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Strong Chaos Without Butter Y E Ect in Dynamical Systems with Feedback

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Strong Chaos Without Butter Y E Ect in Dynamical Systems with Feedback Strong Chaos without Buttery Eect in Dynamical Systems with Feedback 1 2 3 Guido Boetta Giovanni Paladin and Angelo Vulpiani 1 Istituto di Fisica Generale Universita di Torino Via PGiuria I Torino Italy 2 Dipartimento di Fisica Universita del lAquila Via Vetoio I Coppito LAquila Italy 3 Dipartimento di Fisica Universita di Roma La Sapienza Ple AMoro I Roma Italy ABSTRACT We discuss the predictability of a conservative system that drives a chaotic system with p ositive maximum Lyapunov exp onent such as the erratic motion of an asteroid 0 in the gravitational eld of two b o dies of much larger mass We consider the case where in absence of feedback restricted mo del the driving system is regular and completely predictable A small feedback of strength still allows a go o d forecasting in the driving system up to a very long time T where dep ends on the details of the system p The most interesting situation happ ens when the Lyapunov exp onent of the total system is strongly chaotic with practically indep endent of Therefore an exp onential tot 0 amplication of a small incertitude on the initial conditions in the driving system for any co exists with very long predictability times The paradox stems from saturation eects in the evolution for the growth of the incertitude as illustrated in a simple mo del of coupled maps and in a system of three p oint vortices in a disk PACS NUMBERS b It is commonly b elieved that a sensible dep endence on initial condition makes foreca sting imp ossible even in systems with few degrees of freedom This is the socalled buttery eect discovered by Lorenz in a numerical simulation of a mo del of convection with three degrees of freedom Using his words A buttery moving its wings over Brazil might cause the formation of a tornado in Texas In general a dynamical system is considered chaotic when there is an exp onential amplication of an innitesimal p erturbation on the initial conditions with a mean 0 time rate given by the inverse of the maximum Lyapunov exp onent Indeed such a 1 deterministic system is exp ected to b e predictable on times t and to b ehave like a random system on larger times The purp ose of this letter is to show that there exists a wide class of dynamical systems where a large value of the Lyapunov exp onent do es not imply a short predictability time on a physically relevant part of the system In this resp ect one can sp eak of strong chaos without buttery eect In particular we discuss the predictability of a conservative system that drives a stron gly chaotic system with p ositive maximum Lyapunov exp onent In absence of feedback 0 the driving system is regular and completely predictable A small feedback of strength still allows to predict the future of the driving system up to a very long predictability time T that diverges with The Lyapunov exp onent of the total system is and so p tot 0 there is a regime of strong chaos for all values The absence of the buttery eect stems from saturation eects in the evolution laws for the growth of an incertitude on the driven system To b e explicit let us consider a system with evolution given by two sets of equations d F h a dt d G b dt n m with R and R The variables are thus driven by a subsystem represented by the variables with a weak feedback of order A typical physical example is given by an asteroid moving in the gravitational eld generated by two celestial b o dies of much larger mass such as Jupiter and Sun Usually the feedback is neglected and one considers the restricted three b o dy problem ie the variables passively driven However a ner description should take into account even the inuence of the asteroid on the evolution of the other two b o dies ie an active driving where This situation app ears in many other phenomena such as the active advection of a contaminant in a uid a simple example which will b e discussed in this letter The main prop erties of the system in absence of feedback are the following the driver is an indep endent dynamical system that exhibits a regular evolution with zero Lyapunov exp onent the driven subsystem is chaotic with p ositive maximum Lyapunov exp onent say 0 In other terms the b ehavior of the driver variables is completely predictable However as so on as one should consider the total system which is obviously chaotic with a Lyapunov exp onent that a part small correction of order is given by the tot Lyapunov exp onent of the chaotic driven subsystem This means that there is an 0 exp onential amplication of a small incertitude on the knowledge of the initial conditions even in the driver This is an amazing result since it is natural to exp ect that it is p ossible to forecast the b ehavior of the driver for very long times as Actually intuition is correct while the Lyapunov analysis gives completely wrong hints on the predictability problem at dierence with what commonly b elieved The famous buttery eect of Lorenz seems not to forbid the p ossibility of predicting the future of a part of the system The paradox stems from saturation eects in the evolution for the growth of the incertitude To x notation and denitions let us consider the evolution of the total system dx i n+m f x where x R i dt 0 0 The incertitude on its state is t xt x t where x and x are tra jectories starting 0 from close initial conditions ie jx x j In the limit can b e confused 0 0 with the the tangent vector z whose evolution equations are df dz i J t z where J t ik dt dx j x(t) The maximum Lyapunov exp onent is then dened as the exp onential rate of the incertitude growth jtj lim lim lim ln ln jztj t!1 t!1 !0 t t 0 0 It is worth stressing that the full equations for the evolution of an incertitude are nonlinear d 2 J t O dt so that the two limits in cannot b e interchanged The predictability of the system is dened in terms of the allowed maximal ignorance on the state of the system a tolerance parameter which must b e xed according the max necessities of the observer The predictability time is thus 0 0 T sup ft such that jt j for t tg p max t If is well approximated by and the predictability time can b e roughly max identied with the inverse Lyapunov exp onent since max ln T p 0 and the dep endence on the initial error and on the tolerance parameter is only 0 max logarithmic and can b e safely ignored for many practical purp oses ( ) Supp ose now to b e interested only on the incertitude in the driver system When the Lyapunov exp onent of the driver so that it is fully predictable we ( ) have in general T the exp onent dep ending on the particular system while p 0 the driven system has a Lyapunov exp onent However for any the two 0 subsystems are coupled and the global Lyapunov exp onent is exp ected to b e O 0 A direct application of would give ( ) T T p p 0 ( ) ( ) One thus obtains a singular limit lim T T The troubles stem from p p !0 the identication b etween incertitude t and tangent vector zt which is not correct on long time scales It is convenient to illustrate the problem in a simpler context as the main qualitative asp ects of equations a and b can b e repro duced considering only the feedback eect in two coupled maps of the type L h a t+1 t t G b t+1 t 2 where the time t is an integer variable R h h h is a vector function of the 1 2 1 variable R whose evolution is ruled by a chaotic onedimensional map G and L is the linear op erator corresp onding to a rotation of an arbitrary angle cos sin L sin cos When one is left with two indep endent systems one of them regular and of Hamil tonian type the other fully chaotic These maps provide a simple maybe the simplest example of a system with two dierent temp oral regimes A short times where exp t so that it is correct to ignore the nonlinear term in 0 0 so that z B long times where one should consider the full nonlinear equation for the incertitude growth From the observer p oint of view b oth these regimes might b e interesting according his particular exigence If one is interested in forecasting the very ne details of the systems 1 the tolerance threshold could b e quite small hence T In general however a max p 0 system is considered unpredictable when the incertitude is rather large say discrimination b etween sunrain in meteorology and regime B is the relevant one In that case non linear eects in cannot b e neglected and in order to give an analytic estimate of the predictability time we can use a sto chastic mo del of the deterministic equations Indeed the chaotic feedback on the evolution of the driver system can b e simulated by a random vector w ie L w t+1 t t ( ) 0 The incertitude is then given by the dierence of two tra jectories and originated t t t by nearby initial conditions and evolves according the sto chastic map ( ) ( ) L W t t t+1 1 0 where W w w For short times t j ln j one cannot consider the noises w t t 0 t t 0 0 and w as uncorrelated so that the incertitude on the driver grows exp onentially under t ( ) the inuence on the deterministic chaos given by the feedback j j j j exp t t t 1 with O For long times t j ln j the random variables are practically 0 0 0 uncorrelated so that their dierence W still acts as a noisy term As a consequence the t growth of the incertitude is diusive since the formal solution of is t X ( ) t L W L 0 t =0 t and noting that L is a unitary transformation
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