On the Unfolding of a Blowout Bifurcation

Suppose a chaotic attractor A in an invariant subspace loses stability on varying a parameter At the point of loss of stability the most positive Lya punov exponent of the natural measure on A crosses zero at what has been called a blowout bifurcation We introduce the notion of an essential basin of an attractor with an invari ant measure This is the set of points such that the set of measures de ned by the sequence of measures n Pn k fk x has an accumulation point in the support of We characterise supercritical and subcritical scenarios according to whether the Lebesgue measure of the essential basin of A is positive or zero We study a drift di usion model and a model class of piecewise linear map pings of the plane In the supercritical case we nd examples where a Lya punov exponent of the branch of attractors may be positive hyperchaos or negative depending purely on the dynamics far from the invariant subspace For the mappings we nd asymptotically linear scaling of Lyapunov exponents average distance from the subspace and basin size on varying a parameter We conjecture that these are general characteristics of blowout bifurcations Introduction and background There has been a lot of recent interest in the global dynamics of systems with at tractors in invariant subspaces with the recognition that the structure of basins of attraction may be very complicated riddled basins The structure and proper ties of the basins including bifurcations may be understood by examining Lyapunov exponents normal to the invariant subspace Some relevant work is listed in the bib liography but we mention speci cally In this paper we investigate blowout bifurcations of chaotic attractors in invariant subspaces and in particular on o intermittency The term blowout was coined by Ott Sommerer in although the phenomenon was earlier recognised by Pikovsky and Yamada and Fujisaka There are two types of blowout subcritical or hard bifurcation and supercritical or soft bifurcation Although we believe that much of what we nd will be found in dynamical systems satisfying rather general conditions our analytical results centre on model piecewise Department of Mathematical and Computing Sciences University of Surrey Guildford GU XH UK Department of Mathematics UMIST Manchester M QD UK linear planar maps and drift di usion systems In particular we investigate the dy namical and ergodic properties of nearby invariant sets in a class of such models We nd that the Lebesgue measure of the basin subcritical case the normal Lyapunov exponents and the average distance from the invariant subspace supercritical case all scale linearly near the blowout We give examples where the supercritical blowout leads to blownout attractors with either positive or negative normal Lyapunov exponents thus one can get blowout bifurcations that lead to hyperchaos and ones that do not This helps to unify the observations of and who nd hyperchaos with those of Lai who does not The presence or absence of hyperchaos is found to depend only on the dynamics far from the invariant subspace In order to characterise the sub and supercritical bifurcations we introduce the notion of an essential basin of attraction Suppose we have a compact invariant set A for a mapping f of R The essential basin of A Bess A is the set of points which visit each neighbourhood of A not only in nitely often but also in nitely often with a nonvanishing frequency of return To be more precise x Bess A if for each neighbourhood U A lim sup n n fk f x U k ng For the model system we can characterise a blowout as being supercritical if its essential basin has full measure in some neighbourhood of the attractor A at blowout and subcritical if its essential basin has zero measure at blowout In the general case this amounts to a conjecture The paper is organised as follows in Section we discuss some generalities of the dynamics at blowout bifurcation and introduce the essential basin of attraction We discuss the behaviour of the Lyapunov exponents on continuous branches of attract ors In Section we consider a family of maps of the plane leaving the x axis invariant where we can completely understand the dynamics and scaling behaviours In fact we may model the dynamics as a biased random walk on a discrete set of lines or strips Using properties of the random walk model We investigate the system at blowout in detail in particular the properties of the basin and essential basin of attraction of A In the supercritical case we nd that the average distance of a generic point from the invariant subspace scales linearly with the bifurcation parameter when the system exhibits on o intermittency The blownout attractors may be viewed as belonging to a continuous branch of attractors bifurcating from the invariant subspace attractor We nd scenarios where the blowout leads to hyperchaos and scenarios where it does not We recover results of on the distribution of laminar phases in on o intermittency and in the subcritical case before blowout we show that the measure of the basin of attraction also scales linearly with the bifurcation para meter Finally in Section we address further problems and generalisations of this ap proach Essential basins and branching of chaotic at tractors Ott and Sommerer rst noted that a system near a blowout bifurcation char acterised by a normal Lyapunov exponent passing through zero may exhibit on o intermittency riddled basins and a variety of related behaviours They conjectured the existence of two scenarios A hysteretic subcritical scenario where riddled basins before the blowout give rise to a hard loss of stability Pikovsky In other words after blowout almost all points near the invariant subspace eventually move away never to return A non hysteretic supercritical scenario giving a soft loss of stability to an on o intermittent attractor The characteristic feature of this scenario is that trajectories return close to the invariant subspace and remain nearby for long periods of time In it was shown that before blowout bifurcation both scenarios display locally riddled basins in any neighbourhood of the attractor there is a positive measure set of points whose trajectories iterate to a uniform distance from the attractor These trajectories are either re injected to the neighbourhood or are lost to the basin of another attractor For the model in Section we show that we can distinguish between sub and supercritical blowout depending on whether the essential basin of the attractor in the invariant subspace has positive measure or not at the point of blowout Suppose f M M is a locally invertible smooth map on some compact open region M R and M is Lebesgue measure on M For a compact set B M de ne M B f is an f invariant measure with supp B and B g and equip this with the C topology Let x denote the set of accumulation points of ffn x g in M and let x denote the point measure at the point x M We let x be the set of accumulation points of the sequence of measures