Feigenbaum Constant Research Articles

The route to chaos for the Kuramoto-Sivashinsky equation

We present the results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. Our concern is with the asymptotic nonlinear dynamics as the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic (in time) attractor, to another steady fully modal attractor, to another time-periodic attractor, to a steady tetramodal attractor, to another time-periodic attractor having a full sequence of period-doublings (in the parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the lengths of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period-doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4.669201609.... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Feigenvalues for Mandelsets

The authors discuss generalizations of Feigenbaum's constants alpha and delta to complex polynomial maps of degree higher than 2, and present some numerical estimates. Universality classes are found to depend on the nature of the critical points of the polynomial.

A Precise Calculation of the Feigenbaum Constants

A Precise Calculation of the Feigenbaum Constants

THE PARABOLA APPROXIMATION TO THE BIFURCATION THEORY

In this article, we propose a parabola approximation to the theory of bifurcation. From that the process of instability to bifurcation, the effect of stubilization after bifurcation, the limit to wards chaos can be made dear and the Feigenbaum number can be caluclated.

Mutual locking, chaos and devils staircase in two Josephson tunnel junctions with a common resistive shunt

The authors studied the phase-locking in a pair of individually biased, resistively and capacitively shunted Josephson junctions, coupled by a common resistive or capacitive shunt, on an analog computer. Under certain conditions, locking is found to occur at all rational frequency ratios. A critical line is found in parameter space along which the steps form a complete devil's staircase having a fractal dimension of 0.87. Beyond the critical line bifurcations occur on all steps following the Feigenbaum sequence to chaos. The Feigenbaum constants are recovered to a fair accuracy. Preliminary experiments with two Josephson tunnel junctions shunted on the chip by a gold-indium film are discussed. >

Period doubling in six-dimensional symmetric volume-preserving maps

Period doubling in three symmetrically coupled two-dimensional area-preserving maps is numerically studied. It is found that there is a two-dimensional manifold on which all period-2n (n=0, 1, 2, . . .) orbits of a period-doubling bifurcation sequence lie. On this manifold, the universality classes of period doubling are just the three classes for two symmetrically coupled two-dimensional area-preserving maps, each characterised by its own Feigenbaum constants, reported by Mao and Helleman (Phys. Rev. A. vol.35, p.1847, 1987). It is also reported that, for three non-symmetrically coupled two-dimensional area-preserving maps, three maps which are fixed under the renormalisation operator have been found. The relevant eigenvalues of perturbation around the fixed maps are again the same as those found for two non-symmetrically coupled two-dimensional area-preserving maps, by Mao and Greene (Phys. Rev. A vol.35, p.3911, 1987). The above-mentioned numerical and renormalisation results for the six-dimensional maps agree with each other.

Chaotic behavior and period doubling in plasmas.

We report the first experimental observation of chaotic behavior and period doubling in a pulsed plasma discharge sensitive to initial and boundary conditions. The Feigenbaum constants have been measured experimentally in one of the observed chaotic discharge routes.

New Feigenbaum constants for four-dimensional volume-preserving symmetric maps.

We study period doubling in a symmetric four-dimensional volume-preserving quadratic map, i.e., two symmetrically coupled two-dimensional area-preserving H\'enon maps. We must vary two parameters and thus obtain two Feigenbaum constants, ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2}$. It is a very important point that for each region of stability (belonging to some period-q orbit) in this parameter plane we find two regions of stability for the period-2q orbit, four regions for the period-4q orbit, and so on. Hence we have an infinite number of stability regions and infinities of bifurcation ``paths'' through these regions. Almost all self-similar bifurcation paths fall into one of three possible ``universality classes,'' i.e., each class is characterized by its own two Feigenbaum constants, ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2}$. We find ${\ensuremath{\delta}}_{2}$=+4.000. . .,-2.000. . .,-4.404. . ., respectively, for the three classes. These ${\ensuremath{\delta}}_{2}$ values are also recovered here from some approximate (numerical) renormalization scheme. The ${\ensuremath{\delta}}_{1}$ is, in all cases, the same as in two-dimensional area-preserving maps, ${\ensuremath{\delta}}_{1}$=8.721. . . . The ${\ensuremath{\delta}}_{2}$=-15.1. . ., reported in an earlier paper [J. M. Mao, I. Satija, and B. Hu, Phys. Rev. A 32, 1927 (1985)], applies to only two exceptional paths.

One parameter bifurcation diagram for Chua's circuit

The objective of this letter is to report on the usefulness of a 1-dimensional map for characterizing the dynamics of a third-order piecewise-linear circuit. The map is used to reproduce period-doubling birfureations and particularly to compute Feigenbaum's number. The 1-dimensional map gives qualitatively identical results as the circuit equations and proved to be far more computationally efficient.

Observations on period-doubling phenomena in a basic plasma experiment

Observations on period-doubling phenomena by exciting lower hybrid and ion cyclotron fluctuations in the presence of rf power in a plasma experiment are reported. The Feigenbaum number δ is calculated for the first two bifurcations and found to be 4.138. Nonlinear nature of the ion cyclotron oscillations gives rise to the interaction between modes that ultimately generate higher harmonics of the initial frequencies.

Discussion on the first day’s papers

I am grateful to have been invited to contribute to the discussion of the fascinating material we have heard today at this unusual joint meeting of the Royal Society and British Academy on ‘Predictability in Science and Society’. I can add little to the many important specific points brought out in the excellent formal presentations by two economists and two physical scientists and in various contributions from the floor, so I offer a few general remarks under the headings ‘language’, ‘experiments’ and ‘time-series analysis’. Mathematics gives us the operational methods used in predictability work and it also provides concepts, terminology and language, which we have to employ correctly in our efforts to promote working links between various disciplines. The term ‘predictability horizon’ has been used by several speakers today and the irreverent thought crossed my mind that there are several undeniable meteorological consequences of some of the basic theorems of algebraic topology, namely that for all predictability horizons (from minus infinity to plus infinity!): ( a ) the wind-speed must vanish somewhere, ( b ) the wind direction takes all possible values relative to the perimeter of any region of the atmosphere within which the wind-speed is everywhere non-zero, and ( c ) on global contour maps of any meteorological variable such as pressure, the number of ‘highs’ plus the number of 'lows’ minus the number of ‘cols’ is always equal to two! The hard-pressed weather forecaster is unlikely to be impressed by such mathematical gems, but from his perusal of the literature of dynamical meteorology he will already be acquainted with terms from topology and other branches of mathematics with which some of the more theoretical papers in his subject are now liberally sprinkled. ‘Poincaré surface’, ‘strange attractor’, ‘central manifold theorem’, ‘intransitivity’, ‘Hausdorff dimension’, ‘fractal’, ‘bifurcation tree’, ‘Feigenbaum numbers’, ‘Hénon map’, ‘deterministic chaos’, ‘phase portrait’, ‘swallowtail catastrophe’, ‘Cantor set’, ‘Cantor dust’, ‘Sierpinski gasket’, ‘Lorenz equations’, ‘multiple equilibria’, ‘multi-dimensional torus’ and many other terms which I cannot immediately recall are now employed in several disciplines and we must use them with confidence and precision.

Analytical crossover results for the Feigenbaum constants: Crossover from conservative to dissipative systems.

Using renormalization theory, Zisook has shown that at small values of the dissipation (e.g., for a planar map of constant Jacobian B=0.999) the apparent rate of convergence, δ, of the parameter values for period doubling descends (‘‘crosses over’’) from its conservative value (δ=8.721 097 2. . .) to its dissipative value (δ=4.669 201 6. . .) as we consider higher and higher bifurcations. Moreover, this apparent δ at the nth bifurcation is a universal function, depending only on &. Employing a technique of Ghendrih and renormalization theory, I obtain an analytical expression for this function and show that it does not descend monotonically but has a minimum at Be,min≃3×10-6. I also analytically calculate the (orbit) scaling factor α at Be=0 and at Be=1 using some results of a second-order renormalization equation which I derive here. Finally, I obtain an analytical expression for the bifurcation values Cn(B) of a particular map (the Henon map) as a function of n and B.

Approach to chaos: Universal quantitative properties of one-dimensional maps

Using the renormalization-group method, numerical results, and analytical arguments, we obtain the universal metric properties of one-dimensional iterated maps which exhibit period doublings. Such maps can be classified according to their behaviors around the extremum. For maps with the extremum (at $x=0$) of the symmetric form $f(x)\ensuremath{\sim}{|x|}^{z}b(x)$, with $z>1$, where the function $b(x)$ modifies ${|x|}^{z}$ by less than any power of $|x|$, the quantitative universal properties of the perioddoubling approach to chaos is described for asymptotically large $n$, by ${\ensuremath{\delta}}_{n}=\frac{({\ensuremath{\lambda}}_{n}\ensuremath{-}{\ensuremath{\lambda}}_{n\ensuremath{-}1})}{({\ensuremath{\lambda}}_{n+1}\ensuremath{-}{\ensuremath{\lambda}}_{n})}=\ensuremath{\delta}[1+\ensuremath{\epsilon}(n)\ensuremath{\tau}\mathrm{ln}\ensuremath{\alpha}]$ and ${\ensuremath{\alpha}}_{n}=\frac{\ensuremath{-}{d}_{n}}{{d}_{n+1}}=\ensuremath{\alpha}[1+\ensuremath{\epsilon}(n)\ensuremath{\kappa}1\mathrm{n}\ensuremath{\alpha}]$, where ${\ensuremath{\lambda}}_{n}$ is the value of the parameter at the $n$th period-doubling bifurcation, ${d}_{n}$ is the typical distance from the extremum at the $n$th bifurcation point, and $\ensuremath{\epsilon}(n)={[\frac{\ensuremath{\partial}\mathrm{ln}b(x)}{\ensuremath{\partial}\mathrm{ln}|x|}]}_{|x|\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}n}}$, where terms of order ${n}^{\ensuremath{-}2}$ are neglected. Here $\ensuremath{\delta}$, $\ensuremath{\alpha}$, $\ensuremath{\kappa}$, and $\ensuremath{\tau}$ are all functions of $z$ only; they obey the relations $\frac{d\mathrm{ln}\ensuremath{\alpha}}{\mathrm{dz}}=\ensuremath{\kappa}\mathrm{ln}\ensuremath{\alpha}$ and $\frac{d\mathrm{ln}\ensuremath{\delta}}{\mathrm{dz}}=\ensuremath{\tau}\mathrm{ln}\ensuremath{\alpha}$, where $\ensuremath{\alpha}$ and $\ensuremath{\delta}$ are the Feigenbaum constants. In particular, for $f(x)\ensuremath{\approx}{|x|}^{z}{|\mathrm{ln}({x}^{2})|}^{p} {|\mathrm{ln}|\mathrm{ln}({x}^{2})| |}^{q}\ensuremath{\cdots}$, where $z>1$, then ${d}_{n}\ensuremath{\approx}{\ensuremath{\alpha}}^{\ensuremath{-}n}{n}^{\ensuremath{\kappa}p}{(\mathrm{ln}n)}^{\ensuremath{\kappa}q}\ensuremath{\cdots}$ and ${\ensuremath{\lambda}}_{\ensuremath{\infty}}\ensuremath{-}{\ensuremath{\lambda}}_{n}\ensuremath{\approx}{\ensuremath{\delta}}^{\ensuremath{-}n}{n}^{\ensuremath{\tau}p}{(\mathrm{ln}n)}^{\ensuremath{\tau}q}\ensuremath{\cdots}$, where ${\ensuremath{\lambda}}_{\ensuremath{\infty}}$ is the critical value of the parameter beyond which is chaos. For $z=2$, $\ensuremath{\alpha}=2.50291$, $\ensuremath{\delta}=4.66920$, $\ensuremath{\kappa}=\ensuremath{-}0.44453$, and $\ensuremath{\tau}=0.36954$. For the sake of completeness, an analysis of maps with asymmetric extrema is also presented. In this case, $\ensuremath{\alpha}$ and $\ensuremath{\delta}$ are functions of $z$ as well as the size of the asymmetry.

Feigenbaum's number

Feigenbaum's number

Bifurcations and chaos in the ϕ4theory on a lattice

The authors have studied numerically and analytically the discrete phi 4 model defined by Bak and Pokrovsky (see Phys. Rev. Lett., vol.47, no.13, p.958-61, 1981). The model may describe Peierls systems, structural instabilities, metal-insulator transitions, etc., in solid state physics. The phi 4 theory can be formulated as an area-preserving two-dimensional mapping. This mapping exhibits infinite series of period-doubling bifurcation leading to chaos. The bifurcations are characterised by universal numbers delta =8.72109 ... and alpha =4.0180 ..., which appear to be identical to those found by Bountis (1981) for the Henon mapping, but different from the Feigenbaum numbers for dissipative systems. In addition, novel features arise because of marginally stable fixed points and the splitting of one 2-cycle orbit into two 2-cycle orbits.