SIAM Digital Library
 
 
 

SIAM J. on Applied Dynamical Systems

Search Issue | RSS Feeds RSS
Previous Issue

2007

Volume 6, Issue 4, pp. 663-758


When Shil'nikov Meets Hopf in Excitable Systems

Alan R. Champneys, Vivien Kirk, Edgar Knobloch, Bart E. Oldeman, and James Sneyd

SIAM J. Appl. Dyn. Syst. 6, pp. 663-693 (31 pages) | Cited 3 times

Online Publication Date: October 05, 2007

Full Text: | Download PDF

Show Abstract
This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh–Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a U-shaped locus of Hopf bifurcations that corresponds to the onset of small-amplitude linear waves. The central question addressed is how the Hopf and solitary wave (homoclinic orbit in a moving frame) bifurcation curves interact in these “CU systems.” A variety of possible codimension-two mechanisms is reviewed through which such Hopf and homoclinic bifurcation curves can interact. These include Shil'nikov–Hopf bifurcations and the local birth of homoclinic chaos from a saddle-node/Hopf (Gavrilov–Guckenheimer) point. Alternatively, there may be barriers in phase space that prevent the homoclinic curve from reaching the Hopf bifurcation. For example, the homoclinic orbit may bump into another equilibrium at a so-called T-point, or it may terminate by forming a heteroclinic cycle with a periodic orbit. This paper presents the results of detailed numerical continuation results on different CU systems, thereby illustrating various mechanisms by which Hopf and homoclinic curves interact in CU systems. Owing to a separation of time scales in these systems, considerable care has to be taken with the numerics in order to reveal the true nature of the bifurcation curves observed.

The Onset of Oscillations in Microvascular Blood Flow

John B. Geddes, Russell T. Carr, Nathaniel J. Karst, and Fan Wu

SIAM J. Appl. Dyn. Syst. 6, pp. 694-727 (34 pages) | Cited 1 time

Online Publication Date: October 17, 2007

Full Text: | Download PDF

Show Abstract
We explore the stability of equilibrium solution(s) of a simple model of microvascular blood flow in a two-node network. The model takes the form of convection equations for red blood cell concentration, and contains two important rheological effects—the Fåhræus–Lindqvist effect, which governs viscosity of blood flow in a single vessel, and the plasma skimming effect, which describes the separation of red blood cells at diverging nodes. We show that stability is governed by a linear system of integral equations, and we study the roots of the associated characteristic equation in detail. We demonstrate using a combination of analytical and numerical techniques that it is the relative strength of the Fåhræus–Lindqvist effect and the plasma skimming effect which determines the existence of a set of network parameter values which lead to a Hopf bifurcation of the equilibrium solution. We confirm these predictions with direct numerical simulation and suggest several areas for future research and application.

Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators

Peter Ashwin, Gábor Orosz, John Wordsworth, and Stuart Townley

SIAM J. Appl. Dyn. Syst. 6, pp. 728-758 (31 pages) | Cited 5 times

Online Publication Date: December 07, 2007

Full Text: | Download PDF

multimedia

Show Abstract
Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. We investigate such a heteroclinic network between partially synchronized states where the phases cluster into three groups. For the coupling considered there exist 30 different three-cluster states in the case of five oscillators. We study the structure of the heteroclinic network and demonstrate that it is possible to navigate around the network by applying small impulsive inputs to the oscillator phases. This paper shows that such navigation may be done reliably even in the presence of noise and frequency detuning, as long as the input amplitude dominates the noise strength and the detuning magnitude, and the time between the applied pulses is in a suitable range. Furthermore, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states and replace the heteroclinic network by a network of excitable three-cluster states. The resulting “excitable network” has the same structure as the heteroclinic network and navigation around the excitable network is also possible by applying large impulsive inputs. We also discuss features that have implications for related models of neural activity.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Stability of Coupled Map Networks with Delays
  2. Efficient Automation of Index Pairs in Computational Conley Index Theory
  3. Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials
  4. Breathers and Q-Breathers: Two Sides of the Same Coin
  5. Canonical Discontinuous Planar Piecewise Linear Systems
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close