两类Feigenbaum映射的拟极限集的Hausdorff测度

本计算了一类无穷多峰Feigenbaum映射一类单峰Feigenbaum映射的拟极限集的Hausdorff测度。     

3阶Feigenbaum映射的拓扑共轭性

本文讨论3阶Feigenbaum映射限制在非游荡集上的拓扑共轭性．一方面3阶Feigenbaum映射必然产生混沌,混沌的产生使得非游荡集复杂化;另一方面3阶Feigenbaum映射又分为单谷的和非单谷的两类．利用有限型子转移,证明了对任意给定的两个满足一定条件的3阶Feigenbaum映射,限制在其非游荡集上是拓扑共轭．     

3阶非单谷Feigenbaum映射的拟极限集

一个从闭区间到自身的连续映射被称为3阶非单谷Feigenbaum映射,如果它是函数方程f~3(λx)=λf(x)的解．本文讨论了3阶非单谷Feigenbaum映射的拟极限集及其Hausdorff维数．3阶非单谷Feigenbaum映射必然产生混沌,混沌的产生使得拟极限集的存在性问题复杂化．文中采用分形几何中的知识方法证明了此类映射的拟极限集的存在性,并相应的对其Hausdorff维数作出了估计．最后给了一个具体的例子,说明确实存在这样的3阶非单谷Feigenbaum映射．     

单谷Feigenbaum映射的拟极限集

给出了单谷Feigenbaum映射拟极限集的结构以及它的准确的Hausdorff维数的关系式。     

分形拟极限集的可积性

本文讨论了螺线型拟极限集的可积性.作为应用,研究了Feigenbaum乘积映射的分 形拟极限集.     

第二类Feigenbaum函数方程的单谷扩充连续解

本文考虑第二类Feigenbaum函数方程,探讨了由某单谷映射扩充所能得到的一切连续解的性态,并给出构造这类解的可行方法。     

若干Banach空间及有界凸域上的凸映射

本文研究了一类Ｂａｎａｃｈ空间上凸映射的性质。找出了一类Ｂａｎａｃｈ空间上单位球上凸映射的特征，并利用这些结果研究了一类有界凸域上的所有凸映射．     

Fengenbaum映射的搓揉序列与特征集

设f为Feigenbaum映射,亦即函数方程fp(λx)=λf(x)满足一定条件的单峰解．f的搓揉序列为0-1无限序列,f的特征集是临界点轨迹的闭包．本文研究f的性质进而证明．f的搓揉序列是某代换在符号空间中的不动点,f在特征集上的限制是某代换子移位的一个因子．     

第二类Feigenbaum函数方程

近年来Feigenbaum现象有关课题成为动力系研究的一个重要方向。其中关键问题之一是Feigenbaum函数方程解的存在及其性态的考察。本文提出了第二类Feigenbaum函数方程,它和原由Feigenbaum提出的方程能起同等的作用,但几何直观性较强,较便于研究。本文论证了两类方程的等价性,给出了两种构造其单谷连续解的可行方法,指出了C^k解的存在性,最后并提出若干可供进一步探讨的问题。     

一类周期全纯自同构的线性化

该文证明了一类周期全纯自同构映射与线性映射同构，而由这类全纯自同构映射生成的子群在犃狌狋Ｃｎ 中是稠密的．     

A discrete time car following model and the bi-parameter logistic map

A discrete time model for car following behaviour is investigated. It is found that the model can be expressed in the form of a logistic map with multiple control parameter values. The case where the logistic map has a bi-value control parameter is investigated and relevant Feigenbaum diagrams are presented and the behaviour of the Lyapunov exponent is investigated. Divergent behaviour in the chain of vehicles is also considered.     

Period-doubling in discrete relative spatial dynamics and the Feigenbaum sequence

It is shown in this paper that although the period-doubling Feigenbaum sequence and the associated universal numbers in discrete maps of the logistic type hold over parameters, their true nature have them holding over slopes of the corresponding Poincaré maps. This finding enables one to find these Feigenbaum slope sequences in more complex maps. Further, it is demonstrated by an example in discrete relative growth spatial dynamics that a Feigenbaum sequence does not hold over the bifurcation parameter.     

Computing Feigenbaum's δ constant using the Ricker map

The Feigenbaum constant δ is frequently met by students in a first course on chaos, and discussed with reference to the period doubling within the logistic map. The details of the actual calculation of δ are, however, nontrivial, and form the basis for an undergraduate project which may be used to develop skills in discrete maps and numerical methods. This paper considers how the Ricker map can be used to evaluate δ, and also suggests a number of problems which can be solved by students along the way.     

Routes to chaos in continuous mechanical systems: Part 2. Modelling transitions from regular to chaotic dynamics

In second part of the paper both classical and novel scenarios of transition from regular to chaotic dynamics of dissipative continuous mechanical systems are studied. A detailed analysis allowed us to detect the already known classical scenarios of transition from periodic to chaotic dynamics, and in particular the Feigenbaum scenario. The Feigenbaum constant was computed for all continuous mechanical objects studied in the first part of the paper. In addition, we illustrate and discuss different and novel scenarios of transition of the analysed systems from regular to chaotic dynamics, and we show that the type of scenario depends essentially on excitation parameters.     

Thickness of Julia sets of Feigenbaum polynomials with high order critical points

We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point. To cite this article: G. Levin, G. ?wi?tek, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.     

Traveling waves,impulses and diffusion chaos in excitable media

In the present work it is shown, that the FitzHugh–Nagumo type system of partial differential equations with fixed parameters can have an infinite number of different stable wave solutions, traveling along the space axis with arbitrary speeds, and also traveling impulses and an infinite number of different states of spatiotemporal (diffusion) chaos. Those solutions are generated by cascades of bifurcations of cycles and singular attractors according to the FSM theory (Feigenbaum–Sharkovskii–Magnitskii) in the three-dimensional system of ordinary differential equations (ODEs), to which the FitzHugh–Nagumo type system of equations with self-similar change of variables can be reduced.     

The chaotic dynamics of vibrational mechanisms with energy sources of limited power

The dynamics of a vibrational mechanism with an energy source of limited power is considered. A system of two degrees of freedom is reduced to a system of the Lorenz type by the method of averaging. The existence of one of the types of chaotic attractors in a dynamical system which is a vibrational mechanism, that is, a Lorenz attractor, is established by this. The existence of a Feigenbaum attractor and intermittence is also established. Chaotic limit sets determine the chaotic behaviour of the instantaneous frequency of rotation of an asynchronous motor. The qualitative patterns of the rotational characteristic are constructed for different values of the parameters of the system and a physical interpretation of the results is given.