Connected simple systems, transition matrices, and heteroclinic bifurcations

Abstract

Given invariant sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and connecting orbits <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A right-arrow upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A \to B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B right-arrow upper C"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B \to C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we state very general conditions under which they bifurcate to produce an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A right-arrow upper C"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A \to C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.