Periodic-orbit theory of the blowout bifurcationYoshihiko Nagai*and Ying-Cheng Lai†Department of Physics and Astronomy, Kansas Institute for Theoretical and Computational Science, University of Kansas,Lawrence, Kansas 66045~Received 9 May 1997!This paper presents a theory for characterization of the blowout bifurcation by periodic orbits. Blowoutbifurcation in chaotic systems occurs when a chaotic attractor, lying in some symmetric invariant subspace,becomes transversely unstable. We present an analysis and numerical results that indicate that the bifurcationis mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embeddedin the chaotic attractor. There are two distinct groups of periodic orbits: one transversely stable and anothertransversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of thesetwo groups are balanced. Our results thus categorize the blowout bifurcation as a unique type of bifurcationthat involves an infinite number of periodic orbits, in contrast to most previously known bifurcations that aremediated by only a finite number of periodic orbits. @S1063-651X~97!08610-8#PACS number~s!: 05.45.1bI. INTRODUCTIONA central problem in the study of nonlinear dynamicalsystems is to understand how the asymptotic behavior altersas a system parameter changes. Qualitative changes in thesystem’s asymptotic behavior are called bifurcations,whereas the critical parameter values at which the bifurca-tions occur are the bifurcation points. The phenomenon ofbifurcation is extremely common in nonlinear systems. Forinstance, chaos typically arises from a nonchaotic statethrough a series of bifurcations and the number of bifurca-tions involved in the creation of chaos can be as a few as oneor can be as many as infinite. Understanding various types ofthe bifurcations has been one of the focuses in the study ofnonlinear physical systems @1#. Since almost all qualitativechanges in the system’s behavior are due to bifurcations, it isof paramount physical interest to characterize bifurcations interms of fundamental quantities of the system. ‘‘There isnothing more fundamental than to characterize a bifurcationin terms of the periodic orbits embedded in the natural dy-namics of the system.’’ Thus the knowledge of periodic or-bits is the key to understand the bifurcation and, conse-quently, the key to understand the dynamics of the system.Most known bifurcations in nonlinear dynamical systemsinvolve only a finite number of periodic orbits. Examplesinclude the period-doubling bifurcation @2# and the saddle-node bifurcation @1#. In a period-doubling bifurcation, astable periodic orbit of period p becomes unstable and simul-taneously a stable periodic orbit of period 2p is created at thebifurcation @2#. In a saddle-node bifurcation, a pair of peri-odic orbits, one stable and another unstable, is created as theparameter passes through the bifurcation point @1#. Other ex-amples of bifurcations include sudden catastrophic events inchaotic systems such as crises @3# and basin boundary meta-morphoses @4#, which are triggered by the collision of peri-odic orbits, usually of low period, embedded in differentdynamical invariant sets. More recently, an exotic type ofbasin structure was discovered in chaotic systems, that is, thebasin of Wada. Wada basin boundaries are common fractalboundaries of more than two basins of attraction. It wasshown in Ref. @5# that Wada basin boundaries are created bya saddle-node bifurcation on the basin boundary. A directionof intense recent investigation concerns bifurcation in dy-namical systems with one or several symmetric invariantsubspaces. In such systems, it was discovered that the rid-dling bifurcation, a bifurcation that leads to the creation ofriddled basins @6#, is triggered by the loss of the transversestability of some periodic orbit, typically of low period, em-bedded in the chaotic attractor in the invariant subspace @7#.A common feature of all these major bifurcations is thatthere are only one or a few periodic orbits involved.The main purpose of this paper is to present a periodic-orbit theory for a recently discovered bifurcation in chaoticsystems. This is the so-called blowout bifurcation that occursin systems with a simple type of symmetry ~see below for aprecise description!. ‘‘Our main conclusion is that the blow-out bifurcation is fundamentally different from most knownmajor bifurcations in that it involves an infinite number ofperiodic orbits.’’ We provide a quantitative characterizationof the blowout bifurcation in terms of periodic orbits. A shortaccount of this work has been published recently @8#.A fundamental requirement for the blowout bifurcation issymmetry. The existence of symmetry in the system’s equa-tions often leads to a low-dimensional invariant subspace inthe phase space. Denote the invariant subspace by S andassume there is a chaotic attractor in S. Since S is invariant,initial conditions in S generate trajectories that remain in Sforever. Trajectories off S, however, can either be attractedtowards S or be repelled away from it, depending on a sys-tem parameter. The transition from the former to the lattersituations is the blowout bifurcation @9#. Quantitatively, onecan define an infinitesimal vector in the subspace T that istransverse to S. The exponential growth rate of the vector isthe transverse Lyapunov exponent, denoted by LT. When*Electronic address: [email protected]†Also at Department of Mathematics, University of Kansas,Lawrence, KS 66045. Electronic address:[email protected] REVIEW E OCTOBER 1997VOLUME 56, NUMBER 4561063-651X/97/56~4!/4031~11!/$10.00 4031 © 1997 The American Physical SocietyLTis negative, S attracts nearby trajectories transversely andhence the chaotic attractor in S is also an attractor in the fullphase space. If LTis positive, trajectories in the neighbor-hood of S are repelled away from it and consequently theattractor in S is transversely unstable and it is hence not anattractor of the full system. Blowout bifurcation occurs whenLTchanges from negative to positive values. There are in-teresting physical phenomena associated with the blowoutbifurcation. For example, near the bifurcation point whereLTis slightly negative, if there are attractors off S in thephase space, then typically the basin of the chaotic attractorin S is riddled with arbitrarily small holes that