11 papers
An Index Theorem in Relative K-Theory for First-Order Systems
Robert Skiba, Daniel Strzelecki, Nils Waterstraat
Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of pre…
Putnam-like dataset summary: LLMs as mathematical competition contestants
Bartosz Bieganowski, Daniel Strzelecki, Robert Skiba +1
In this paper we summarize the results of the Putnam-like benchmark published by Google DeepMind. This dataset consists of 96 original problems in the spirit of the Putnam Competit…
Evans function, parity and nonautonomous bifurcations
Christian Pötzsche, Robert Skiba
The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the…
At most n-valued maps
Daciberg Lima Goncalves, Robert Skiba, P. Christopher Staecker
This paper concerns various models of ``at-most--valued maps''. That is, multivalued maps for which has cardinality at most for each . We consider…
Global bifurcation of homoclinic solutions
Iacopo P. Longo, Christian Pötzsche, Robert Skiba
In the analysis of parametrized nonautonomous evolutionary equations, bounded entire solutions are natural candidates for bifurcating objects. Appropriate explicit and sufficient c…
The Index Bundle for Selfadjoint Fredholm Operators and Multiparameter Bifurcation for Hamiltonian Systems
Robert Skiba, Nils Waterstraat
The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generali…
Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory
Robert Skiba, Nils Waterstraat
In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and fa…
Global continuation of homoclinic solutions
Christian Potzsche, Robert Skiba
When extending bifurcation theory of dynamical systems to nonautonomous problems, it is a central observation that hyperbolic equilibria persist as bounded entire solutions under s…
The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems
Robert Skiba, Nils Waterstraat
We develop a -theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions.…
Global bifurcation of homoclinic trajectories of discrete dynamical systems
Jacobo Pejsachowicz, Robert Skiba
We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized b…
Topology and Homoclinic Trajectories of Discrete Dynamical Systems
Jacobo Pejsachowicz, Robert Skiba
We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solut…