27,449 research outputs found
On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations
At a border-collision bifurcation a fixed point of a piecewise-smooth map
intersects a surface where the functional form of the map changes. Near a
generic border-collision bifurcation there are two fixed points, each of which
exists on one side of the bifurcation. A simple eigenvalue condition indicates
whether the fixed points exist on different sides of the bifurcation (this case
can be interpreted as the persistence of a single fixed point), or on the same
side of the bifurcation (in which case the bifurcation is akin to a saddle-node
bifurcation). A similar eigenvalue condition indicates whether or not there
exists a period-two solution on one side of the bifurcation. Previously these
conditions have been combined to obtain five distinct scenarios for the
existence and relative coexistence of fixed points and period-two solutions
near border-collision bifurcations. In this Letter, it is shown that one of
these scenarios, namely that two fixed points exist on one side of the
bifurcation and a period-two solution exists on the other side of the
bifurcation, cannot occur. The remaining four scenarios are feasible. Therefore
there are exactly four distinct scenarios for fixed points and period-two
solutions near border-collision bifurcations.Comment: 8 pages, 1 figure, submitted to Appl. Math. Let
The structure of mode-locking regions of piecewise-linear continuous maps
The mode-locking regions of a dynamical system are the subsets of the
parameter space of the system within which there exists an attracting periodic
solution. For piecewise-linear continuous maps, these regions have a curious
chain structure with points of zero width called shrinking points. In this
paper we perform a local analysis about an arbitrary shrinking point. This is
achieved by studying the symbolic itineraries of periodic solutions in nearby
mode-locking regions and performing an asymptotic analysis on one-dimensional
slow manifolds in order to build a comprehensive theoretical framework for the
local dynamics. We obtain leading-order quantitative descriptions for the shape
of nearby mode-locking regions, the location of nearby shrinking points, and
the key properties of these shrinking points. We apply the results to the
three-dimensional border-collision normal form, nonsmooth Neimark-Sacker-like
bifurcations, and grazing-sliding bifurcations in a model of a dry friction
oscillator
Scaling laws for large numbers of coexisting attracting periodic solutions in the border-collision normal form
A wide variety of intricate dynamics may be created at border-collision
bifurcations of piecewise-smooth maps, where a fixed point collides with a
surface at which the map is nonsmooth. For the border-collision normal form in
two dimensions, a codimension-three scenario was described in previous work at
which the map has a saddle-type periodic solution and an infinite sequence of
stable periodic solutions that limit to a homoclinic orbit of the saddle-type
solution. This paper introduces an alternate scenario of the same map at which
there is an infinite sequence of stable periodic solutions due to the presence
of a repeated unit eigenvalue in the linearization of some iterate of the map.
It is shown that this scenario is codimension-four and that the sequence of
periodic solutions is unbounded, aligning with eigenvectors corresponding to
the unit eigenvalue.
Arbitrarily many attracting periodic solutions coexist near either scenario.
It is shown that if denotes the number of attracting periodic solutions,
and denotes the distance in parameter space from one of the two
scenarios, then in the codimension-three case scales with
, where denotes the unstable stability multiplier
associated with the saddle-type periodic solution, and in the codimension-four
case scales with . Since decays significantly
slower than , large numbers of attracting periodic solutions
coexist in open regions of parameter space extending substantially further from
the codimension-four scenarios than the codimension-three scenarios.Comment: 37 pages, 5 figures, submitted to: Int. J. Bifurcation Chao
On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations
A two-fold singularity is a point on a discontinuity surface of a
piecewise-smooth vector field at which the vector field is tangent to the
surface on both sides. Due to the double tangency, forward evolution from a
two-fold is typically ambiguous. This is an especially serious issue for
two-folds that are reached by the forward orbits of a non-zero measure set of
initial points. The purpose of this paper is to explore the concept of
perturbing the vector field so that forward evolution is well-defined, and
characterising the perturbed dynamics in the limit that the size of the
perturbation tends to zero. This concept is applied to a two-fold in two
dimensions. Three forms of perturbation: hysteresis, time-delay, and noise, are
analysed individually. In each case, the limit leads to a novel probabilistic
notion of forward evolution from the two-fold
A general framework for boundary equilibrium bifurcations of Filippov systems
As parameters are varied a boundary equilibrium bifurcation (BEB) occurs when
an equilibrium collides with a discontinuity surface in a piecewise-smooth
system of ODEs. Under certain genericity conditions, at a BEB the equilibrium
either transitions to a pseudo-equilibrium (on the discontinuity surface) or
collides and annihilates with a coexisting pseudo-equilibrium. These two
scenarios are distinguished by the sign of a certain inner product. Here it is
shown that this sign can be determined from the number of unstable directions
associated with the two equilibria by using techniques developed by Feigin. A
new normal form is proposed for BEBs in systems of any number of dimensions.
The normal form involves a companion matrix, as does the leading order sliding
dynamics, and so the connection to the stability of the equilibria is explicit.
In two dimensions the parameters of the normal form distinguish, in a simple
way, the eight topologically distinct cases for the generic local dynamics at a
BEB. A numerical exploration in three dimensions reveals that BEBs can create
multiple attractors and chaotic attractors, and that the equilibrium at the BEB
can be unstable even if both equilibria are stable. The developments presented
here stem from seemingly unutilised similarities between BEBs in discontinuous
systems (specifically Filippov systems as studied here) and BEBs in continuous
systems for which analogous results are, to date, more advanced
Twenty Hopf-like bifurcations in piecewise-smooth dynamical systems
For many physical systems the transition from a stationary solution to
sustained small amplitude oscillations corresponds to a Hopf bifurcation. For
systems involving impacts, thresholds, switches, or other abrupt events,
however, this transition can be achieved in fundamentally different ways. This
paper reviews 20 such `Hopf-like' bifurcations for two-dimensional ODE systems
with state-dependent switching rules. The bifurcations include boundary
equilibrium bifurcations, the collision or change of stability of equilibria or
folds on switching manifolds, and limit cycle creation via hysteresis or time
delay. In each case a stationary solution changes stability and possibly form,
and emits one limit cycle. Each bifurcation is analysed quantitatively in a
general setting: we identify quantities that govern the onset, criticality, and
genericity of the bifurcation, and determine scaling laws for the period and
amplitude of the resulting limit cycle. Complete derivations based on
asymptotic expansions of Poincare maps are provided. Many of these are new,
done previously only for piecewise-linear systems. The bifurcations are
collated and compared so that dynamical observations can be matched to
geometric mechanisms responsible for the creation of a limit cycle. The results
are illustrated with impact oscillators, relay control, automated balancing
control, predator-prey systems, ocean circulation, and the McKean and
Wilson-Cowan neuron models
Grazing-sliding bifurcations creating infinitely many attractors
As the parameters of a piecewise-smooth system of ODEs are varied, a periodic
orbit undergoes a bifurcation when it collides with a surface where the system
is discontinuous. Under certain conditions this is a grazing-sliding
bifurcation. Near grazing-sliding bifurcations structurally stable dynamics are
captured by piecewise-linear continuous maps. Recently it was shown that maps
of this class can have infinitely many asymptotically stable periodic solutions
of a simple type. Here this result is used to show that at a grazing-sliding
bifurcation an asymptotically stable periodic orbit can bifurcate into
infinitely many asymptotically stable periodic orbits. For an abstract ODE
system the periodic orbits are continued numerically revealing subsequent
bifurcations at which they are destroyed
Equilibrium-Independent Dissipativity with Quadratic Supply Rates
Equilibrium-independent dissipativity (EID) is a recently introduced system
property which requires a system to be dissipative with respect to any forced
equilibrium configuration. This paper is a detailed examination of EID with
quadratic supply rates for a common class of nonlinear control-affine systems.
We provide an algebraic characterization of EID for such systems in the spirit
of the Hill-Moylan lemma, where the usual stability condition is replaced by an
incremental stability condition. Based on this characterization, we state
results concerning internal stability, feedback stability, and absolute
stability of EID systems. Finally, we study EID for discrete-time systems,
providing the relevant definitions and an analogous Hill-Moylan-type
characterization. Results for both continuous-time and discrete-time systems
are illustrated through examples on physical systems and convex optimization
algorithms.Comment: Revised version, 15 page
The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations
An attractor of a piecewise-smooth continuous system of differential
equations can bifurcate from a stable equilibrium to a more complicated
invariant set when it collides with a switching manifold under parameter
variation. Here numerical evidence is provided to show that this invariant set
can be chaotic. The transition occurs locally (in a neighbourhood of a point)
and instantaneously (for a single critical parameter value). This phenomenon is
illustrated for the normal form of a boundary equilibrium bifurcation in three
dimensions using parameter values adapted from of a piecewise-linear model of a
chaotic electrical circuit. The variation of a secondary parameter reveals a
period-doubling cascade to chaos with windows of periodicity. The dynamics is
well approximated by a one-dimensional unimodal map which explains this
bifurcation structure. The robustness of the attractor is also investigated by
studying the influence of nonlinear terms
Dimension reduction for slow-fast, piecewise-smooth, continuous systems of ODEs
The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems
of ODEs occurs on critical manifolds that are piecewise-linear. At points of
non-differentiability, such manifolds are not normally hyperbolic and so the
fundamental results of geometric singular perturbation theory do not apply. In
this paper it is shown that if the critical manifold is globally stable then
the system is forward invariant in a neighbourhood of the critical manifold. It
follows that in this neighbourhood the dynamics is given by a regular
perturbation of the dynamics on the critical manifold and so dimension
reduction can be achieved. If the attraction is instead non-global, additional
dynamics involving canards may be generated. For boundary equilibrium
bifurcations of piecewise-smooth, continuous systems, the results are used to
establish a general methodology by which such bifurcations can be analysed.
This approach is illustrated with a three-dimensional model of ocean
circulation
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