Stability Relative to A(F) and Bifurcation
As for ordinary differential equations, the primary objective in the qualitative theory of RFDEs is to study the dependence of the flow Фt = Ф t F on F. This implicitly requires the existence of a criterion for deciding when two RFDEs are equivalent. A study of the dependence of the flow on changes of the RFDE through the use of a notion of equivalence based on a comparison of all orbits is very difficult and is likely to give too small equivalence classes. The difficulty is associated with the infinite dimensionality of the phase space and the associated smoothing properties of the solution operator. In order to compare all orbits of two RFDEs one needs to take into account the changes in the range of the solution map Фt, for each fixed t, a not so easy task due to the difficulties associated with backward continuation of solutions. Therefore, it is reasonable to begin the study by considering a notion of equivalence which ignores some of the orbits of the RFDEs to be compared. We restrict ourselves to RFDEs defined by functions F ∈ X1.
KeywordsPeriodic Orbit Equilibrium Point Hopf Bifurcation Unstable Manifold Local Stable Manifold
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