Otto E. Rössler

An Equation for Hyperchaos

Abstract A simple four-variable oscillator containing but one quadratic term produces a higher form of chaos with two (rather than one) directions of hyperbolic instability on the attractor. The topology is simple, a model difference equation complicated. Turbulence may be hyperchaos of high order.

CONTINUOUS CHAOS—FOUR PROTOTYPE EQUATIONS

If oscillation is the typical behavior of 2-dimensional dynamical systems (Euclidean and on manifolds), then chaos, in the same way, characterizes 3-dimensional continuous systems. First a method t o obtain chaos in degenerate (relaxation type) dynamical systems in two variables is outlined whereby five basic flow patterns emerge. Second, following a piecewise linear degenerate equation, four prototypically simple quadratic differential equations in three variables that realize nondegenerate analogs of those five flows are presented. Finally a possible equation for an even higher type of qualitative behavior beyond chaos is proposed.

Chaotic Behavior in Simple Reaction Systems

Deterministic nonperiodic flow (of “chaotic” or “strange” or “tumbling” type, respectively) was first observed, in a 3-component differential system, by E. N. Lorenz in 1963. A 3-component abstract reaction system showing the same qualitative behavior is indicated. It consists of (1) an ordinary 2-variable chemical oscillator and (2) an ordinary single-variable chemical hysteresis system. According to the same dual principle, many more analogous systems can be devised, no matter whether chemical, biochemical, biophysical, ecological, sociological, economic, or electronic in nature. Their dynamics are determined by the presence of a “folded” Poincaré map. Under numerical simulation, the proposed chemical system provides an almost ideal illustration to the underlying dynamical prototype, the “3-dimensional blender”. Thus, continuous Euklidean dynamics (and with it chemical kinetics) proves to be of equal interest in studying chaos as discrete dynamical systems already have.

The Chaotic Hierarchy

Abstract 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 second 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.

Chaos in the Zhabotinskii reaction

THE Belousov–Zhabotinskii reaction is a chemical Bonhoeffer–van der Pol circuit, that is, a relaxation oscillator that can be run as both an astable and a monostable ‘flip-flop’1–3. Apparently the reaction also belongs to the slightly more complicated class of‘universal circuits’4 as introduced by Khaikin5–6. Oscillators of this type not only show ‘smooth’ and ‘relaxation type’ oscillations5–6, but also ‘chaotic’ oscillations4. As evidenced by the simplest equation of this type7, both ‘spiral type’ and ‘screw type’ chaos20 are possible in such systems. We present here preliminary evidence for the occurrence of screw-type chaos in the Zhabotinskii reaction.

A robust, locally interpretable algorithm for Lyapunov exponents

Abstract An enhanced version of the well known Wolf algorithm for the estimation of the Lyapunov characteristic exponents (LCEs) is proposed. It permits interpretation of the local behavior of non-linear flows. The new variant allows for reliable calculation of the non-uniformity-factors (NUFs). The NUFs can be interpreted as standard deviations of the LCEs. Since the latter can also be estimated by the Wolf algorithm, however, without local information on the flow, the new version ensures local interpretability and therefore allows the calculation of the NUFs. The local contributions to the LCEs which we call “local LCEs” can at least be calculated up to three dimensions. Application of the modified method to a hyperchaotic flow in four dimensions shows that an extension to many dimensions is possible and promises new insight into so far not fully understood high-dimensional non-linear systems.

Chemical Automata in Homogeneous and Reaction-Diffusion Kinetics

A finite automaton or, synonymously, a finite state machine is in the simplest case a triple (X,I,λ), whereby X is a finite set of states, I is a finite set of inputs, and λ is the next-state mapping, such that λ : X x I → X (cf. Arbib, 1969). For example, if x1 and x2 are two state variables each possessing two possible states, the whole automaton has four possible states, and λ specifies the transitions between these states in dependence on a given input. If the input is constant, one speaks of an autonomous automaton, otherwise of a nonautonomous automaton .

Horseshoe-map chaos in the Lorenz equation

Abstract The Lorenz equation of turbulence-generation and irregular laser spiking shows at least 3 types of chaos: Lorenzian chaos, horseshoe map chaos of the walking-stick type, and “original” horseshoe map chaos. It is the first 3-variable differential equation that realizes Smales horseshoe map.

Does a Centralized Clock for Ageing Exist

It is proposed that a centralized clock controlling ageing is located in the pineal gland with the calcification process occurring there providing a highly accurate bio-inorganic timing mechanism and the secreted melatonin carrying a signal to all cells in the organism. An explicit programme of data gathering and experiments suitable for the falsification of the proposal, which is consistent with presently known anatomical and physiological facts, is presented. The underlying motivation comes from evolutionary biology and the invariance, that is the allometry, of life expectancy curves.

Deductive biology—Some cautious steps

For certain environments, the Darwinian model allows unique prediction of a function that any surviving system adapted to such an environment has to perform. This is the case for those environments that determine a “survival functional” of position in space-time of known shape. Purely temporal survival functionals can be distinguished from spatial and mixed ones. In each case, there exists an optimum path in combined physical and (reduced) metabolic space. Dependent on the admissible error, approximate solutions of different complexity are sufficient. All solutions possess an afferent, a central, and an efferent part. Within this general frame, specific, “probably simplest”, solutions are proposed for adaptive chemotaxis, insect locomotion, lower vertebrates locomotion, higher vertebrates locomotion, chronobiological systems, and immune systems, respectively—or rather, for the underlying functionals.