Research in future

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The following gives an incomplete list of topics, intended as a shortterm research
projects involved (2) General theorem about the existence and global continuation of periodic solutions in symmetric delay differential equations. (3) Pattern formation in symmetric oncenter offsurround neural networks with delay. (4) In a coupled cell system with nearest neighbor and next nearest neighbor coupling
in a ring of $n$ identical (5) As in systems of ordinary differential equations with $O(2)$ symmetry, the travelling wave solutions in functional differential equations with $O(2)$ symmetry maybe occur at a Hopf bifurcation on a branch of trivial states. By using the isotropy subgroups of group $O(2)\times S^1$ with the two dimensional fixed point subspaces, we can probably consider the switching from the branch of trivial steady state solutions at a Hopf bifurcation to branches of the travelling wave solutions and the standing wave solutions, and study a probably torus bifurcation on travelling wave solutions by using the canonical coordinates transformation. (6) Relative periodic orbits are periodic solutions of a flow induced by an equivariant
vector field on a space of group orbits. In applications they typically appear as
oscillations of a system which are periodic when viewed in some rotating or translating
frame. Existing theoretical work on Hamiltonian relative periodic (7) In systems with symmetry, the Hopf bifurcation from a nontrivial steady state solutions probably gives rise to a branch of direction reversing wave solutions. Further bifurcation from these time periodic solutions maybe lead to a branch of modulated travelling solutions. The standard Hopf theorem establioshed by Golubitsky and coauther cannot be applied in this situation since there is a zero eigenvalue of the Jacobian at every nontrivial steady state solution, due to the group orbit of solutions. In this case, the degeneracy can be dealt with by splitting the vector field into two parts, one tangent to the group orbit and one normal to it. A standard bifurcation analysis can then be performed on the normal vector field and the results are then interpreted for the whole vector field. (8) Recent results in dynamical systems theory have shown the coexistence of chaos and
symmetry. This coexistence seems to be a paradox because symmetry represents order and
regularity, while chaosdisorder and impredictability. It will be interesting to
investigate coexistence of chaos and symmetry in system of (9) To understand information processing in the brain, behaviors of entirely inphase or partially inphase periodic solutions in coupled neuronal models with delayed synaptic transmissions should be intensively investigated, they are usually restricted to some invariant subspaces. Accordingly, one analysis for a phaselocked periodic solution can be reduced to the analysis for periodic solutions observed in a simplified system with a delayed mutual and selfcoupling. According to their symmetrical properties, periodic solutions observed were classified into many different types. We can also calculate bifurcations of periodic solutions with various kinds of synchronization. We can probably find not only various types of periodic solutions but also chaotic oscillations. (10) A structurepreserving generalized LyapunovSchmidt reduction and normal form theory for bifurcations of periodic solutions from a fixed point should be established in order to be applied to functional differential equations. In particular we can study bifurcation of periodic orbits of a given period. The reduction leads to a similar problem on a lower dimensional space with an additional $\mathbb{D}_n$symmetry. (11) We need to exploit the $\mathbb{D}_n$symmetry and the normal forms to prove existence of subharmonic bifurcations at resonances and also in a simple case of multiple resonance. Here the fixed point is called resonant when the derivative at the fixed point has roots of unity as eigenvalues. We have multiple resonance when a pair of complex conjugate resonant eigenvalues has higher multiplicity or when there is more than one such pair of resonant eigenvalues. (12) A general point of contact between the present study and neuroscience lies in the
relationship between explicitly modelling individual neurons and their couplings, and
averaging the behavior of a (sub)population into a single connectionisttype {\it unit}.
Domain of attraction and probability density results for (13) Stability properties for the bifurcating periodic solutions can be discussed briefly either by using the reduction and normal form results, or by the Floquet theory and monotonic dynamics theory. 