Feigenbaum Constant Research Articles

Death of period-doublings: locating the homoclinic-doubling cascade

This paper studies a natural mechanism, called a homoclinic-doubling cascade, for the disappearance of period-doubling cascades in vector fields. Simply put, an entire period-doubling cascade collides with a saddle-type equilibrium. Homoclinic-doubling cascades are known to have self-similar structure. In contrast to the well-known Feigenbaum constant, the scaling constants for homoclinic-doubling depend on the eigenvalues of the saddle equilibrium. Specifically, we present here for the first time a detailed study of homoclinic-doubling cascades in a smooth vector field, namely a three-dimensional polynomial model proposed by Sandstede. A numerical algorithm is presented for computing homoclinic-doubling cascades in general vector fields, which makes use of the program AUTO/ HomCont. This allows us to compute two types of homoclinic-doubling cascades, one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation. Our results bring out the scaling constants in good agreement with analytical estimates obtained from one-dimensional maps.

Estimation of the component amplitudes in oscillation spectra and the Feigenbaum constant in solutions of the Rossler set of equations

A method based upon approximation of the sequence of counts by a first-order trigonometric polynomial with variable frequencies of the harmonic functions is proposed for evaluation of the parameters of components in oscillation spectra. This approach was used to estimate the parameters of bifurcational period doubling and the Feigenbaum constant in solutions of the Rossler set of equations. A possibility of decreasing the level of side components of an intense oscillation by using a difference spectrum in estimating a weak spectral component is considered.

Chaotic behaviour in the Carotid–Kundalini map function

Bifurcations and chaotic behaviour, via the period doubling route, are shown for the Carotid–Kundalini map function. This function is particularly interesting since it is not simply unimodal in the range [−1⋯1]. Calculations of the Feigenbaum constants α and δ are made which do not differ greatly from the established values. The exponential divergence of nearby trajectories is used to demonstrate the existence of chaotic behaviour.

A New Feature in Some Quasi-discontinuous Systems

Many systems can display a very short, rapid changing stage (quasi-discontinuous region) inside a relatively very long and slowly changing process. A quantitative definition for the "quasi-discontinuity" in these systems has been introduced. We have shown by a simplified model that extra-large Feigenbaum constants can be found inside some period-doubling cascades due to the quasi-discontinuity. As an example, this phenomenon has also been observed in Rose-Hindmash model describing neuron activities.

Superstrings, Cantorian-fractal Spacetime and Quantum-like Chaos in Atmospheric Flows

Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with the inverse power law form for spectra of temporal fluctuations. Such non-local connections are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. A recently developed cell dynamical system model for atmospheric flows predicts the observed self-organized criticality as a natural consequence of quantum-like mechanics governing flow dynamics. The model is based on the concept that spatial integration of enclosed small scale fluctuations results in the formation of large eddy circulations. The model predicts the following: (a) The flow structure consists of an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. (b) Conventional power spectrum analysis will resolve such spiral trajectories as a continuum of eddies with progressive increase in phase. (c) Increments in phase are concomitant with increases in period length and also represent the variance, a characteristic of quantum systems identified as Berrys phase. (d) The universal algorithm for self-organized criticality is expressed in terms of the universal Feigenbaum constants, a and d, as 2 a 2= πd, where the fractional volume intermittency of occurrence πd contributes to the total variance 2 a 2 of fractal structures. (e) The Feigenbaum constants are expressed as functions of the golden mean. ( f) The quantum mechanical constants fine structure constant and ratio of proton mass to electron mass, which are pure numbers and are obtained by experimental observations only, are now derived in terms of the Feigenbaum constant, a. (g) Atmospheric flow structure follows Keplers third law of planetary motion. Therefore, Newtons inverse square law for gravitation also applies to eddy masses. The centripetal acceleration representing the inertial masses (of eddies) are equivalent to gravitational masses. The fractal-Cantorian structure of spacetime can also be visualized as a nested continuum of vortex (eddy) circulations, whose inertial masses obey Newtons inverse square law of gravitation. The model concept resembles a superstring model for subatomic dynamics which incorporates gravitational forces.

THE PROCESS EQUATION

The iteration of the simple equation A A g sin A generates t 1 t t fundamental numerical constants, ''biotic'' patterns with dynamic features observed in empirical recordings of physiological data but not in chaos , bifurcations, chaos, an infinite number of periodicities, and multiple flights toward infinity. For g 2, the equation converges to pi. At g 2, outcomes bifurcate and diverge. In a significant union of opposites, one path reaches the Fibonacci ratio describing spiral order when the opposite path achieves the Feigenbaum number describing chaos-inducing bifurcations. Chaotic patterns start when g approximates Feigenbaum's point 3.56 ... . Biotic patterns start at g 4.6 Feigenbaum's constant. Pointing to numerical cosmic forms, significant integer 2 n and irrational numbers occur as both outcomes and gain g. The equation embodies the basic postulates of process theory: 1 iteration models the linear flow of action i.e., time; 2 the sine function models the cycling of complementary opposites generating positive and negative feedback.

Scaling of potential function derived from parameter renormalization of maps

A systematic way for deriving the parameter renormalization equation for one-dimensional maps is presented and the critical behavior of periodic-doubling is investigated. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function derived from the parameter renormalization equation shows scaling in the parameter space with the universal convergent rate at the accumulation point, but with a different size scaling factor from the Feigenbaum constant.

MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS

The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main interest of the present paper lies in the occurrence of successive homoclinic doubling bifurcations under an appropriate condition, which is a part of the entire bifurcation for multiple homoclinic orbits. Since this is a totally global bifurcation, we need the aid of numerical experiments for which we must choose a concrete set of ordinary differential equations that exhibits the desired bifurcation. In this paper we employ a family of continuous piecewise-linear vector fields for such a model equation. In order to explain the cascade of homoclinic doubling bifurcations theoretically, we also derive a two-parameter family of unimodal maps as a singular limit of the Poincaré maps along homoclinic orbits. We locate bifurcation curves for this family of unimodal maps in the two-dimensional parameter space, which basically agree with those for the piecewise-linear vector fields. In particular, we show, using a standard technique from the theory of unimodal maps, that there exists an infinite sequence of doubling bifurcations which corresponds to the sequence of homoclinic doubling bifurcations for the piecewise-linear vector fields described above. Since our unimodal map has a singularity at a boundary point of its domain of definition, the doubling bifurcation is slightly different from that for standard quadratic unimodal maps, for instance the Feigenbaum constant associated with the accumulation of the doubling bifurcations is different from the standard value 4.6692.…

A continuation algorithm for discovering new chaotic motions in forced Duffing systems

We study a dynamical system described by the Duffing equation under static plus large periodic excitation. The bifurcation diagram in terms of the amplitude of the periodic excitation reveals that there are at least six cascades of period doubling bifurcations with the other parameters fixed at specific values. These six sequences are further investigated by a continuation algorithm which is based on the principles of the shooting method combined with the Newton method for solving nonlinear equations. A Runge-Kutta-Hûta method has been used for solving the system of differential equations. The final conclusion is that each of the six sequences is governed by Feigenbaum's number δ = 4.6692 from Universality Theory. Beyond the limit values derived for the amplitude of the periodic excitation new strange attractors are found.

Chaos in a pendulum with forced horizontal support motion: a tutorial

This contribution presents a tutorial on chaos in forced pendulum motion. A pendulum system with its support forced along a horizontal line is considered. The bifurcation diagram in function of the amplitude of the forced periodic motion shows several period doubling subharmonic cascades, chaotic motions and windows of periodic motions. It is illustrated that the cascades are governed by Feigenbaum's number from Universality Theory. The chaotic attractors are investigated for their characteristics such as its evolution in the phase plane, the computation of the Liapounov exponents, the Liapounov dimension, the power spectrum and the winding number.

MULTI-PARAMETER CRITICALITY IN CHUA'S CIRCUIT AT PERIOD-DOUBLING TRANSITION TO CHAOS

Investigation of non-Feigenbaum types of period-doubling universality is undertaken for a single Chua's circuit and for two systems with a unidirectional coupling. Some codimension-2 critical situations are found numerically that were known earlier for bimodal 1D maps. However, the simplest of them (tricritical) does not survive in a strict sense when the exact dynamical equations are used instead of the 1D map approximation. In coupled systems double Feigenbaum's point and bicritical behavior are found and studied. Scaling properties that are the same as in two logistic maps with a unidirectional coupling are illustrated.

Scaling symmetries in nonlinear dynamics. A view from parameter space

The orbit of the critical point of a discrete nonlinear dynamical system defines a family of polynomials in the parameter space of that system. We show here that for the important class of quadratic-like maps, these polynomials become indistinguishable under a suitable scaling transformation. The universal representation of these polynomials produced by such a scaling leads directly to accurate approximations for those parameter values where windows of order appear, the sizes of such windows, measures of window distortion, and the characterization of the internal structure of the windows in terms of generalized Feigenbaum numbers.

Entropy of the symbolic sequence for critical circle maps.

The rabbit sequence, which is related to the golden mean and the Fibonacci numbers, is a self-similar infinite sequence of 0's and 1's that occurs in a variety of contexts in physics and mathematics. For instance, it represents the symbolic dynamics of the nonlinear circle map at the transition from periodicity to chaos, and it appears in mathematical models of quasicrystals. Here, the block entropy for the rabbit sequence is derived analytically. It has already been argued that the sequence exhibits long-range order. Our results confirm this conjecture: The entropy per bit, a direct measure for the long-range order in symbolic sequences, decays logarithmically. The same behavior has been found in a heuristic way for the symbolic dynamics generated by the logistic map at the Feigenbaum point.

METRIC UNIVERSALITY OF ORDER IN ONE-DIMENSIONAL DYNAMICS

The orbit of the critical point of a nonlinear dynamical system defines a family of functions in the parameter space of the system. For unimodal maps a renormalization makes these functions indistinguishable over a wide range of parameter values. The universal representation of these functions leads directly to a number of interesting results: (1) the positions in the parameter space of the windows of order; (2) the sizes of the windows of order; (3) measures of distortion in the window structure; and (4) various generalized Feigenbaum numbers. We explicitly discuss the examples of the quadratic and sine maps.

Introduction to chaos and coherence

Preface. Introduction. Fractals: A cantor set. The Koch triadic island. Fractal dimensions. The logistic map: The linear map. The fixed points and their stability. Period two. The period doubling route to chaos. Feigenbaum's constants. Chaos and strange attractors. The critical point and its iterates. Self similarity, scaling and univerality. Reversed bifurcations. Crisis. Lyapunov exponents. Dimensions of attractors. Tangent bifurcations and intermittency. Exact results at ^D*l = 1. Poincar^D'e maps and return maps. The circle map: The fixed points. Circle maps near K = 0. Arnol'd tongues. The critical value K = 1. Period two, bimodality, superstability and swallow tails. Where can there be chaos? Higher dimensional maps: Linear maps in higher dimensions. Manifolds. Homoclinic and hetroclinic points. Lyapunov exponents in higher dimensional maps. The Kaplan-Yorke conjecture. The Hopf bifurcation. Dissipative maps in higher dimensions: The Henon map. The complex logistic map. Two dimensional coupled logistic map. Conservative maps: The twist map. The KAM theorem. The rings of Saturn. Cellular automata. Ordinary differential equations: Fixed points. Linear stability analysis. Homoclinic and hetroclinic orbits. Lyapunov exponents for flows. Hopf bifurcations for flows. The Lorenz model. Time series analysis: Fractal dimension from a time series. Autoregressive models. Rescaled range analysis. The global temperature - an example. Appendices. Further reading. Index.

Numerical Evidence of Feigenbaum's Number δ in Non-linear Oscillations

Numerical Evidence of Feigenbaum's Number δ in Non-linear Oscillations

Period doubling and multifractals in 1-D interative maps

A highly accurate method to compute the period doubling bifurcation solutions of a general one-dimensional iterative map is presented. The technique consists of constructing similar structures of the period doubling solutions and then applying a renormalization procedure to evaluate the appropriate length scaling factors. For period-doubling bifurcation solutions leading to chaos, this approach yields multifractal results of very high precision compared to the usual multifractal analysis alone. If the critical parameter associated with the Feigenbaum number is employed, the fractal characteristic parameters calculated will be exact. The stability status of the computed solutions can also be easily determined. An example is solved to demonstrate and to assess the accuracy of the procedure. As shown, this technique yields solutions that represent a marked improvement over the currently available results in literature.

Chaotic behavior of reaction systems: parallel cubic autocatalators

A model of a reaction system consisting of two parallel, isothermal autocatalytic reactions in a CSTR has been examined. It is shown that self-sustained chaotic behavior can occur in this system. The region of chaos is entered and exited according to period-doubling and halving sequences, with both sets of bifurcations giving rise to Feigenbaum's number. Power spectral density calculations show that the nature of the chaotic behavior depends quite strongly on the parameter values. From a calculation of the Lyapunov exponents it is found that the Lyapunov dimension of the strange attractor is only slightly greater than that of a two-periodic torus.

Predicting chaos for infinite dimensional dynamical systems: the Kuramoto-Sivashinsky equation, a case study.

The results of extensive computations are presented to accurately characterize transitions to chaos for the Kuramoto-Sivashinsky equation. In particular we follow the oscillatory dynamics in a window that supports a complete sequence of period doubling bifurcations preceding chaos. As many as 13 period doublings are followed and used to compute the Feigenbaum number for the cascade and so enable an accurate numerical evaluation of the theory of universal behavior of nonlinear systems, for an infinite dimensional dynamical system. Furthermore, the dynamics at the threshold of chaos exhibit a self-similar behavior that is demonstrated and used to compute a universal scaling factor, which arises also from the theory of nonlinear maps and can enable continuation of the solution into a chaotic regime. Aperiodic solutions alternate with periodic ones after chaos sets in, and we show the existence of a period six solution separated by chaotic regions.

A precise calculation of the Feigenbaum constants

The Feigenbaum constants arise in the theory of iteration of real functions. We calculate here to high precision the constants $\alpha$ and $\delta$ associated with period-doubling bifurcations for maps with a single maximum of order