| Abstract: |
| Nonlinear boundary value problems often exhibit rich bifurcation structures and may possess multiple solutions. From a dynamical systems perspective, such problems can be interpreted as orbit problems: solutions correspond to trajectories of an associated differential equation that satisfy prescribed boundary conditions. In this formulation, determining all solutions amounts to identifying all admissible solution orbits of the underlying dynamical system.
In this talk, we present a computer-assisted framework for the rigorous global search of all such solution orbits. A central difficulty in this search is that the parameters determining trajectories, such as initial conditions in a shooting formulation, typically range over a noncompact set, making direct exploration infeasible.
To overcome this difficulty, we first derive a priori bounds that confine all possible solutions to a compact set of initial conditions. This significantly reduces the set of candidates. On this set, we use the kv library to rigorously prove the existence or nonexistence of solutions.
This approach allows us to systematically enumerate all solution orbits within a given parameter range. As an application, we present results for nonlinear elliptic problems including H\`{e}non-type equations and discuss the resulting bifurcation structures. |
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