Connected simple systems, transition matrices, and heteroclinic bifurcations

Abstract

Abstract. Given invariant sets A , B , and C , and connecting orbits A -» B and B -> C , we state very general conditions under which they bifurcate to produce an A -» C connecting orbit. In particular, our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits, and for which the Conley index of the invariant sets is the same as that of a hyperbolic critical point. Our proof depends on the connected simple system associated with the Conley index for isolated invariant sets. Furthermore, we show how this change in connected simple systems can be associated with transition matrices, and hence, connection matrices. This leads to some simple examples in which the nonuniqueness of the connection matrix can be explained by changes in the connected simple system.

Algebraic transition matrices in the Conley index theory

“…In an attempt to better understand singular transition matrices and for the purpose of applications in which (4) provided little information, C. McCord and the second author developed a topological transition matrix [13]. The starting point for this theory is a refinement of the homotopy Conley index, known as a connected simple system.…”

“…The answer is yes and worth explaining since it justifies the need for the algebraic machinery we are attempting to develop. (5) was used in [13] to establish the existence of global bifurcations. This result, in turn, was exploited to prove the existence of specific travelling waves for a general family of predatorprey systems [25].…”

Directional transition matrix

“…In order to remove the artificial dependence on the slow parameter drift, McCord and Mischaikow [6] introduced the notion of topological transition matrix. The topological transition matrix can be defined only from the parametrized system at ε = 0, and detects the change of the topological nature of connecting orbits among Morse sets when the parameter varies from y = 0 to y = 1.…”

“…The topological transition matrix is lower triangular and shares the same property as the singular transition matrix, namely its off diagonal nonzero entry implies the existence of connecting orbits between appropriate Morse sets for various y ∈ (0, 1). See [6] for more details. Furthermore McCord and Mischaikow [7] showed the equivalence of the singular and topological transition matrices.…”

“…For this situation, either singular or topological transition matrix is not useful since they are both essentially unidirectional. In order to illustrate the difficulty, let us consider a variant of the connecting orbit problem studied in [6]. See Figure 1.…”