Symmetrically coupled cell systems

2020 ◽  
pp. 81-100
Author(s):  
Michael Field
Keyword(s):  
Cell Systems ◽  
Coupled Cell Systems

STABILITY COMPUTATIONS FOR NILPOTENT HOPF BIFURCATIONS IN COUPLED CELL SYSTEMS

2007 ◽  
Vol 17 (08) ◽  
pp. 2595-2603 ◽  
Author(s):  
M. GOLUBITSKY ◽  
M. KRUPA
Keyword(s):  
Hopf Bifurcations ◽  
Cell Systems ◽  
Codimension One ◽  
Partial Stability ◽  
Branching Patterns ◽  
Coupled Cell Systems ◽  
The Stability ◽  
Identity Transformations ◽  
Partial Synchrony ◽  
Stability Results

Vanderbauwhede and van Gils, Krupa, and Langford studied unfoldings of bifurcations with purely imaginary eigenvalues and a nonsemisimple linearization, which generically occurs in codimension three. In networks of identical coupled ODE these nilpotent Hopf bifurcations can occur in codimension one. Elmhirst and Golubitsky showed that these bifurcations can lead to surprising branching patterns of periodic solutions, where the type of bifurcation depends in part on the existence of an invariant subspace corresponding to partial synchrony. We study the stability of some of these bifurcating solutions. In the absence of partial synchrony the problem is similar to the generic codimension three problem. In this case we show that the bifurcating branches are generically unstable. When a synchrony subspace is present we obtain partial stability results by using only those near identity transformations that leave this subspace invariant.

Heteroclinic Networks in Coupled Cell Systems

1999 ◽  
Vol 148 (2) ◽  
pp. 107-143 ◽  
Author(s):  
Peter Ashwin ◽  
Michael Field
Keyword(s):  
Cell Systems ◽  
Coupled Cell Systems

SYNCHRONY-BREAKING HOPF BIFURCATION IN A MODEL OF ANTIGENIC VARIATION

2013 ◽  
Vol 23 (02) ◽  
pp. 1350021 ◽  
Author(s):  
BERNARD S. CHAN ◽  
PEI YU
Keyword(s):  
Plasmodium Falciparum ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Antigenic Variation ◽  
Cross Reactivity ◽  
In Vivo Model ◽  
Main Attention ◽  
Cell Systems ◽  
Coupled Cell Systems

In this paper, we will analyze the bifurcation dynamics of an in vivo model of Plasmodium falciparum. The main attention of this model is focused on the dynamics of cross-reactivity from antigenic variation. We apply the techniques of coupled cell systems to study this model. It is shown that synchrony-breaking Hopf bifurcation occurs from a nontrivial synchronous equilibrium. In proving the existence of a Hopf bifurcation, we also discover the condition under which possible 2-color synchrony patterns arise from the bifurcation. The dynamics resulting from the bifurcation are qualitatively similar to known behavior of antigenic variation. These results are discussed and illustrated with specific examples and numerical simulations.

Nilpotent Hopf Bifurcations in Coupled Cell Systems

2006 ◽  
Vol 5 (2) ◽  
pp. 205-251 ◽  
Author(s):  
Toby Elmhirst ◽  
Martin Golubitsky
Keyword(s):  
Hopf Bifurcations ◽  
Cell Systems ◽  
Coupled Cell Systems

Topology and bifurcations in Hamiltonian coupled cell systems

2016 ◽  
Vol 32 (1) ◽  
pp. 23-45
Author(s):  
B. S. Chan ◽  
P. L. Buono ◽  
A. Palacios
Keyword(s):  
Cell Systems ◽  
Coupled Cell Systems

scholarly journals CYCLING CHAOS IN ONE-DIMENSIONAL COUPLED ITERATED MAPS

2002 ◽  
Vol 12 (08) ◽  
pp. 1859-1868 ◽  
Author(s):  
ANTONIO PALACIOS
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Fixed Points ◽  
Cell Systems ◽  
One Dimensional ◽  
Iterated Maps ◽  
Systems Of Differential Equations ◽  
Coupled Cell Systems ◽  
Cycling Chaos ◽  
Cycling Behavior

Cycling behavior involving steady-states and periodic solutions is known to be a generic feature of continuous dynamical systems with symmetry. Using Chua's circuit equations and Lorenz equations, Dellnitz et al. [1995] showed that "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets, can also be found generically in coupled cell systems of differential equations with symmetry. In this work, we use numerical simulations to demonstrate that cycling chaos also occurs in discrete dynamical systems modeled by one-dimensional maps. Using the cubic map f (x, λ) = λx - x3 and the standard logistic map, we show that coupled iterated maps can exhibit cycles connecting fixed points with fixed points and periodic orbits with periodic orbits, where the period can be arbitrarily high. As in the case of coupled cell systems of differential equations, we show that cycling behavior can also be a feature of the global dynamics of coupled iterated maps, which exists independently of the internal dynamics of each map.

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