Loss-of-hyperbolicity and jump bifurcations in scalar nonautonomous d-concave ODEs

Abstract

The paper studies bifurcation for additively parametrized scalar nonautonomous ODEs of the form $x'=f(t,x)+λ$, where $f$ is d-concave and coercive in the state variable. By formulating the problem through the hull of $f$ and the associated skewproduct flow, the autonomous notion of bifurcation can be extended in terms of loss of hyperbolicity of copies of the base. A fairly complete classification of the possible bifurcation diagrams is provided. The notion of a jump bifurcation, linked to abrupt changes in the global attractor, is introduced and related to the occurrence of critical transitions. Several nontrivial, genuinely nonautonomous examples are presented, illustrating the different bifurcation diagrams that may arise from the lack of unique ergodicity on the hull and showing how critical transitions can result from an underlying jump bifurcation point. The framework has potential applications in climate dynamics, ecology, circuits, optics, biology, and other models involving abrupt transitions.