A competition between heteroclinic cycles
Competition between co-existing heteroclinic cycles that have a common heteroclinic connection is considered. A simple model problem, consisting of a system of ordinary differential equations in R4…
Noisy heteroclinic networks
- D. Armbruster, E. Stone, V. Kirk
- Environmental Science
- 17 January 2003
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching…
When Shil'nikov Meets Hopf in Excitable Systems
- A. Champneys, V. Kirk, E. Knobloch, Bart E. Oldeman, J. Sneyd
- MathematicsSIAM Journal on Applied Dynamical Systems
- 5 October 2007
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…
Models of Calcium Signalling
This book presents the general modelling theory as well as a large number of specific case examples to show how mathematical modelling can interact with experimental approaches, in an interdisciplinary and multifaceted approach to the study of an important physiological control mechanism.
Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle
- A. Champneys, V. Kirk, E. Knobloch, Bart E. Oldeman, J. Rademacher
- EngineeringSIAM Journal on Applied Dynamical Systems
- 23 September 2009
An idle detector and safety apparatus for use in an internal combustion engine includes a magnet which is mounted to the carburetor piston valve and a magnetically actuatable switch positioned…
On the dynamical structure of calcium oscillations
It is shown that Class I Ca2+ oscillations have a common dynamical structure, irrespective of the oscillation period, which allows the construction of a simple canonical model that incorporates this underlying dynamical behavior.
Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics
- E. Harvey, V. Kirk, M. Wechselberger, J. Sneyd
- Environmental ScienceJournal of nonlinear science
- 10 May 2011
Analysis of the number and nature of the distinct timescales in a model allows us to make useful predictions about the dynamics associated with the model, and may give us more information about the model dynamics than a classification according to the modelling assumptions made about different cellular mechanisms in deriving the models.
Merging of resonance tongues
- V. Kirk
- Mathematics
- 15 July 1993
The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit
- V. Kirk, A. Rucklidge
- Physics
- 1 August 2006
The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known…
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