Resumo: We provide a functional-analytical approach to the bifurcation theory of bounded entire solutions of differential equations. On the one hand, it is rather specific in the sense that it applies only to unstable solutions of at least planar and necessarily nonautonomous problems. On the other hand, it can provide a detailed insight into the set of bounded entire solutions.

Initial point of our endeavor is the Evans function, an established concept for the stability analysis of traveling wave solutions to evolutionary partial differential equations. We relate this to the parity [1], which is a crucial tool in the abstract bifurcation theory of nonlinear Fredholm operators. It is illustrated that isolated zeros of the Evans function imply that critical spectral intervals of the Sacker-Sell (dichotomy) spectrum split, while sign changes of the Evans function are sufficient for local and global bifurcations of whole continua of bounded entire solutions. Previously known criteria for such nonautonomous bifurcations result as special cases.

This is a joint work with Iacopo Longo (Imperial College, London, UK) and Robert Skiba (Nicolaus Copernicus University, Toruń, Poland)

References:
[1] J. Pejsachowicz, P. Rabier. Degree theory for $C^1$ Fredholm mappings of index 0, Journal d'Analyse Math{\'e}matique 76(1), 289–319, 1998.
[2] C. Pötzsche. Nonautonomous bifurcation of bounded solutions: Crossing curve situations, Stoch. Dyn. 12(2), 1150017, 2012.
[3] B. Sandstede. Stability of travelling waves, in Handbook of Dynamical Systems 2 (ed. B. Fiedler), Elsevier Science, Amsterdam, 983--1055, 2002.