Bifurcations, Normal Forms and their Applications
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The first part is a study of an ecological model with one herbivore and $N$ plants. The system has a new type of functional response due to the speculation that the plants compete with each other and have different levels of toxin which inhibit the herbivore's ability to eat up to a certain amount. We first derive the model mathematically and then investigate, both analytically and numerically, the possible dynamics for this model, including the bifurcation and chaos. We also discuss the conditions under which all the species can coexist. The second part is a study in the normal form theory. In particular, we study the relations between the normal forms and the first integrals in analytic vector fields. We are able to generalize one of Poincare's classical results on the nonexistence of first integrals in an autonomous system. Then in the space of 2n-dimensional analytic autonomous systems with exactly n resonances and n functionally independent first integrals, we obtain some results related to the convergence and generic divergence of the normalizations. Lastly we give a new proof of the necessary and sufficient conditions for a planar Hamiltonian system to have an isochronous center.