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| Let $X$ be a 4-dimensional vector field with an equilibrium point where the 1-jet of $X$ is linearly conjugated to
$$
y\frac{\partial}{\partial x}-\omega v\frac{\partial}{\partial u}+\omega u\frac{\partial}{\partial v}.
$$
with $\omega\neq 0$. We refer to $X$ as a Hopf-Bogdanov-Takens (HBT in the sequel) singularity. We will discuss local bifurcations arising in generic unfoldings of HBT singularities of codimension 3. The catalogue includes several types of Hopf-Zero and Hopf-Hopf bifurcations. Some global aspects of the bifurcation diagram will also be treated.
Our interest in HBT singularities is due to their role in applications. They arise in a natural way when planar systems displaying Hopf bifurcations are coupled by simple mechanisms. For instance, they are present in a model consisting of two brusselators linearly coupled by diffusion. They also play a crucial role in the understanding of the unfolding of a 4-dimensional nilpotent singularity of codimension four. Complex dynamical behaviour emerging from HBT singularities will be shown in the context of such applications. |
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