Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system

A variety of different dynamical regimes involving strange nonchaotic attractors (SNAs) can be observed in a quasiperiodically forced delayed system. We describe some numerical experiments giving evidences of intertwined basin boundaries (smooth, non-Wada fractal and Wada property) for SNAs. In particular, we show that Wada property, fractality and smoothness can be intertwined on arbitrarily fine scales. This suggests that SNAs can exhibit the final state sensitivity and unpredictable behaviors. An interesting dynamical transition of SNAs together with associated mechanisms from non-Wada fractal to Wada intertwined basin boundaries is examined. A scaling exponent is used to characterize the intertwined basin boundaries.     

Dynamics of coin tossing is predictable

The dynamics of the tossed coin can be described by deterministic equations of motion, but on the other hand it is commonly taken for granted that the toss of a coin is random. A realistic mechanical model of coin tossing is constructed to examine whether the initial states leading to heads or tails are distributed uniformly in phase space. We give arguments supporting the statement that the outcome of the coin tossing is fully determined by the initial conditions, i.e. no dynamical uncertainties due to the exponential divergence of initial conditions or fractal basin boundaries occur. We point out that although heads and tails boundaries in the initial condition space are smooth, the distance of a typical initial condition from a basin boundary is so small that practically any uncertainty in initial conditions can lead to the uncertainty of the results of tossing.     

Strange saddles and the dimensions of their invariant manifolds

Nonattracting chaotic sets play a fundamental role in typical dynamical systems. They occur, for example, in the form of chaotic transient sets and fractal basin boundaries. The subject of this paper is the dimensions of these sets and of their stable and unstable manifolds. Numerical experiments are performed to determine these dimensions. The results are consistent with a conjectured formulae expressing the dimensions in terms of Lyapunov exponents and the transient life-time associated with the strange saddle.     

Multiple coexisting attractors,Basin boundaries and basic sets

Orbits initialized exactly on a basin boundary remain on that boundary and tend to a subset on the boundary. The largest ergodic such sets are called basic sets. In this paper we develop a numerical technique which restricts orbits to the boundary. We call these numerically obtained orbits “straddle orbits”. By following straddle orbits we can obtain all the basic sets on a basin boundary. Furthermore, we show that knowledge of the basic sets provides essential information on the structure of the boundaries. The straddle orbit method is illustrated by two systems as examples. The first system is a damped driven pendulum which has two basins of attraction separated by a fractal basin boundary. In this case the basic set is chaotic and appears to resemble the product of two Cantor sets. The second system is a high-dimensional system (five phase space dimensions), namely, two coupled driven Van der Pol oscillators. Two parameter sets are examined for this system. In one of these cases the basin boundaries are not fractal, but there are several attractors and the basins are tangled in a complicated way. In this case all the basic sets are found to be unstable periodic orbits. It is then shown that using the numerically obtained knowledge of the basic sets, one can untangle the topology of the basin boundaries in the five-dimensional phase space. In the case of the other parameter set, we find that the basin boundary is fractal and contains at least two basic sets one of which is chaotic and the other quasiperiodic.     

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.     

What Can One Learn About Self-Organized Criticality from Dynamical Systems Theory?

We develop a dynamical system approach for the Zhang model of self-organized criticality, for which the dynamics can be described either in terms of iterated function systems or as a piecewise hyperbolic dynamical system of skew-product type. In this setting we describe the SOC attractor, and discuss its fractal structure. We show how the Lyapunov exponents, the Haussdorf dimensions, and the system size are related to the probability distribution of the avalanche size via the Ledrappier–Young formula.     

Using invariants to change detection in dynamical system with chaos

Change detection is the crucial subject in dynamical systems. There are suitable methods for detecting changes for linear systems and some methods for nonlinear systems, but there is a lack of methods concerning chaotic systems. This paper presents change detection techniques for dynamical systems with chaos. We consider the dynamical system described by the time series which originated from ordinary differential equation and real-world phenomena. We assume that the change parameters are unknown and the change could be either slight or drastic. The process of change detection is based on characteristic dynamical system invariants. Changes in the invariants’ values of the dynamical systems are the indicators of change. We propose a method of change detection based on the fractal dimension and recurrence plot. The automatic detection is provided by control charts. Methods were checked by using small data sets and stream data.     

The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.Research in part supported by AFOSR and by the Department of Energy (Scientific Computing Staff Office of Energy Research)     

Output functions and fractal dimensions in dynamical systems

We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the output function evaluation (OFE) method. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with D<0.5, where D is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition.     

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     

Hamiltonian dynamics,nanosystems, and nonequilibrium statistical mechanics

An overview is given of recent advances in nonequilibrium statistical mechanics on the basis of the theory of Hamiltonian dynamical systems and in the perspective provided by the nanosciences. It is shown how the properties of relaxation toward a state of equilibrium can be derived from Liouville's equation for Hamiltonian dynamical systems. The relaxation rates can be conceived in terms of the so-called Pollicott–Ruelle resonances. In spatially extended systems, the transport coefficients can also be obtained from the Pollicott–Ruelle resonances. The Liouvillian eigenstates associated with these resonances are in general singular and present fractal properties. The singular character of the nonequilibrium states is shown to be at the origin of the positive entropy production of nonequilibrium thermodynamics. Furthermore, large-deviation dynamical relationships are obtained, which relate the transport properties to the characteristic quantities of the microscopic dynamics such as the Lyapunov exponents, the Kolmogorov–Sinai entropy per unit time, and the fractal dimensions. We show that these large-deviation dynamical relationships belong to the same family of formulas as the fluctuation theorem, as well as a new formula relating the entropy production to the difference between an entropy per unit time of Kolmogorov–Sinai type and a time-reversed entropy per unit time. The connections to the nonequilibrium work theorem and the transient fluctuation theorem are also discussed. Applications to nanosystems are described.     

Exterior dimension of fat fractals

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which we call the exterior dimension. In addition, it is shown that the exterior dimension is related to the “uncertainty exponent” previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.     

Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD)method,we study bifurcations governing the escape of periodically forced oscillators in a potential well,in which a chaotic saddle plays an extremely important role.Int this paper,we find the chaotic saddle,and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property,that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins.The chaotic saddle in the Wada fractal boundary,by colliding with a chaotic attractor,leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system.We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary particularly concentrating on its discontinuous bifurcations(metamorphoses),We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries.After a final escape bifurcation,there only exists the attractor at infinity;a chaotic saddle with a beautiful pattern is left behind in phase space.     

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.     

Adaptively locating unknown steady states: Formalism and basin of attraction

The adaptive technique, which includes both dynamical estimators and coupling gains, has been recently verified to be practical for locating the unknown steady states numerically. This Letter, in the light of the center manifold theory for dynamical systems and the matrix spectrum principle, establishes an analytical formalism of this adaptive technique and reveals a connection between this technique and the original adaptive controller which includes only the dynamical estimator. More interestingly, in study of the well-known Lorenz system, the selections of the estimator parameters and initial values are found to be crucial to the successful application of the adaptive technique. Some Milnor-like basins of attraction with fractal structures are found quantitatively. All the results obtained in the Letter can be further extended to more general dynamical systems of higher dimensions.     

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.     

Dynamic facilitation picture of a higher-order glass singularity

We show that facilitated spin mixtures with a tunable facilitation reproduce, on a Bethe lattice, the simplest higher-order singularity scenario predicted by the mode-coupling theory (MCT) of liquid-glass transition. Depending on the facilitation strength, they yield either a discontinuous glass transition or a continuous one, with no underlying thermodynamic singularity. Similar results are obtained for facilitated spin models on a diluted Bethe lattice. The mechanism of dynamical arrest in these systems can be interpreted in terms of bootstrap and standard percolation and corresponds to a crossover from a compact to a fractal structure of the incipient spanning cluster of frozen spins. Theoretical and numerical simulation results are fully consistent with MCT predictions.     

Boolean delay equations: A simple way of looking at complex systems

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time; such BDEs can be seen therefore as metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil’s staircases and “fractal sunbursts.” All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades of loading and failure in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid-earth problems. The former have used small systems of BDEs, while the latter have used large hierarchical networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (“partial BDEs”) and discuss connections with other types of discrete dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.