Analytics of period doubling

Approximate analytic solutions for periodic orbits of the quadratic mapxrx(1–x) are developed using algebraic methods. These solutions form the basis of an exact algorithm which predicts the quantitative order of periodic points characteristic of the Feigenbaum scenario. The algorithm holds for any one dimensional unimodal map. A general procedure is developed which permits calculation of period doubling parameters for large period orbits from those of low period to any desired degree of accuracy. Explicit equations are given through second order.     

Approximate scaling of period doubling windows

An experimental test is made of a prediction that subharmonic windows which display period doubling should exhibit global scaling. It is shown that this is approximately true in a multidimensional system.     

Corrections to scaling for period doubling

We have studied a multiple scaling which describes corrections to scaling. For the period doubling in one-dimensional dissipative maps, two-dimensional areapreserving maps, and four-dimensional symplectic maps, the multiple scaling is seen to be well-obeyed, and new scaling factors have been found. The multiple scaling is also seen to be a very powerful tool for searching for scaling behavior.     

Bounds on the unstable eigenvalue for period doubling

Bounds are given for the unstable eigenvalue of the period-doubling operator for unimodal maps of the interval. These bounds hold for all types of behaviour |x| ^r of the interval map near its critical point. They are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point.     

Stability and torsion in the period doubling cascade

We present expressions for the largest Lyapunov exponent and the generalized winding number within the period doubling cascade of 3D continuous dynamical systems.     

Universality of the topology of period doubling dynamical systems

The evolution of the topology of the invariant manifolds of the attractors of 3-D autonomous dynamical systems during period doubling is shown to be universal. The overall topology of the nth attractor is shown to depend only on the topology of the first attractor at birth.     

Oscillation period doubling chaos in a laser

We observe chaotic emission from a continuous He-Ne laser, approached by oscillation period doubling.     

A horseshoe for the doubling operator: Topological dynamics for metric universality

The doubling operator, properly defined on the space of smooth maps on the interval at the boundary of chaos, yields a dynamical system in this function space. Even if one restricts oneself to the space of real analytic maps, there is evidence that the dynamics of the doubling operator contains a horseshoe whose symbolic dynamics is described by the one-sided shift on two symbols. We indicate also how some of the global aspects of this dynamics could be recognized in a physical experiment on the transition to chaos.     

Coordinate change eigenvalues for bimodal period doubling renormalisation

The coordinate change eigenvalues for the MacKay and van Zeijts period doubling renormalisation operator for bimodal 1 D maps are derived. They are found in numerical computations of the spectrum at all the periodic orbits of renormalisation of period up to five.     

Non-asymptotic corrections to orbital scaling in period doubling maps

An extension of Feigenbaum's scaling laws for orbital sequences is presented for one-dimensional maps with quadratic maxima. We find that two universal constants, and combinations of these with the Feigenbaum scaling factors, characterize this generalized approach to chaos. Suggestions for future investigations are presented.     

电流模式SEPIC变换器倍周期分岔现象研究

以SEPIC变换器中前置电感电流为控制对象,建立了SEPIC变换器断续电流模式下的离散时间模型, 并分析了不动点的稳定性.通过相轨图、功率谱图以及分岔图, 发现电路中存在一种特殊的倍周期分岔现象——电路的运行状态经历了1倍周期、 2倍周期、4倍周期再到2倍周期、4倍周期,最终进入混沌状态.实验结果与仿真结果相一致, 证实了SEPIC变换器在断续电流模式下存在这种倍周期分岔现象.     

Subcritical period doubling in the Duffing equation-type 3 intermittency,attractor crisis

We study period doubling bifurcations, subcritical period doubling, and type 3 intermittency of the Duffing equation. The Floquet exponents of the linearization show a remarkably ordered structure and serve as a guidance of the solutions when control parameters are changed.     

Deterministic chaos in one-dimensional maps—the period doubling and intermittency routes

This paper is a review of the present status of studies relating to occurrence of deterministic chaos and its characterization in one-dimensional maps. As our primary aim is to introduce the nonspecialists into this fascinating world of chaos we start from very elementary concepts and give sufficient arguments for clarity of ideas. The two main scenarios during onset of chaos viz. the period doubling and intermittency are dealt with in detail. Although the logistic map is often discussed by way of illustration, a few more interesting maps are mentioned towards the end.     

The trajectory scaling function for period doubling bifurcations in flows

We extend the trajectory scaling function as defined for maps to flows whose dynamics is governed by ordinary differential equations. The results are obtained for the Duffing oscillator and are expected to be the same for other dissipative flows as well.     

Spectral properties of a tight binding Hamiltonian with period doubling potential

We study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence. We prove that its spectrum is purely singular continuous and supported on a Cantor set of zero Lebesgue measure, for all nonzero values of the potential strength. Moreover, we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength.     

Structure of the attraction zones of the final states in the presence of dynamical period doubling bifurcations

The structure of the attraction zones of the final states associated with dynamical period doubling bifurcations is investigated. It is found that on the “initial value—transition rate” plane the attraction zones of the two possible final states alternate with each other and that a subdivision of the attraction regions occurs with a decrease in the transition rate. It is shown that the boundaries of the attraction zones are smeared out because of the effect of noise and in this situation the fine structure of the attraction zones is destroyed. As analytical and numerical calculations have shown, the critical value of the noise variance, corresponding to the boundary between the dynamical (or predictable) and stochastic (or unpredictable) modes, has a power-law dependence on the transition rate with a typical exponent value of one. The existence of “noise” invariants is also observed: the integrated (over all initial values) probability of achieving the final state is invariant with respect to the noise level. Zh. éksp. Teor. Fiz. 113, 369–380 (January 1998)