Methods of qualitative theory in nonlinear dynamics
- L. Shilnikov, A. Shilnikov, D. Turaev, L. Chua
- Mathematics
- 1998
Structurally Stable Systems Bifurcations of Dynamical Systems The Behavior of Dynamical Systems on Stability Boundaries of Equilibrium States The Behavior of Dynamical Systems on Stability Boundaries…
Methods of Qualitative Theory in Nonlinear Dynamics (Part II)
- L. Shilnikov, A. Shilnikov, D. Turaev, L. Chua
- Physics
- 27 September 2001
Bifurcation and predictability analysis of a low-order atmospheric circulation model
- A. Shilnikov, G. Nicolis, C. Nicolis
- Physics
- 1 December 1995
A comprehensive bifurcation analysis of a low-order atmospheric circulation model is carried out. It is shown that the model admits a codimension-2 saddle-node-Hopf bifurcation. The principal…
Mechanism of bistability: tonic spiking and bursting in a neuron model.
- A. Shilnikov, R. Calabrese, G. Cymbalyuk
- BiologyPhysical review. E, Statistical, nonlinear, and…
- 31 May 2005
It is argued that the Lukyanov-Shilnikov bifurcation of a saddle-node periodic orbit with noncentral homoclinics may initiate a bistability observed in a model of a leech heart interneuron under defined pharmacological conditions.
Methods of the Qualitative Theory for the Hindmarsh-rose Model: a Case Study - a Tutorial
- A. Shilnikov, M. Kolomiets
- PhysicsInternational Journal of Bifurcation and Chaos in…
- 1 August 2008
It is shown that a modified model can exhibit the blue sky bifurcation, as well as, a bistability of the coexisting tonic spiking and bursting activities.
NORMAL FORMS AND LORENZ ATTRACTORS
- A. Shilnikov, L. Shilnikov, D. Turaev
- Mathematics
- 1 October 1993
Normal forms for eleven cases of bifurcations of codimension-3 are considered, basically, in systems with a symmetry, which can be reduced to one of the two three-dimensional systems. The first…
Origin of Chaos in a Two-Dimensional Map Modeling Spiking-bursting Neural Activity
- A. Shilnikov, N. Rulkov
- PhysicsInternational Journal of Bifurcation and Chaos in…
- 1 November 2003
This work studies the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases in a simple two-dimensional map model.
Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe.
- A. Shilnikov, G. Cymbalyuk
- PhysicsPhysical Review Letters
- 31 January 2005
A continuous and reversible transition between periodic tonic spiking and bursting activities in a neuron model constitutes a biophysically plausible mechanism for the regulation of burst duration that increases with no bound like 1/square root alpha-alpha0 as the transition value alpha0 is approached.
Electrogenic properties of the Na⁺/K⁺ ATPase control transitions between normal and pathological brain states.
- G. Krishnan, Gregory N Filatov, A. Shilnikov, M. Bazhenov
- BiologyJournal of Neurophysiology
- 1 May 2015
The study demonstrates the profound role of the current mediated by Na(+)/K(+) ATPase on the stability of neuronal dynamics that was previously unknown.
Key Bifurcations of Bursting Polyrhythms in 3-Cell Central Pattern Generators
- J. Wojcik, J. Schwabedal, R. Clewley, A. Shilnikov
- BiologyPLoS ONE
- 28 December 2013
This work identifies and describes the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG) and provides a powerful qualitative approach to studying detailed models of rhythmic behavior.
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