December  2009, 2(4): 829-850. doi: 10.3934/dcdss.2009.2.829

Homoclinic clusters and chaos associated with a folded node in a stellate cell model


Department of Mathematics and Statistics, University of Sydney, Sydney, Australia

Received  September 2008 Revised  May 2009 Published  September 2009

Acker et al (J. Comp. Neurosci., 15, pp.71-90, 2003) developed a model of stellate cells which reproduces qualitative oscillatory patterns known as mixed mode oscillations observed in experiments. This model includes different time scales and can therefore be viewed as a singularly perturbed system of differential equations. The bifurcation structure of this model is very rich, and includes a novel class of homoclinic bifurcation points. The key to the bifurcation analysis is a folded node singularity that allows trajectories known as canards to cross from a stable slow manifold to an unstable slow manifold as well as a node equilibrium of the slow flow on the unstable slow manifold. In this work we focus on the novel homoclinic orbits within the bifurcation diagram and show that the return of canards from the unstable slow manifold to the funnel of the folded node on the stable slow manifold results in a horseshoe map, and therefore gives rise to chaotic invariant sets. We also use a one-dimensional map to explain why many homoclinic orbits occur in "clusters'' at exponentially close parameter values.
Citation: Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023


Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191


Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015


Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263


Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409


Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933


Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1


Hsin-Lun Li. Mixed Hegselmann-Krause dynamics. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021084


Qiang Lin, Yang Xiao, Jingju Zheng. Selecting the supply chain financing mode under price-sensitive demand: Confirmed warehouse financing vs. trade credit. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2031-2049. doi: 10.3934/jimo.2020057


Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511


Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381


Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021097


Xiaochun Gu, Fang Han, Zhijie Wang, Kaleem Kashif, Wenlian Lu. Enhancement of gamma oscillations in E/I neural networks by increase of difference between external inputs. Electronic Research Archive, , () : -. doi: 10.3934/era.2021035


Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781


Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004


Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010


Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137


Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222


Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009


Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, , () : -. doi: 10.3934/era.2021032

2019 Impact Factor: 1.233


  • PDF downloads (74)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]