441 research outputs found
Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial
A new computational technique based on the symbolic description utilizing
kneading invariants is proposed and verified for explorations of dynamical and
parametric chaos in a few exemplary systems with the Lorenz attractor. The
technique allows for uncovering the stunning complexity and universality of
bi-parametric structures and detect their organizing centers - codimension-two
T-points and separating saddles in the kneading-based scans of the iconic
Lorenz equation from hydrodynamics, a normal model from mathematics, and a
laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201
Models of Multifunctional Central Pattern Generators: Polyrhythmic Bursting
We demonstrate a motif of three reciprocally inhibitory cells that is able to produce multiple patterns of bursting rhythms. Through the examination of the qualitative geometric structure of two-dimensional maps for phase lag between the cells we reveal the organizing centers of emergent polyrhythmic patterns and their bifurcations, as the asymmetry of the synaptic coupling is varied. The presence of multistability and the types of attractors in the network are shown to be determined by the duty cycle of bursting. This analysis does not require knowledge of the equations that model the system, and so provides a powerful new approach to studying regulatory networks. Thus, the approach is applicable to a variety of biological phenomena beyond motor control
Symbolic Toolkit for Chaos Explorations
New computational technique based on the symbolic description utilizing
kneading invariants is used for explorations of parametric chaos in a two
exemplary systems with the Lorenz attractor: a normal model from mathematics,
and a laser model from nonlinear optics. The technique allows for uncovering
the stunning complexity and universality of the patterns discovered in the
bi-parametric scans of the given models and detects their organizing centers --
codimension-two T-points and separating saddles.Comment: International Conference on Theory and Application in Nonlinear
Dynamics (ICAND 2012
Subthreshold oscillations in a map-based neuron model
Self-sustained subthreshold oscillations in a discrete-time model of neuronal
behavior are considered. We discuss bifurcation scenarios explaining the birth
of these oscillations and their transformation into tonic spikes. Specific
features of these transitions caused by the discrete-time dynamics of the model
and the influence of external noise are discussed.Comment: To be published in Physics Letters
Blue sky catastrophe in singularly perturbed systems
We show that the blue sky catastrophe, which creates a stable periodic orbit whose length increases with no bound, is a typical phenomenon for singularly-perturbed (multi-scale) systems with at least two fast variables. Three distinct mechanisms of this bifurcation are described. We argue that it is behind a transition from periodic spiking to periodic bursting oscillations
On some mathematical topics in classic synchronization
A few mathematical problems arising in the classical synchronization theory are discussed, especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies done by Cartwright and Littlewood. Today we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive descriptiob of bifurcational phenomena in the two-parameter case. Of a particular value among ones is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori
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