Progress of Theoretical Physics, Vol. 71, No.1, January 1984 Fractal Basin Structure Shinji T AKESUE and Kunihiko KANEKO Department of Physics, University of Tokyo, Tokyo 113 (Received August 1, 1983) The basin structure of a class of one-dimensional mappings is studied. In a model proposed by the present authors, the basin of attraction is separated into uncountable pieces of open intervals and has a kind of self-similarity. Symbolic sequences are used to elucidate the structure. The way of breakdown of the structure due to a noise is studied on the basis of the analysis using symbolic sequences. The effect of noise on a crisis is discussed. § 1. Introduction 35 Revealing novel and essential aspects in nonlinear physics is one of the most impor- tant problems in our age_ The recent advance in this field has thrown light on various phenomena, such as the period-doubling, intermittency and collapse of torL There remains, however, a fundamental problem, characteristic of nonlinear phenomena. That is the analysis of basin structure in multibasin systems. 1)-11) More than. one attractors can coexist in a lot of nonlinear systems. Typical and simple examples appear in certain one-dimensional mappings 2 )-S),*) and in two-dimensional mappings: 6 )-Il) Mandelbrof) has shown that the basin boundary of a certain analytic map on the complex plane is fractal, which has attracted many authors and has been investigated. 9 ),10) On the other hand, one of the present authors (K. K.)Il) has found the "self-similar basin structure" in a two-dimensional map and has related it to the stretching at the periodic .saddle. In the present paper, we focus our attention mainly on the basin structure itself, not on the basin boundary. / In §2, "fractal basin structures" are revealed by using a class of one-dimensional mappings. The structure is caused by a topological chaos 12 ) present between coexisting attractors. It has infinitesimal structures and has a kind of scaling property .. We analyze the structure utilizing symbolic sequences, i.e., so-called "itineraries" .13) Each unit of the basin structure corresponds. to a symbolic sequence and the characteristic features of the fractal basin structures are understood by the itineraries. In §3, the effect of noise on the structure is analyzed. We introduce the notion of probability into the basin of attraction. The probability to go to one attractor is calculat- ed on the basis of the itinerary analysis. The small structure is destroyed by a small noise, while the large one survives. The scaling between the size of the structure and the strength of the noise is represented by a "stability number", which is given through the above analysis. In that section we make use of some properties of linear stochatic difference equations, shown in the Appendix. Section 4 is devoted to the critical behavior near the crisis. 14 ) The effect of noise on the lifetime of the chaotic transiene S ) is calculated. *) Multibasin can appear in a one-dimensional map on an interval with multi-humps or positive Schwarzian derivative. Progress of Theoretical Physics, Vol. 71, No.1, January 1984 Fractal Basin Structure Shinji T AKESUE and Kunihiko KANEKO Department of Physics, University of Tokyo, Tokyo 113 (Received August 1, 1983) The basin structure of a class of one-dimensional mappings is studied. In a model proposed by the present authors, the basin of attraction is separated into uncountable pieces of open intervals and has a kind of self-similarity. Symbolic sequences are used to elucidate the structure. The way of breakdown of the structure due to a noise is studied on the basis of the analysis using symbolic sequences. The effect of noise on a crisis is discussed. § 1. Introduction 35 Revealing novel and essential aspects in nonlinear physics is one of the most impor- tant problems in our age_ The recent advance in this field has thrown light on various phenomena, such as the period-doubling, intermittency and collapse of torL There remains, however, a fundamental problem, characteristic of nonlinear phenomena. That is the analysis of basin structure in multibasin systems. 1)-11) More than. one attractors can coexist in a lot of nonlinear systems. Typical and simple examples appear in certain one-dimensional mappings 2 )-S),*) and in two-dimensional mappings: 6 )-Il) Mandelbrof) has shown that the basin boundary of a certain analytic map on the complex plane is fractal, which has attracted many authors and has been investigated. 9 ),10) On the other hand, one of the present authors (K. K.)Il) has found the "self-similar basin structure" in a two-dimensional map and has related it to the stretching at the periodic .saddle. In the present paper, we focus our attention mainly on the basin structure itself, not on the basin boundary. / In §2, "fractal basin structures" are revealed by using a class of one-dimensional mappings. The structure is caused by a topological chaos 12 ) present between coexisting attractors. It has infinitesimal structures and has a kind of scaling property .. We analyze the structure utilizing symbolic sequences, i.e., so-called "itineraries" .13) Each unit of the basin structure corresponds. to a symbolic sequence and the characteristic features of the fractal basin structures are understood by the itineraries. In §3, the effect of noise on the structure is analyzed. We introduce the notion of probability into the basin of attraction. The probability to go to one attractor is calculat- ed on the basis of the itinerary analysis. The small structure is destroyed by a small noise, while the large one survives. The scaling between the size of the structure and the strength of the noise is represented by a "stability number", which is given through the above analysis. In that section we make use of some properties of linear stochatic difference equations, shown in the Appendix. Section 4 is devoted to the critical behavior near the crisis. 14 ) The effect of noise on the lifetime of the chaotic transiene S ) is calculated. *) Multibasin can appear in a one-dimensional map on an interval with multi-humps or positive Schwarzian derivative. Downloaded from https://academic.oup.com/ptp/article/71/1/35/1851018 by guest on 28 June 2022