6,974 research outputs found
Spontaneous Emergence of Spatio-Temporal Order in Class 4 Automata
We report surprisingly regular behaviors observed for a class 4 cellular
automaton, the totalistic rule 20: starting from disordered initial
configurations the automaton produces patterns which are periodic not only in
time but also in space. This is the first evidence that different types of
spatio-temporal order can emerge under specific conditions out of disorder in
the same discrete rule based algorithm.Comment: 5 pages, 6 color figures, Proceedings Medyfinol 2004, Physica A in
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Self-similarities in the frequency-amplitude space of a loss-modulated CO laser
We show the standard two-level continuous-time model of loss-modulated CO
lasers to display the same regular network of self-similar stability islands
known so far to be typically present only in discrete-time models based on
mappings. For class B laser models our results suggest that, more than just
convenient surrogates, discrete mappings in fact could be isomorphic to
continuous flows.Comment: (5 low-res color figs; for ALL figures high-res PDF:
http://www.if.ufrgs.br/~jgallas/jg_papers.html
Monogenic period equations are cyclotomic polynomials
We study monogeneity in period equations, psi(e)(x), the auxiliary equations introduced by Gauss to solve cyclotomic polynomials by radicals. All monogenic psi(e)(x) of degrees 4 = 4, we conjecture all monogenic period equations to be cyclotomic polynomials. Totally real period equations are of interest in applications of quadratic discrete-time dynamical systems
Field discriminants of cyclotomic period equations
We show that several orbital equations and orbital clusters of the quadratic (logistic) map coincide surprisingly with cyclotomic period equations, polynomials whose roots are Gaussian periods. An analytical expression for the field discriminant of period equations is obtained and applied to discover and to fill gaps in number field databases constructed by numerical search processes. Such expression allows easy assess to inessential divisors of conventional discriminants and sheds light into why numerical construction of databases is a hard problem. It also provides significant information about the organization of periodic orbits of the quadratic map
Orbital carriers and inheritance in discrete-time quadratic dynamics
Explicit formulas for orbital carriers of periods 4, 5 and 6 are reported for discrete-time quadratic dynamics. A systematic investigation of orbital inheritance for periods as high as k <= 12 is also reported. Inheritance means that unknown orbits may be obtained by nonlinear transformations of known orbits. Such nested orbit within orbit stratification shows orbits not to be necessarily independent of each other as generally assumed. Orbital stratification is potentially significant to rearrange trajectories sums in trace formulas underlying modern semiclassical interpretations of atomic physics spectra. The stratification seems to dominate as the orbital period grows
Zip and velcro bifurcations in competition models in ecology and economics
During the last six years or so, a number of interesting papers discussed systems with line segments of equilibria, planes of equilibria, and with more general equilibrium configurations. This note draws attention to the fact that such equilibria were considered previously by Miklos Farkas (1932-2007), in papers published in 1984-2005. He called zip bifurcations those involving line segments of equilibria, and velcro bifurcations those involving planes of equilibria. We briefly describe prototypical situations involving zip and velcro bifurcations
Accumulation horizons and period-adding in optically injected semiconductor lasers
We study the hierarchical structuring of islands of stable periodic
oscillations inside chaotic regions in phase diagrams of single-mode
semiconductor lasers with optical injection. Phase diagrams display remarkable
{\it accumulation horizons}: boundaries formed by the accumulation of infinite
cascades of self-similar islands of periodic solutions of ever-increasing
period. Each cascade follows a specific period-adding route. The riddling of
chaotic laser phases by such networks of periodic solutions may compromise
applications operating with chaotic signals such as e.g. secure communications.Comment: 4 pages, 4 figures, laser phase diagrams, to appear in Phys. Rev. E,
vol. 7
Preperiodicity and systematic extraction of periodic orbits of the quadratic map
Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree 4020, while for the 52 377 period-20 orbits the degree rises already to 1 047 540. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map
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