large amplitude periodic orbits
exist. The addition of small noise induces mixed-mode oscillations (MMOs) in
the vicinity of the canard point. We determine the effect of each model
parameter on the stochastically driven MMOs. In particular we show that any
parameter variation (such as a modification of the piecewise-linear function in
the model) that leaves the ratio of noise amplitude to time-scale separation
unchanged typically has little effect on the width of the interval of the
primary bifurcation parameter over which MMOs occur. In that sense, the MMOs
are robust. Furthermore we show that the piecewise-linear model exhibits MMOs
more readily than the classical FitzHugh-Nagumo model for which a cubic
polynomial is the only nonlinearity. By studying a piecewise-linear model we
are able to explain results using analytical expressions and compare these with
numerical investigations.Comment: 25 pages, 10 figure
On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in
In this paper we use the blow up method of Dumortier and Roussarie
\cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to
Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of
singularities of piecewise smooth dynamical systems
\cite{filippov1988differential} in . Using the regularization
method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the
power of our approach by considering the case of a fold line. We quickly
recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a
simple manner. Then, for the two-fold singularity, we show that the regularized
system only fully retains the features of the singular canards in the piecewise
smooth system in the cases when the sliding region does not include a full
sector of singular canards. In particular, we show that every locally unique
primary singular canard persists the regularizing perturbation. For the case of
a sector of primary singular canards, we show that the regularized system
contains a canard, provided a certain non-resonance condition holds. Finally,
we provide numerical evidence for the existence of secondary canards near
resonance.Comment: To appear in SIAM Journal of Applied Dynamical System
- …