8,103 research outputs found
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
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.
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
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
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
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