3,858 research outputs found
Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics
This paper presents an overview of physical ideas and mathematical methods
for implementing non-smooth and discontinuous substitutions in dynamical
systems. General purpose of such substitutions is to bring the differential
equations of motion to the form, which is convenient for further use of
analytical and numerical methods of analyses. Three different types of
nonsmooth transformations are discussed as follows: positional coordinate
transformation, state variables transformation, and temporal transformations.
Illustrating examples are provided.Comment: 15 figure
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
Observer design for piecewise smooth and switched systems via contraction theory
The aim of this paper is to present the application of an approach to study
contraction theory recently developed for piecewise smooth and switched
systems. The approach that can be used to analyze incremental stability
properties of so-called Filippov systems (or variable structure systems) is
based on the use of regularization, a procedure to make the vector field of
interest differentiable before analyzing its properties. We show that by using
this extension of contraction theory to nondifferentiable vector fields, it is
possible to design observers for a large class of piecewise smooth systems
using not only Euclidean norms, as also done in previous literature, but also
non-Euclidean norms. This allows greater flexibility in the design and
encompasses the case of both piecewise-linear and piecewise-smooth (nonlinear)
systems. The theoretical methodology is illustrated via a set of representative
examples.Comment: Preprint accepted to IFAC World Congress 201
Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems
Switches in real systems take many forms, such as impacts, electronic relays,
mitosis, and the implementation of decisions or control strategies. To
understand what is lost, and what can be retained, when we model a switch as an
instantaneous event, requires a consideration of so-called hidden terms. These
are asymptotically vanishing outside the switch, but can be encoded in the form
of nonlinear switching terms. A general expression for the switch can be
developed in the form of a series of sigmoid functions. We review the key steps
in extending the Filippov's method of sliding modes to such systems. We show
how even slight nonlinear effects can hugely alter the behaviour of an
electronic control circuit, and lead to `hidden' attractors inside the
switching surface.Comment: 12 page
Synchronization of electrically coupled resonate-and-fire neurons
Electrical coupling between neurons is broadly present across brain areas and
is typically assumed to synchronize network activity. However, intrinsic
properties of the coupled cells can complicate this simple picture. Many cell
types with strong electrical coupling have been shown to exhibit resonant
properties, and the subthreshold fluctuations arising from resonance are
transmitted through electrical synapses in addition to action potentials. Using
the theory of weakly coupled oscillators, we explore the effect of both
subthreshold and spike-mediated coupling on synchrony in small networks of
electrically coupled resonate-and-fire neurons, a hybrid neuron model with
linear subthreshold dynamics and discrete post-spike reset. We calculate the
phase response curve using an extension of the adjoint method that accounts for
the discontinuity in the dynamics. We find that both spikes and resonant
subthreshold fluctuations can jointly promote synchronization. The subthreshold
contribution is strongest when the voltage exhibits a significant post-spike
elevation in voltage, or plateau. Additionally, we show that the geometry of
trajectories approaching the spiking threshold causes a "reset-induced shear"
effect that can oppose synchrony in the presence of network asymmetry, despite
having no effect on the phase-locking of symmetrically coupled pairs
Parameter switching in a generalized Duffing system: Finding the stable attractors
This paper presents a simple periodic parameter-switching method which can
find any stable limit cycle that can be numerically approximated in a
generalized Duffing system. In this method, the initial value problem of the
system is numerically integrated and the control parameter is switched
periodically within a chosen set of parameter values. The resulted attractor
matches with the attractor obtained by using the average of the switched
values. The accurate match is verified by phase plots and Hausdorff distance
measure in extensive simulations
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