The logistic map and the birth of period-3 cycle

The goal of this paper is to present a proof that for the logistic map the period-3 begins at . The third-iterate map is the key for understanding the birth of the period-3 cycle. Any point in a period-3 cycle repeats every three iterates by definition. Such points satisfy the condition ,and they are therefore fixed points of the third-iterate map. This fact and the so called tangent bifurcation for the logistic map, as well as the fixed points definition, are used for finding the value. The algebraic treatment utilizes some properties of symmetric polynomials in three variables. For the purposes of this paper, the bifurcation diagram for the logistic map is also presented, as well as a program in Mathematica for its construction

Stability of real parametric polynomial discrete dynamical systems

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter $\lambda$, and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real $m$-th degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of Canonical Polynomial Maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates, and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.Comment: 23 pages, 4 figures, now published in Discrete Dynamics in Nature and Societ

Density function associated with nonlinear bifurcating map

In the class of nonlinear one-parameter real maps we study those with bifurcation that exhibits period doubling cascade. The fixed points of such a map form a finite discrete real set with dimension (2^n)m, where m is the (odd) number of "primary branches" of the map in the non-chaotic region and n is a non-negative integer. We associate with this map a nonlinear dynamical system whose Hamiltonian matrix is real, tridiagonal and symmetric. The density of states of the system is calculated and shown to have a number of separated bands equals to (2^n-1)m for n not equal 0, in which case the density has m bands. The location of the bands depends only on the map parameter and the odd/even ordering of the fixed points in the set. Polynomials orthogonal with respect to this density (weight) function are constructed. The logistic map is taken as an illustrative example.Comment: 8 pages of text, 9 figures (one in color

Milnor’s Conjecture on Monotonicity of Topological Entropy: results and questions

This note discusses Milnor’s conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof which was obtained in joint work with Henk Bruin, see [BvS09]. At the end of this note we explore some related conjectures and questions

Hierarchy of Chaotic Maps with an Invariant Measure

We give hierarchy of one-parameter family F(a,x) of maps of the interval [0,1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of these maps analytically, where the results thus obtained have been approved with numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain parameter values, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameter whose Lyapunov characteristic exponent begins to be positive.Comment: 18 pages (Latex), 7 figure