Non-smooth saddle-node bifurcations III: Strange attractors in continuous time

Abstract

Non-smooth saddle-node bifurcations give rise to minimal sets of interesting geometry built of so-called strange non-chaotic attractors. We show that certain families of quasiperiodically driven logistic differential equations undergo a non-smooth bifurcation. By a previous result on the occurrence of non-smooth bifurcations in forced discrete time dynamical systems, this yields that within the class of families of quasiperiodically driven differential equations, non-smooth saddle-node bifurcations occur in a set with non-empty C 2 -interior. † actions of linear cocycles [7,32,33,35,44]. 1 A natural setting for the creation of SNA's are non-smooth saddle-node bifurcations of oneparameter families of driven one-dimensional systems (see Section 1.2). Here, they occur as the outcome of the collision of two continuous invariant curves. The present work shows that the property of undergoing such a non-smooth bifurcation has-analogously to the discrete time case-non-empty interior in the C 2 -topology in the class of qpf families of one-dimensional ode's (see Theorem 2.2).The proof of this-in a sense-abstract fact has a consequence which is important from the applied point of view introduced above: our core idea is to consider the logistic differential equation with quasiperiodic additive forcing and reduce its dynamics-by means of a suitable Poincaré section-to those of qpf maps that verify the assumptions of Theorem 1.11, that is, to maps for which there exists an SNA. Now, an easy argument shows that the respective reduced system possesses an SNA if and only if the original system does (see Section 3). While the main work thus happens to be the rather technical analysis of the qpf logistic ode (carried out in Section 4), the robustness of non-smooth bifurcations in the continuous time case comes as a by-product of the application of Theorem 1.11. Furthermore, with little extra effort, we can carry over the geometric findings from the discrete time setting in [15] to the present situation (see Theorem 2.3).Our main results are contained in Section 2. Their proofs can be found in the last section of this article. In the remainder of the current section, we introduce some basic notation and review some facts and definitions from non-autonomous bifurcation theory, fractal geometry and the discrete time analogue to what we consider in this work.

Rate-Induced Tipping and Saddle-Node Bifurcation for Quadratic Differential Equations with Nonautonomous Asymptotic Dynamics

“…The bifurcation occurs when the set of bounded solutions is empty at one side of a certain value of the parameter and contains a classical attractor-repeller pair on the other side. Detailed descriptions of this situation for some cases of nonautonomous quadratic differential equations are given in [20] and [13], where the possibility of occurrence of strange nonchaotic attractors is carefully analyzed.…”

On the effect of forcing of fold bifurcations and early-warning signals in population dynamics

“…In addition, the simulations in Figures 2(b) and Figure 4(e)-(h) provide similar numerical evidence for the existence of nonsmooth bifurcations in the qpf Allee model (2) and (3) below. These findings are backed up by rigorous results in [Fuh14,Fuh16], showing that non-smooth fold bifurcations occur for open sets of parameter families of quasiperiodically forced scalar ODE's. They can therefore be robust and persistant under small perturbations of the system.…”

“…(19) In fact, (19) a priori only yields a local flow where trajectories may diverge and hence not be defined for all times t ∈ R. As we will only deal with bounded solutions (see also Lemma 4.5), this issue is not of further importance. We refer the interested reader to [Fuh16] for more details. Now, in order to apply the above statements to flows defined by equations of the form (19), it is crucial that the validity of the assumptions can be read off directly from the differential equations.…”

“…Finally, the convexity of the fibre maps required in (vi) is a consequence of the concavity of F β in the considered region. We refer to [AJ12,Fuh16] for further details, as well as to the discussion of the application to the forced Allee model in Section 2.3.…”

Li–Yorke chaos in nonautonomous Hopf bifurcation patterns—I

“…The choice of each one of these categories defines a different research line, and all of them have shown their relevance. The works of Braaksma et al [9], Alonso and Obaya [2], Johnson and Mantellini [24], Fabbri et al [14], Langa et al [29,30], Rasmussen [43,44], Núñez and Obaya [37], Pötzsche [41,42], Anagnostopoulou and Jäger [3], Fuhrmann [16], and Caraballo et al [12], develop some of these lines, providing nonautonomous transcritical, saddle-node and pitchfork bifurcation patterns. In some cases these nonautonomous phenomena admit a dynamical description analogous to that of the autonomous case.…”