The non-smooth pitchfork bifurcation: a renormalization analysis

Abstract

We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions, self-similar behaviour of the attractor on and near the bifurcation curve can be observed, which corresponds to a periodic orbit of an underlying renormalization operator. We examine the scaling properties for various parameter choices including the so-called pitchfork critical point. Finally, we study the autocorrelation function for the system and show that it is equivalent to that present in symmetric barrier billiards.

Renormalization of correlations in a quasiperiodically forced two-level system for a general class of modulation function

“…The work also provides a link between the two-level system and other quasi-periodically forced systems. We have shown that the renormalization strange set occurring for symmetric barrier billiards can be reproduced in the two level system, and in [1] we found the same set occurring in the study of correlations of strange non-chaotic attractors, thus making it a somewhat universal set in the study of correlations for this class of system. This work can be trivially generalised to a class of quadratic irrational frequency as has been done in the past [15], although the problem of generalising to general irrational frequencies remains challenging.…”

“…In future works we hope to be able to produce an analytic proof of the chaoticity of the correlations for symmetric barrier billiards (which lie on the set shown in Figure 4) by conjugating the renormalization operator to a sub-shift of finite type. If successful, this will naturally carry over to the work presented here for the two-level system and Figure 15: Renormalization strange set which appears in the limit κ → 0. the work presented in [1] for the correlations of SNAs.…”

“…In [9] the so-called "blow-out birth of an SNA" is studied for the inverse golden mean frequency, and it is shown that at and near the critical boundary the attractor is self-similar following scaling laws determined by the renormalization equations. We have extended this study in [1] to the more general non-smooth pitchfork bifurcation. In [11] the renormalization of correlations of the original SNA first discovered in [7] are studied.…”

Dimensions of structurally stable and critical strange non-chaotic attractors

“…The models under consideration are 'pinched skew-products' (a term used by Glendinning in [6]), and undergo a transition to SNA at a critical parameter value via the non-smooth pitchfork bifurcation which we have previously studied in [1]. The work in [1] was motivated by [14], in which the bifurcation was labelled 'the blow-out birth' of an SNA.…”

“…The models under consideration are 'pinched skew-products' (a term used by Glendinning in [6]), and undergo a transition to SNA at a critical parameter value via the non-smooth pitchfork bifurcation which we have previously studied in [1]. The work in [1] was motivated by [14], in which the bifurcation was labelled 'the blow-out birth' of an SNA. In both papers a renormalization approach is presented and scaling properties of the attractor both at and near the critical point of transition are determined showing its self-similarity for certain choices of initial phase.…”