Spectral sequences and computation of parametric normal forms of differential equations

Parametric normal form theory is the only theory of normal forms that is useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life problems. It can provide the transformations between the original parametric system and its parametric normal form.;Recently several researchers have developed an efficient computing method for parametric normal forms and have applied it to several singularities. It is proved that parametric normal form theory requires time rescaling and reparametrization alongside of changes of state variable. We develop two new methods for parametric normal forms of vector fields. One uses the method of spectral sequences on locally finite graded parametric vector fields while the other employs the notion of formal decompositions and the multi-Lie bracket method. We introduce suitable algebraic structures for computation of parametric normal forms of several singularities. This includes parametric state space, parametric time space and parameter space. Parametric time space represents a locally finite graded local ring while parametric state space is represented by a Lie algebra as well as a module over the ring of parametric time space. Parameter space describes a locally finite graded vector space while the near identity reparametrization maps form a group acting on parametric state space. It is known that the near identity changes of state variable generated from the flows of vector fields with no linear part form a group, called the Campbell-Hausdorff group, acting on the parametric state space. Therefore, the near identity time rescaling, reparametrization and changes of state variable can be unified by considering them all as some groups acting on the parametric state space. We prove that all three groups are subgroups of filtration preserving automorphisms of the parametric state space. Indeed, they are presented as direct products of each other in this representation. This plays the key role in the method of spectral sequences on parametric normal forms. It also explains the main reason for our claim that time rescaling and reparametrization are required alongside the change of state variable. We introduce these three group structures for parametric systems of Hopf, generalized Hopf and Bogdanov-Takens types and apply them to obtain parametric normal forms of these singularities. Some generic conditions are assumed regarding parameters for all three cases in addition to an extra generic condition for the Bogdanov-Takens singularity.