The classification of bifurcations with hidden symmetries

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an N-dimensional rectangle under Neumann boundary conditions

Dynamics of neural systems with discrete and distributed time delays

In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings