26 research outputs found
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
Canards existence in the Hindmarsh-Rose model
In two previous papers we have proposed a new method for proving the existence of "canard solutions" on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381-431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of "canard solutions" in the Hindmarsh-Rose model
Canards existence in the Hindmarsh-Rose model
In two previous papers, we have proposed a new method for proving the existence of "canard solutions" on one hand for three- and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand, for four-dimensional singularly perturbed systems with two fast variables; see [4, 5]. The aim of this work is to extend this method, which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of "canard solutions" in the Hindmarsh-Rose model
Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling
This paper investigates the dependence of synchronization transitions of
bursting oscillations on the information transmission delay over scale-free
neuronal networks with attractive and repulsive coupling. It is shown that for
both types of coupling, the delay always plays a subtle role in either
promoting or impairing synchronization. In particular, depending on the
inherent oscillation period of individual neurons, regions of irregular and
regular propagating excitatory fronts appear intermittently as the delay
increases. These delay-induced synchronization transitions are manifested as
well-expressed minima in the measure for spatiotemporal synchrony. For
attractive coupling, the minima appear at every integer multiple of the average
oscillation period, while for the repulsive coupling, they appear at every odd
multiple of the half of the average oscillation period. The obtained results
are robust to the variations of the dynamics of individual neurons, the system
size, and the neuronal firing type. Hence, they can be used to characterize
attractively or repulsively coupled scale-free neuronal networks with delays.Comment: 15 pages, 9 figures; accepted for publication in PLoS ONE [related
work available at http://arxiv.org/abs/0907.4961 and
http://www.matjazperc.com/
Leonid Shilnikov and mathematical theory of dynamical chaos
This Focus Issue Global Bifurcations, Chaos, and Hyperchaos Theory and Applications is dedicated to the 85th anniversary of the great mathematician, one of the founding fathers of dynamical chaos theory, Leonid Pavlovich Shilnikov