Elsevier

Physica D: Nonlinear Phenomena

Volume 268, 1 February 2014, Pages 100-105
Physica D: Nonlinear Phenomena

Computation of true chaotic orbits using cubic irrationals

https://doi.org/10.1016/j.physd.2013.11.003Get rights and content

Highlights

  • We propose a method for computing true orbits of 1D piecewise linear fractional maps.

  • The method uses cubic irrationals as numbers and involves only integer arithmetic.

  • True orbits generated by the method display the same properties as typical orbits.

  • We can simulate maps whose simulation has been difficult, such as the Bernoulli map.

Abstract

We introduce a method that enables us to generate long true orbits of discrete-time dynamical systems defined by one-dimensional piecewise linear fractional maps with integer coefficients. The method uses cubic irrationals to represent numbers and involves only integer arithmetic to compute true orbits. By applying the method to the Bernoulli map and a modified Bernoulli map, we show that it successfully generates true chaotic and intermittent orbits, respectively, in contrast with conventional simulation methods. We demonstrate through simulations concerning invariant measures and the power spectrum that the statistical properties of the true orbits generated agree well with those of typical orbits of the two maps.

Introduction

Chaotic phenomena occur in diverse areas of nonlinear science, and simulations using digital computers are essential for studying these phenomena  [1], [2], [3]. The accuracy of simulations is an important element in determining the validity of such studies. However, the difficulty of generating true (exact) chaotic orbits using digital computers is also well known. This difficulty involves the following factors when discrete-time dynamical systems are considered.

If we use a representation of numbers with fixed precision (i.e., a representation where the total number of bits used to represent the numbers is fixed), such as the double-precision floating-point numbers that are usually used in numerical computations, then it is impossible to generate true chaotic orbits due to inevitable round-off errors. In addition, because of the fixed bit length, all orbits become eventually periodic in principle. Accordingly, we can say that number representations with fixed precision are, in the first place, incompatible with the fundamental properties of chaos: sensitive dependence on initial conditions and aperiodicity. In fact, the validity of simulations of chaotic phenomena has been a matter of serious concern for decades (see, e.g., Refs.  [4], [5], [6], [7]). There is indeed a view (see, e.g., Refs.  [4], [5]) that the qualitative or statistical properties of a system are preserved, even if one performs discretization using a number representation with fixed precision, especially when the total number of possible discrete values is sufficiently large. However, it is also known that in some cases the use of fixed-precision number representations can destroy even the system’s macroscopic, qualitative properties, as exemplified in spatial bifurcation phenomena in open flow systems [8], [9].

For the next standard number representation other than that of fixed-precision numbers, we can cite the fractional representation of numbers. A digital computer can perform integer arithmetic, and thus the four arithmetic operations on rational numbers, without errors. Therefore, if we consider a dynamical system described by a rational map with integer coefficients and choose rational numbers as a number representation, we can generate errorless true orbits of the map in principle (the initial value(s) are to be chosen from the rational numbers). However, this scheme is also known to present two difficulties.

  • 1.

    For a general rational map with integer coefficients, the representation of a point on a true orbit grows huge very rapidly. In fact, the numerator(s) and denominator(s) of the coordinate(s) of the point grow superexponentially, making the cost of the true orbit computation extremely high  [10]. For example, even with the most powerful computers available, only the first few tens of time steps of the true orbit can be generated for the logistic map with generic rational parameters and initial values.

  • 2.

    In the case of piecewise linear or linear fractional maps [11], [12], the cost of computing true orbits is relatively low. However, there is still a danger of generating only atypical ones  [2], [13]. For example, for the Bernoulli map (i.e., the 2x modulo 1 map), the orbit from the initial value given by a rational number becomes eventually periodic, which makes the Bernoulli map a well-known map that is difficult to simulate. The fact that true orbits are not necessarily typical constitutes another difficulty with true orbit computation because it is usually intended for simulations to reproduce the typical behavior of a target system.

In the above discussion, we restricted the numbers used for the true orbit generation of a rational map to the rational numbers, but we can also consider expanding them to algebraic numbers (the coefficients of a rational map can also be expanded to algebraic numbers). Even if we expand numbers in this manner, it is still true as in the case of rational numbers that, for a general rational map, the computational cost necessary for generating true orbits is extremely high. However, mathematically, such dynamics over algebraic sets has been extensively investigated in the field of arithmetic dynamics. (See, e.g., Ref.  [10] for an introduction to this developing field of research.) In particular, a related study was conducted on the evolution of (roots of) polynomials under a general rational map  [14]. The method provided in Ref.  [14] is theoretical but of high generality. For example, it can take arbitrary algebraic numbers as the coefficients of both a rational map and a polynomial. In our study, we will focus on piecewise linear fractional maps (including piecewise linear maps), for which we can expect to generate long true orbits by using integer arithmetic. More specifically, we will deal with one-dimensional piecewise linear fractional maps with integer coefficients, whose eventually periodic points are usually rational numbers and quadratic irrationals (i.e., the roots of irreducible quadratic polynomials with integer coefficients). Consequently, if we use algebraic numbers whose degree is greater than 2, we can usually guarantee the aperiodicity of the generated true orbits.

In this paper, by using real cubic irrationals having complex conjugates as numbers, we propose a true orbit generation method involving only integer arithmetic and allowing exact simulations of discrete-time dynamical systems defined by one-dimensional piecewise linear fractional maps with integer coefficients. Moreover, we simulate examples of the Bernoulli map and a modified Bernoulli map and demonstrate that long true orbits obtained by this method display the same properties as typical orbits. We also confirm at most linear growth of the number representation, which is a distinctive characteristic of our method of computation.

Before we proceed to the main subject, it is worth pointing out the relation of our method to the issue of shadowing. Shadowability has been focused on as a requirement for ensuring the validity of computer-generated (pseudo-)orbits  [15]. For hyperbolic dynamical systems, the shadowing lemma assures the existence of true orbits that trace pseudo-orbits for arbitrarily long times [16], [17]. Typical physical systems, however, are considered nonhyperbolic, and for this reason researchers have been concerned with the question of to what extent (e.g., for how long) pseudo-orbits remain traced by true orbits in the case of nonhyperbolic systems (see, e.g., Refs.  [18], [19], [20]). In contrast, the method that we report here can directly generate true orbits regardless of whether a system is hyperbolic or not. Consequently, unlike before, there is no need with our method to verify the shadowability of generated orbits.

Section snippets

Preliminaries

Let us consider a cubic equation, px3+qx2+rx+s=0, where the coefficients p(0), q, r, s are integers and the polynomial px3+qx2+rx+s is irreducible. We suppose that Eq. (1) has a single real root α. In other words, α is a real cubic irrational having complex conjugates. We denote by S the set of such real cubic irrationals having complex conjugates. A number αS satisfying Eq. (1) is denoted by a vector (p,q,r,s)Z4. Notice that this representation of α is not unique, e.g.,  (1,1,1,1)=(2,2,2

Simulations

In the following, we report results concerning the application of our true orbit generation method to two maps: the Bernoulli map and a modified Bernoulli map. We also explain why conventional simulation methods have serious difficulties in generating orbits of these maps.

Conclusions

By using real cubic irrationals having complex conjugates and integer arithmetic, we have introduced a true orbit generation method enabling exact simulations of discrete-time dynamical systems defined by one-dimensional piecewise linear fractional maps with integer coefficients. We have demonstrated that the true orbits generated by our method show clear chaotic and intermittent behaviors for the Bernoulli map and modified Bernoulli map, respectively, in contrast with the case for conventional

Acknowledgments

We thank E. Brooks-Pollock, R.S. MacKay, J. Tamura, and S. Yasutomi for their suggestions. This research was supported by the JST PRESTO program and by a Grant-in-Aid for Young Scientists(B) 21700256 from MEXT of Japan.

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    1

    Present address: Toho University, 2-2-1 Miyama, Funabashi, Chiba 274-8510, Japan.

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