Metamorphoses: Sudden jumps in basin boundaries

In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are calledbasin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.This research was supported in part by grants and contracts from the Defense Advanced Research Projects Agency, the Consiglio Nazionale delle Ricerche (Comitato per le Matematiche), and the National Science Foundation     

Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map

The partially Wada basin boundaries are referred to the coexistence of Wada points and non-Wada points in the same basin boundary. We demonstrate two types of Wada bifurcations and analyze the transitions from totally Wada basins to partially Wada basins and from totally Wada basins to totally Wada basins in a two-dimensional cubic map. We describe some numerical experiments giving the evidence of partially Wada basin boundaries. Our results show that the basin cell erosion and the basin cell bifurcation can induce the Wada basin boundary metamorphoses.     

A special type of explosion of basin boundary

We study a special type of explosion of a basin boundary set in an archetypal oscillator. A typical feature is that the basin boundaries change the number of basins separating at the same time. Before the explosion, a basin boundary contains some Wada points of ten basins. After the explosion, the basin boundary contains some Wada points of eighteen basins. The underlying mechanism for the explosion is investigated by the heteroclinic tangency and Lambda lemma. Basin entropy and boundary basin entropy are also used to describe the nature of basins of attraction and the basin boundary explosion.     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

Multifractal structure of a riddled basin

For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum f(gamma), which characterizes the "skeletons" of the riddled basin, is introduced. With f(gamma), the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a "boundary" for the riddled basin. A conjecture on the relation between f(gamma) and the "stable sets" of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed. (c) 2001 American Institute of Physics.     

Fractal basin boundaries

Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.     

Fractal basin boundary of a two-dimensional cubic map

The fractal basin boundary of a two-dimensional cubic map and the origin of the fractal boundary are investigated. The fractal dimensions of the boundary and of the set of homoclinic points are computed.     

Fluctuational transitions through a fractal basin boundary

Fluctuational transitions between two coexisting chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the transition mechanism is determined by a hierarchy of homoclinic points. The most probable escape path from a chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.     

两个周期激励下Duffing-van der Pol系统的混沌瞬态和广义激变

运用广义胞映射图方法研究两个周期激励作用下Duffing-van der Pol系统的全局特性.发现了系统的混沌瞬态以及两种不同形式的瞬态边界激变, 揭示了吸引域和边界不连续变化的原因. 瞬态边界激变是由吸引域内部或边界上的混沌鞍和分形边界上周期鞍的稳定流形碰撞产生.第一种瞬态边界激变导致吸引域突然变小, 吸引域边界突然变大; 第二种瞬态边界激变使两个不同的吸引域边界合并成一体.此外, 在瞬态合并激变中两个混沌鞍发生合并, 最后系统的混沌瞬态在内部激变中消失. 这些广义激变现象对混沌瞬态的研究具有重要意义. 关键词： 广义胞映射图方法 Duffing-van der Pol 混沌瞬态 广义激变     

Stalactite basin structure of dynamical systems with transient chaos in an invariant manifold

Dynamical systems with invariant manifolds occur in a variety of situations (e.g., identical coupled oscillators, and systems with a symmetry). We consider the case where there is both a nonchaotic attractor (e.g., a periodic orbit) and a nonattracting chaotic set (or chaotic repeller) in the invariant manifold. We consider the character of the basins for the attracting nonchaotic set in the invariant manifold and another attractor not in the invariant manifold. It is found that the boundary separating these basins has an interesting structure: The basin of the attractor not in the invariant manifold is characterized by thin cusp shaped regions ("stalactites") extending down to touch the nonattracting chaotic set in the invariant manifold. We also develop theoretical scalings applicable to these systems, and compare with numerical experiments. (c) 2000 American Institute of Physics.     

Fractal basin boundary in dynamical systems and the Weierstrass-Takagi functions

It is shown that the basin boundary of the complex maps Z_n+1 = Z^q_n + C (q⩾2 is an integer and |C| ⪡ 1) is expressible with the Weierstrass function which is continuous but nowhere differentiable. The relation between the Weierstrass function and the Takagi function is discussed, and these functions are extended in a general situation. The fractal basin boundaries expressed by the generalized Weierstrass-Takagi functions are investigated.     

Thermotectonic and fault dynamic analysis of Precambrian basement and tectonic constraints with the Parana basin

The Precambrian crystalline basement of southeast Brazil is affected by many Phanerozoic reactivations of shear zones that developed during the end of the Neoproterozoic in the Brasiliano orogeny. These reactivations with specific tectonic events, a multidisciplinary study was done, involving geology, paleostress, and structural analysis of faults, associated with apatite fission track methods along the northeastern border of the Parana basin in southeast Brazil.The results show that the study area consists of three main tectonic domains, which record different episodes of uplift and reactivation of faults. These faults were brittle in character and resulted in multiple generations of fault products as pseudotachylytes and ultracataclasites, foliated cataclasites and fault gouges.Based on geological evidence and fission track data, an uplift of basement rocks and related tectonic subsidence with consequent deposition in the Parana basin were modeled.The reactivations of the basement record successive uplift events during the Phanerozoic dated via corrected fission track ages, at (Ordovician); (Triassic); (Jurassic), (Early Cretaceous); (Late Cretaceous) and (Late Cretaceous). These results indicate differential uplift of tectonic domains of basement units, probably related to Parana basin subsidence. Six major sedimentary units (supersequences) that have been deposited with their bounding unconformities, seem to have a close relationship with the orogenic events during the evolution of southwestern Gondwana.     

The attractor—basin portrait of a cellular automaton

Local space-time structures, such as domains and the intervening dislocations, dominate a wide class of cellular automaton (CA) behavior. For such spatially-extended dynamics regular domains, vicinities, and attractors are introduced as organizing principles to identify the discretized analogs of attractors, basins, and separatrices: structures used in classifying dissipative continuous-state dynamical systems. We describe the attractor-basin portrait of nonlinear elementary CA rule 18, whose global dynamics is largely determined by a single regular attracting domain. The latter's basin is analyzed in terms of subbasin and portal structures associated with particle annihilation. The conclusion is that the computational complexity of such CA is more apparent than real. Transducer machines are constructed that automatically identify domain and dislocation structures in space-time, count the number of dislocations in a spatial pattern, and implement an isomorphism between rule 18 and rule 90. We use a transducer to trace dislocation trajectories, and confirm that in rule 18, isolated dislocation trajectories, as well as a dislocation gas, agree extremely well with the classical model of annihilating diffusive particles. The CA efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space-time.     

一类新的边界激变现象:混沌的边界激变

混沌吸引子的激变是一类普遍现象.借助于广义胞映射图论(generalized cell mapping digraph)方法发现了嵌入在分形吸引域边界内的混沌鞍,这个混沌鞍由于碰撞混沌吸引子导致混沌吸引子完全突然消失,是一类新的边界激变现象,称为混沌的边界激变.可以证明混沌的边界激变是由于混沌吸引子与分形吸引域边界上的混沌鞍相碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混沌吸引子连同它的吸引域突然消失,同时这个混沌鞍也突然增大 关键词： 广义胞映射 有向图 激变 混沌鞍     

基于卷积神经网络的气体绝缘组合开关盆式 绝缘子螺栓松动检测方法*

盆式绝缘子是GIS的关键绝缘器件,它与两侧气室法兰通过螺栓进行紧固连接,当螺栓松动时会导致盆式绝缘子应力分布不均,严重时会引起绝缘子破裂,从而影响GIS运行的安全性和可靠性。文章搭建了盆式绝缘子螺栓松动超声波检测系统,以获取不同螺栓不同工况下的超声信号,基于卷积神经网络对超声信号进行特征提取,并且与BP神经网络的训练结果进行对比分析。实验结果表明,卷积神经网络可以自动提取GIS盆式绝缘子螺栓松动特征量,对十种螺栓松动工况的识别准确率达到100%,相比于BP神经网络具有较高的识别准确率,该方法可以直接用于盆式绝缘子螺栓松动检测。     

Mean temperature in a closed basin by remote sensing

Summary In this paper a procedure to determine the mean vertical temperature in a closed shallow basin will be presented. The procedure is based on the remotely observed surface temperature implemented by a calibration point at which usual meteorological measurements are performed. By the energy conservation equation applied at the calibration point the behaviour of the mean vertical temperature is obtained. Field measurement have been performed (June 1987) in a basin called Comacchio Valley (Italia). Experimental results are shown. This work has been in part supported by the Consiglio Nazionale delle Ricerche, Italy. The VAX 11/750 of the Centro Interdipartimentale di Calcolo Automatico ed Informatica Applica (CICAIA) of the University of Modena, Italy, has been used.     

An ALE-based numerical technique for modeling sedimentary basin evolution featuring layer deformations and faults

In this paper we present a numerical tool to simulate dynamics of stratified sedimentary basins, i.e. depressions on the Earth’s surface filled by sediments. The basins are usually complicated by crustal deformations and faulting of the sediments. The balance equations, the non-Newtonian rheology of the sediments, and the depth-porosity compaction laws describe here a model of basin evolution. We propose numerical schemes for the basin boundary movement and for the fault tracking. In addition, a time splitting algorithm is employed to reduce the original model into some simpler mathematical problems. The numerical stability and the other features of the developed methodology are shown using simple test cases and some realistic configurations of sedimentary basins.     

Centrifugal transformation of the transverse structure of freely propagating paraxial light beams

Superpositions of coaxial Laguerre-Gaussian modes with common waists and the same Gaussian envelopes are known to form beams whose transverse structures experience the self-similar transformation during free beam propagation: the beam shape remains the same except for the divergence and rotation around the propagation axis. We show that under certain conditions this transformation can be represented as if every point of the beam cross section performs a centrifugal straight-line fly off.     

The size of the sync basin

We suggest a new line of research that we hope will appeal to the nonlinear dynamics community, especially the readers of this Focus Issue. Consider a network of identical oscillators. Suppose the synchronous state is locally stable but not globally stable; it competes with other attractors for the available phase space. How likely is the system to synchronize, starting from a random initial condition? And how does the probability of synchronization depend on the way the network is connected? On the one hand, such questions are inherently difficult because they require calculation of a global geometric quantity, the size of the "sync basin" (or, more formally, the measure of the basin of attraction for the synchronous state). On the other hand, these questions are wide open, important in many real-world settings, and approachable by numerical experiments on various combinations of dynamical systems and network topologies. To give a case study in this direction, we report results on the sync basin for a ring of n > 1 identical phase oscillators with sinusoidal coupling. Each oscillator interacts equally with its k nearest neighbors on either side. For k/n greater than a critical value (approximately 0.34, obtained analytically), we show that the sync basin is the whole phase space, except for a set of measure zero. As k/n passes below this critical value, coexisting attractors are born in a well-defined sequence. These take the form of uniformly twisted waves, each characterized by an integer winding number q, the number of complete phase twists in one circuit around the ring. The maximum stable twist is proportional to n/k; the constant of proportionality is also obtained analytically. For large values of n/k, corresponding to large rings or short-range coupling, many different twisted states compete for their share of phase space. Our simulations reveal that their basin sizes obey a tantalizingly simple statistical law: the probability that the final state has q twists follows a Gaussian distribution with respect to q. Furthermore, as n/k increases, the standard deviation of this distribution grows linearly with square root of n/k. We have been unable to explain either of these last two results by anything beyond a hand-waving argument.