27,738 research outputs found

    Dynamics of two classes of recursive sequences

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    AbstractWe investigate the dynamics of two classes of recursive sequences: xn+1=∑j=0k∑(i0,i1,…,i2j)∈A2jxn−i0xn−i1⋯xn−i2j+b∑j=1k∑(i0,i1,…,i2j−1)∈A2j−1xn−i0xn−i1⋯xn−i2j−1+1+b, and xn+1=∑j=1k∑(i0,i1,…,i2j−1)∈A2j−1xn−i0xn−i1⋯xn−i2j−1+1+b∑j=0k∑(i0,i1,…,i2j)∈A2jxn−i0xn−i1⋯xn−i2j+b. We prove that their unique equilibrium x¯=1 is globally asymptotically stable. In addition, a new approach is presented to research the theory of recursive sequences

    DNA sequences classification and computation scheme based on the symmetry principle

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    The DNA sequences containing multifarious novel symmetrical structure frequently play crucial role in how genomes work. Here we present a new scheme for understanding the structural features and potential mathematical rules of symmetrical DNA sequences using a method containing stepwise classification and recursive computation. By defining the symmetry of DNA sequences, we classify all sequences and conclude a series of recursive equations for computing the quantity of all classes of sequences existing theoretically; moreover, the symmetries of the typical sequences at different levels are analyzed. The classification and quantitative relation demonstrate that DNA sequences have recursive and nested properties. The scheme may help us better discuss the formation and the growth mechanism of DNA sequences because it has a capability of educing the information about structure and quantity of longer sequences according to that of shorter sequences by some recursive rules. Our scheme may provide a new stepping stone to the theoretical characterization, as well as structural analysis, of DNA sequences

    Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction

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    Consider a sequence (ηN(t):t0)(\eta^N(t) :t\ge 0) of continuous-time, irreducible Markov chains evolving on a fixed finite set EE, indexed by a parameter NN. Denote by RN(η,ξ)R_N(\eta,\xi) the jump rates of the Markov chain ηtN\eta^N_t, and assume that for any pair of bonds (η,ξ)(\eta,\xi), (η,ξ)(\eta',\xi') arctan{RN(η,ξ)/RN(η,ξ)}\arctan \{R_N(\eta,\xi)/R_N(\eta',\xi')\} converges as NN\uparrow\infty. Under a hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a recursive procedure which provides a sequence of increasing time-scales \theta^1_N, \dots, \theta^{\mf p}_N, θNjθNj+1\theta^j_N \ll \theta^{j+1}_N, and of coarsening partitions \{\ms E^j_1, \dots, \ms E^j_{\mf n_j}, \Delta^j\}, 1\le j\le \mf p, of the set EE. Let \phi_j: E \to \{0,1, \dots, \mf n_j\} be the projection defined by \phi_j(\eta) = \sum_{x=1}^{\mf n_j} x \, \mb 1\{\eta \in \ms E^j_x\}. For each 1\le j\le \mf p, we prove that the hidden Markov chain XNj(t)=ϕj(ηN(tθNj))X^j_N(t) = \phi_j(\eta^N(t\theta^j_N)) converges to a Markov chain on \{1, \dots, \mf n_j\}

    Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences

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    We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.Comment: This version: updated figures and added discussion of generalised time quasilattices. 17 pages, 4 figure

    Geometry and Topology of Escape II: Homotopic Lobe Dynamics

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    We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each endpoint of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an ``Epistrophe Start Rule'': a new epistrophe is spawned Delta = D+1 iterates after the segment to which it converges, where D is the minimum delay time of the complex.Comment: 13 pages, 8 figures, to appear in Chaos, second of two paper
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