35 research outputs found
A stiction oscillator under slowly varying forcing: Uncovering small scale phenomena using blowup
In this paper, we consider a mass-spring-friction oscillator with the
friction modelled by a regularized stiction model in the limit where the ratio
of the natural spring frequency and the forcing frequency is on the same order
of magnitude as the scale associated with the regularized stiction model. The
motivation for studying this situation comes from \cite{bossolini2017b} which
demonstrated new friction phenomena in this regime. The results of Bossolini et
al 2017 led to some open problems, that we resolve in this paper. In
particular, using GSPT and blowup we provide a simple geometric description of
the bifurcation of stick-slip limit cycles through a combination of a canard
and a global return mechanism. We also show that this combination leads to a
canard-based horseshoe and are therefore able to prove existence of chaos in
this fundamental oscillator system
The dud canard: Existence of strong canard cycles in
In this paper, we provide a rigorous description of the birth of canard limit
cycles in slow-fast systems in through the folded saddle-node of
type II and the singular Hopf bifurcation. In particular, we prove -- in the
analytic case only -- that for all there is a family of
periodic orbits, born in the (singular) Hopf bifurcation and extending to
cycles that follow the strong canard of the folded saddle-node.
Our results can be seen as an extension of the canard explosion in , but in contrast to the planar case, the family of periodic orbits in
is not explosive. For this reason, we have chosen to call the
phenomena in , the ``dud canard''. The main difficulty of the
proof lies in connecting the Hopf cycles with the canard cycles, since these
are described in different scalings. As in , we use blowup to
overcome this, but we also have to compensate for the lack of uniformity near
the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation
in the limit . In the present paper, we do so by imposing
analyticity of the vector-field. This allows us to prove existence of an
invariant slow manifold, that is not normally hyperbolic
Periodic orbits near a bifurcating slow manifold
This paper studies a class of -degree-of-freedom Hamiltonian
systems with a slowly varying phase that unfolds a Hamiltonian pitchfork
bifurcation. The main result of the paper is that there exists an order of
-many periodic orbits that all stay within an -distance from the union of the normally elliptic slow
manifolds that occur as a result of the bifurcation. Here
measures the time scale separation. These periodic orbits are predominantly
unstable. The proof is based on averaging of two blowup systems, allowing one
to estimate the effect of the singularity, combined with results on asymptotics
of the second Painleve equation. The stable orbits of smallest amplitude that
are {persistently} obtained by these methods remain slightly further away from
the slow manifold being distant by an order .Comment: To appear in JD
Canards in stiction: on solutions of a friction oscillator by regularization
We study the solutions of a friction oscillator subject to stiction. This
discontinuous model is non-Filippov, and the concept of Filippov solution
cannot be used. Furthermore some Carath\'eodory solutions are unphysical.
Therefore we introduce the concept of stiction solutions: these are the
Carath\'eodory solutions that are physically relevant, i.e. the ones that
follow the stiction law. However, we find that some of the stiction solutions
are forward non-unique in subregions of the slip onset. We call these solutions
singular, in contrast to the regular stiction solutions that are forward
unique. In order to further the understanding of the non-unique dynamics, we
introduce a regularization of the model. This gives a singularly perturbed
problem that captures the main features of the original discontinuous problem.
We identify a repelling slow manifold that separates the forward slipping to
forward sticking solutions, leading to a high sensitivity to the initial
conditions. On this slow manifold we find canard trajectories, that have the
physical interpretation of delaying the slip onset. We show with numerics that
the regularized problem has a family of periodic orbits interacting with the
canards. We observe that this family has a saddle stability and that it
connects, in the rigid body limit, the two regular, slip-stick branches of the
discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure
An iterative method for the approximation of fibers in slow-fast systems
In this paper we extend a method for iteratively improving slow manifolds so
that it also can be used to approximate the fiber directions. The extended
method is applied to general finite dimensional real analytic systems where we
obtain exponential estimates of the tangent spaces to the fibers. The method is
demonstrated on the Michaelis-Menten-Henri model and the Lindemann mechanism.
The latter example also serves to demonstrate the method on a slow-fast system
in non-standard slow-fast form. Finally, we extend the method further so that
it also approximates the curvature of the fibers.Comment: To appear in SIAD
Slow divergence integral in regularized piecewise smooth systems
In this paper we define the notion of slow divergence integral along sliding
segments in regularized planar piecewise smooth systems. The boundary of such
segments may contain diverse tangency points. We show that the slow divergence
integral is invariant under smooth equivalences. This is a natural
generalization of the notion of slow divergence integral along normally
hyperbolic portions of curve of singularities in smooth planar slow-fast
systems. We give an interesting application of the integral in a model with
visible-invisible two-fold of type . It is related to a connection
between so-called Minkowski dimension of bounded and monotone "entry-exit"
sequences and the number of sliding limit cycles produced by so-called canard
cycles
Singularly Perturbed Boundary-Focus Bifurcations
We consider smooth systems limiting as to piecewise-smooth
(PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable
local normal form, we study the dynamics for the smooth system with using a combination of geometric singular perturbation theory
and blow-up. We show that the type of BF bifurcation in the PWS system
determines the bifurcation structure for the smooth system within an
dependent domain which shrinks to zero as ,
identifying a supercritical Andronov-Hopf bifurcation in one case, and a
supercritical Bogdanov-Takens bifurcation in two other cases. We also show that
PWS cycles associated with BF bifurcations persist as relaxation cycles in the
smooth system, and prove existence of a family of stable limit cycles which
connects the relaxation cycles to regular cycles within the
dependent domain described above. Our results are applied to models
for Gause predator-prey interaction and mechanical oscillation subject to
friction