A Chaotic Hierarchy

The following sections are included:

• PREFACE

• CONTENTS

”Dynamics is spatial intuition made rigorous” O.E. Rössler [1].

The complexity of dynamical behavior possible in nonlinear (for example, electronic) systems depends only on the number of state variables involved. Single variable dissipative dynamical systems (like the single-transistor flip flop) can only possess point attractors. Two-variable systems (like an LC-oscillator) can possess a one-dimensional attractor (limit cycle). Three-variable systems admit two even more complicated types of behavior: a toroidal attractor (of doughnut shape) and a chaotic attractor (which looks like an infinitely often folded sheet). The latter is easier to obtain. In four variables, we analogously have the hyper-toroidal and the hyper-chaotic attractor, respectively, and so forth. In every higher-dimensional case, all of the lower forms are also possible as well as ”mixed cases” (like a combined hypertoroidal and chaotic motion, for example). Ten simple ordinary differential equations, most of them easy to implement electronically, are presented to illustrate the hierarchical tree. A sedond tree, in which one more dimension is needed for every type, is called the weak hierarchy because the chaotic regimes contained cannot be detected physically and numerically. The relationship between the two hierarchies is posed as an open question. It may be approached empirically — using electronic systems, for example …

The phase entrainment of coupled oscillators, often confused with frequency entrainment, was first described by Huygens in 1665 as the ”sympathy of clocks”. As it refers to the independence of the equilibrium phase difference from variations in initial conditions or the strength of the coupling, we refer to this phenomenon as phase locking. In the context of ensembles of coupled oscillators, it has many important applications. Recently, Vassalo Pereira has given a derivation of the sympathy of clocks based on Andronov’s model for the pendulum clock, showing that it is the ”tick-tock” of the escapements, rather than the swings of the pendula, which are responsible for the phase regulation. Here, we generalize Vassalo Pereira’s result to arbitrary coupled oscillators, to obtain a geometric theory of phase regulation due to pulsatile forces. The extension of this geometric theory to chaotic attractors is indicated as well …

We comment on the sequences of periodic and chaotic states observed in some experiments on the continuously fed Belousov-Zhabotinskii reaction. We present numerical results obtained with a seven-variable Oregonator model which reproduce most of the features of the experimental sequences. We discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity. We construct Poincaré map models which reduce to either multi-humped or multi-branched 1D maps in the strong area contraction limit. We report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of ”spiral-type” strange attractor satisfies the symbolic dynamics predicted by Shil’nikov’s theory of homoclinic chaos …

This document is an informal overview of work being done using the program SNA.EXE, which implements Stoichiometric Network Analysis on a personal computer …

Determinism was formulated as a philosophical doctrine by the ancient greeks. Later it found support in physics. Recently ”deterministic chaos” became the catchword of a vigorous interdisciplinary school of scientists who study non-linear dynamics. In the present paper we ask the question: Can determinism be reconciled with chaos? The question is discussed in terms of non-invertible maps …

We describe experiments on an electronic analogue of the membrane of the giant axon of the squid. As a parameter corresponding to the external potassium concentration is changed, one sees a complex sequence of periodic and aperiodic behaviours in the transmembrane potential. We interpret these time series in terms of trajectories in the phase space of the system. The concept of ”displaced reinjection” discussed by Rüssler is crucial in interpreting the phase-space portraits. At this time, we do not know whether the aperiodic behaviours seen are a reflection of chaotic dynamics in the system …

Evolution processes described adequately by autonomous fourth order ODE’s are examined by constructive analytical methods, which permit qualitative and quantitative comparisons with experimentally observed or numerically computed data. The study is restricted to ODE’s possessing at least one static equilibrium point with two pairs of complex conjugate eigenvalues, whose real parts vanish or have absolutely small values. Since the ratio of the imaginary parts is then as rule an irrational number, any linearized oscillation near that fixed point contains two incommensurable frequencies. In the presence of nonlinearities steady states are either periodic or more complex. The latter provide specimens for a ”chaos taxonomy” …

Vertebrate neural systems have a modular, hierarchical anatomical structure, which is viewed as providing a hierarchical motor control or sensory analysis function. If each level in such a structural-functional hierarchy is considered as a dynamical system, its behaviour may be classified in a dynamical hierarchy, from equilibria, through periodicity, quasi-periodicity to chaos and higher order structured irregularities. These stuctural-functional and dynamical concepts of hierarchy may be combined if the spatio-temporal patterning of neural activity is considered. Spatio-temporal behaviour in a system which may be represented by a nonlinear partial differential system is often dominated by only a few degrees of freedom, and so may be approximated by nonlinear systems of low dimensionality. The nervous system is spatially extensive, but is a hierarchy of interacting subsystems. Each subsystem may be approximated by a nonlinear system of low dimensionality. Such a hierarchical network of coupled interacting subsystems generates a richness of patterned and irregular behaviours that is coherent over several levels and is determined by the network connectivity rather than the component and subsystem dynamics. The behaviour itself has a hierarchical structure, whose elements are not a simple mapping of the elements of the structural-functional or dynamical hierarchies …

We discuss some examples of chaos and hyperchaos in chemically reacting systems which require four dimensions for their description. A few phenomena which can arise in model systems of four ordinary differential equations are first described. We then discuss some experimental results from electrochemical reactions, viz., the route to chaos via period doubling of tori and a chaos of apparent higher dimension …

Periodic orbits in the neighborhood of heteroclinic trajectories in the Chua oscillator are treated. These trajectories are found in “periodic windows” of the parameter space. A numerical scan of the α-β parameter plane using a 1-D map approximation of the continuous flow indicates that all periodic windows of the above type have the shape of a swallow tail and both their linear dimensions and their mutual distances show a scaling property. The qualitative behavior inside the periodic windows is presented in terms of the full two-dimensional “transfer and return maps” induced by the flow of the system. The canonical parameters of these maps also directly provide a first approximation to the scaling factors of the swallow tails …

A brief review on the climbing-up of dynamical hierarchy is presented, inspired by Otto Rössler’s philosophy. The hierarchy rises from chaos to hyperchaos, and then to (hyper)^∞ chaos. (Hyper)^∞ chaos is studied in coupled map lattices as a model for spatiotemporal chaos. A further step of hierarchy from (hyper)^∞ chaos leads to the hierarchical dynamics, as is discussed in globally coupled maps. This hierarchical dynamics is suggested to be relevant to neural nets, the economics, and the evolution. As a next step of hierarchy, chaotic itinerancy is discussed in globally coupled maps. It leads to a dynamical system whose effective degrees of freedom changes with time. Possible relation of this dynamics with coherent structure in turbulence is discussed. As a further step of hierarchy, the evolutionary dynamical process is discussed. The dynamics of punctuated equlibrium may be related with chaotic itinerancy. However, other essentially novel dynamics is necessary to include the novelty pressure and genetic fusion. How high should we climb up the dynamical hierarchy to reach intelligence?.

Dynamical systems with chaotic properties have been studied in both discrete and continuous time domains. Both approaches have advantages in demonstrating the mechanisms which generate chaotic motion. In this paper we discuss an example of each type and how they can be connected through smooth suspensions. We focus on the similarity in the underlying dynamics of a class of continuous flows in three-space and a class of iterated mappings in the plane.

In particular, we examine the geometrical similarities between the dynamics of a certain class of suspended Cremona transformations and the dynamics of Rössler’s Band. They both are characterized by Rössler’s ”walking-stick diffeomorphisms”.

As the first step on his ladder to higher chaos, Rössler shows how the walking-stick diffeomorphism can be naturally extended to the three-dimensional folded-towel diffeomorphism. We present a version of the folded-towel diffeomorphism for which the different stages of the transformation can be studied explicitly. With the help of interactive computer graphics, we have analyzed the manifold structure as well as local transformations close to the attractors of these chaotic dynamical systems …

Nonlinear recursive relations are examined under variation of a control parameter between two values A and B. Different periodic sequences (e.g. ABAB…, AABAB AABAB…, A⁶ B⁶ A⁶B⁶…) as well as random sequences are assumed. The graphical representation of the Lyapunov exponent on the A–B-plane yields an overwhelming richness of structures and allows the straightforward identification of several interesting features: multiple oscillatory states, self-similar periodic regions, closed and crossing superstable lines, order arising counterintuitively for processes which are chaotic for A alone and for B alone, and chaos at surprisingly low parameter values (”early chaos”) …

The purpose of this paper is to show that a chaotic hierarchy appears also by consideration of the simplest one dimensional map T(x→x² –λ), x being a real variable, λ a real parameter. This hierarchy results from the fractal properties of the box-within-a-box bifurcations structure of T, inducing different levels of self similarities on the x-axis, when repulsive cycles merge into critical points, and accumulations of such situations.

By embedding T into a two-dimensional diffeomorphism, one obtains a transfer of this hierarchy into a wider class of problems …

Biochemists generally assume that the metabolic state of the cell is a stable steady state, although they also acknowledge that such a steady state may change with time. However, in early 1976 Otto Rössler demonstrated that simple models of chemical and biochemical reactions may exhibit dynamics in which metabolites fluctuate wildly in time in an apparently random fashion with no recurrency. Similar behavior was reported a decade earlier by Edward Lorenz in his studies of a simple model of the weather. Chaos as this irregular behavior was coined soon became a popular research topic in almost all scientific disciplines. Shortly after its discovery in chemical models, chaos was observed experimentally in both biochemical and chemical reactions. The peroxidase–oxidase reaction, in which reduced nicotinamide adenine dinucleotide (NADH) is oxidised by molecular oxygen, was the first biochemical system to exhibit chaotic behavior. Although only a single enzyme, peroxidase, is involved, the reaction has similar regulatory properties as many biochemical pathways, which involve many enzymes, and hence it may be considered as a simple model of cell metabolism. We describe a few of the many dynamic behaviors found experimentally and in models of the peroxidase–oxidase reaction. On the basis of these observations we propose that a stable steady state is the least likely dynamic possibility of cell metabolism …

Different kinds of nonlinear dynamical behavior ranging from a nonoscillatory up to a hyperchaotic state are reviewed for a semiconductor experiment. The affirmed chaotic hierarchy can be attributed to nonlinear coupling between competing localized oscillation centers intrinsic to the multicomponent system considered …

In this contribution a new approach to calculate the Lypunov exponents from experimental time series is discussed. This approach is shown to give more direct information on the dynamics of the underlying system while allowing a faster calculation with a smaller amount of data than the implementations used so far, such that a new class of physical systems can now be treated. The algorithm is applied to simulated data of some well known nonlinear systems. It is hoped that this algorithm will be of particular help in the evaluation of Lyapunov exponents from small time series or higher dimensional embedding spaces and in modelling experimental systems …

Being based on the observation that a conjugate pair of representations, or dual logic, is a necessity under the presence of chaos, a new interpretation of quantum theory is proposed as describing proto-chaos. This chaos has to be a result of basic nonlinearity in the dynamic structure, of which, however, the nonchaotic phase seems to lie outside the reach of experimental technique, thus the term proto-chaos. Nevertheless, assuming no extra degrees of freedom, the interpretation clarifies a number of riddles posed hitherto and throws some light on the overall hierarchical structure of our understanding and description of nature …

The qualitative dynamical behavior of more than three-dimensional dynamical systems is still beset with unknowns. Four specific questions are posed, two about dissipative and two about conservative systems. They concern the existence of nowhere differentiable attractors, self-similar nowhere differentiable basin boundaries and two different types of unit-cell boundaries, respectively …

The following sections are included:

• O.E. RÖSSLER PUBLICATION LIST ON CHAOS