Period 2k Research Articles

Scaling and fine structure of superstable periodic orbits in the logistic map

The logistic map, defined by the recurrence xn+1 = αxn(1 − xn), has been a major tool in the emergence of chaos theory. When the parameter α is varied from 0 to 4, various bifurcations occur, with cycles of period 2, 4, 8 and so on, appearing successively, till cycles of period 2k are all unstable and chaos emerges. Even after chaos has emerged, stable periodic cycles of period p can appear in some parameter α windows, which include one superstable cycle, characterized by a zero first derivative. In this paper, all superstable values of α are obtained numerically for period p up to 21, and the two smallest superstable values are established for p values up to 1000, plus p = 1024 and p = 2048. These superstable parameters exhibit discrete fine structures, often different for even and odd p, combined with global and local scaling properties. The number of superstable orbits increases like 2p−1/p for large values of p, with a distribution of values approximately following a gamma distribution with mean αΜ = 4–0.0753 + 0.0026p. Our solution for the largest superstable value of α at a period p, written α = 4−εp, follows Lorenz scaling prediction εp = π2/4p, to an accuracy better than 10−3 %. The next superstable values α = 4−βp,j follow the (1 + 2j)2 multiplier to εp predicted by Simó and Tatjer up to values of j increasing with p. The width of the superstable orbits, the xn domain around initial point 1/2 with periodicity p, determined numerically, obeys a scaling law β/2p, for β sufficiently small. For p larger than 15, the periodicity of the calculated time-series sometimes displays numerical instabilities. Thus, the numerical logistic map differs from the mathematical ideal, which invites for caution, but it also provides a generic imperfect model closer to reality and experimental data when investigating the role of higher order periodic orbits as underlying structures of chaotic behaviours.

Eventually Periodic Solutions of a Max-Type System of Difference Equations of Higher Order

We study in this paper the following max-type system of difference equations of higher order: xn=max{A,yn-k/xn-1} and yn=max{B,xn-k/yn-1}, n∈{0,1,2,…}, where A≥B>0, k≥1, and the initial conditions x-k,y-k,x-k+1,y-k+1,…,x-1,y-1∈(0,+∞). We show that (1) if AB>1, then every solution of the above system is periodic with period 2 eventually. (2) If AB=1>B, then every solution of the above system is periodic with period 2k or 2 eventually. (3) If A=B=1 or AB<1, then the above system has a solution which is not periodic eventually.

TOWARDS SOME GENERAL ECOLOGICAL PRINCIPLES

Discrete deterministic age-structured, stage-structured and difference delay equation population models are analysed and compared with respect to stability and nonstationary behaviour. All three models show that species with iteroparous life histories tend to be more stable than species with semelparous life histories which allow us to conclude that this must be a fairly general ecological principle. Considering iteroparity, the precocious case appears to be more stable than the delayed case. The nonstationary dynamics shows a great deal of resemblance too, but when the number of age classes are even there is a mismatch between the dynamical outcomes of the age- and stage-structured case whenever the survival probabilities are large or moderate. Regarding semelparous species the analysis of the age-structured and the difference delay equation model clearly suggest that precocious semelparous species are more stable than delayed semelparous species and, moreover, that the transfer from stability to instability goes through a Hopf bifurcation. This is in great contrast to the outcome of the stage-structured model. In this case we find that the delayed case is more stable than the precocious and in unstable parameter regions there are orbits of period 2k, k > 1, which we do not find when the life history is precocious.

Periodic Character of a Class of Difference Equation

In this note, we prove that every positive solution of the difference equationwhere and the initial conditions are positive real numbers, converges to a periodic solution of period 2k. This result confirms conjectures 11.4.10, 11.4.16 (a) and 11.4.17 in Kulenović and Ladas (2001) Dynamics of Second Order Rational Difference Equations (Chapman and Hall/CRC).

Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems

We investigate cascades of isochronous pitchfork bifurcations of straight-line librating orbits in some two-dimensional Hamiltonian systems with mixed phase space. We show that the new bifurcated orbits, which are responsible for the onset of chaos, are given analytically by the periodic solutions of the Lam\'e equation as classified in 1940 by Ince. In Hamiltonians with C_${2v}$ symmetry, they occur alternatingly as Lam\'e functions of period 2K and 4K, respectively, where 4K is the period of the Jacobi elliptic function appearing in the Lam\'e equation. We also show that the two pairs of orbits created at period-doubling bifurcations of touch-and-go type are given by two different linear combinations of algebraic Lam\'e functions with period 8K.

Spatio-temporal sub-harmonic cascade in a non-linear interferometer with a liquid-crystal light valve

A bounded spatio-temporal period-doubling sequence is shown to occur in the Ahkmanov et al. non-linear interferometer. This happens when the pattern is a rotating 2k-armed spiral. First, all arms are identical and the period is the time required for any arm to rotate by the angle 2 pi /2k. As the control parameter increases, the time trace at some location on the pattern displays period doubling as the arms develop progressively distinct spatial modulations. Period 2 behaviour occurs when the nearest patterns similar to arm 'i' are arms 'i+or-2', period 4 behaviour occurs when arm 'i' has arms 'i+or-4' as its nearest similar neighbours, and so on until period 2k when all arms are distinct and the pattern must complete a full revolution for the time series to repeat. Such a cascade is shown to be qualitatively modelled by a mapping.

Examples of expanding maps with some special properties

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

Chains and fixing blocks in irreducible binary sequences

A fixing block in an irreducible word is a square, segment of the form XX, and the period is the length of X. An occurrence of a block C in an irreducible word, W = XCY, is called a chain if XA having an infinite irreducible extension implies A is an initial segment of C. The length of such a chain is the length of C. It is shown that every fixing block has period equal 2k or 3(2k), and fixing blocks are produced having these periods. To each of these fixing blocks is associated a chain. The length of the chain which corresponds to the fixing block with period 2k is 3(2k−1)−1, and the length of the chain which corresponds to the fixing block of length 3(2k) is 2k+2−1. Moreover, these chains occur in words which contain no square longer than the associated fixing block.

Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities

The restricted eight-vertex solid-on-solid (SOS) model is an exactly solvable class of two-dimensional lattice models. To each sitei of the lattice there is associated an integer heightl i restricted to the range 1⩽l i ⩽r−1. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a parameterη. In an earlier paper we considered the caseη=K/r. Here we generalize those considerations to the caseη=sK/r, s an integer relatively prime tor. We are again led to generalizations of the Rogers-Ramanujan identities.

Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

The eight-vertex model is equivalent to a “solid-on-solid” (SOS) model, in which an integer heightl i is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1⩽l i⩽4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1⩽l i⩽r−1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponentsβ, α, $$\bar \alpha $$ are obtained.

XI.—The Zeta Function of Jacobi. A seven-decimal table of Z(u/m) at interval 0·01 for u and for values of m(= k2) from 0·1 to 1·0

The elliptic integral of the second kindcan, by substitution, be made to depend uponWriting E(K) = E, Jacobi's Zeta function is defined bywhereZ(u) is an odd function of u with period 2K.