A number of 3-variable chemical and other systems capable of showing 'nonperiodic' oscillations are governed by walking-stick shaped maps as Poincare cross-sections in state space. The 2-dimensional simple walking-stick diffeomorphism contains the one-dimensional 'single-humped' Li-Yorke map (known to be chaos producing) as a 'degenerate' special case. To prove that chaos is possible also in strictly 2-dimensional walking-stick maps, it suffices to show that a homoclinic point (and hence an in finite number of periodic solutions) is possible in these maps. Such a point occurs in the second iterate at a certain (modest) 'degree of overlap' of the walking-stick map. At a slightly larger degree, a 'nonlinear horseshoe map' is formed in the second iterate. It implies presence of periodic trajectories of all even periodicities (at least) in the walking-stick map. At the same time, two major, formerly disconnected, chaotic subregimes merge into one. Diagnostic criterion: presence of 'syncopes' in an otherwise non-monotone sequence of amplitudes.
An example of a chemical reaction system producing a type of oscillation that is close to a quasi-periodic oscillation is presented. Such system may help clarify the relationship between quasi-periodicity and chaos.
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.
Three types of abstract chemical reaction systems are described: 1. The generalized catalytical system, 2. the generalized autocatalytical system, 3. a spontaneously evolving chemical system. The significance of the second and third system for a very early phase of pre-biological evolution is discussed.
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.
Nonperiodic oscillation ("chaos"), formerly found in several 3-variable homogeneous abstract reaction systems, is also possible in 2-morphogen compartmental systems. In a 2-cellular (and hence 4-variable) symmetrical morphogenetic system of Rashevsky-Turing type, a nonperiodic return toward an (almost) undifferentiated state is observed under numerical simulation. The system hereby shows a novel, "bi-chaotic" (rather than bistable), type of behavior. The 2 chaotic regimes are of the screw type each. They are separated by a symmetrical saddle-limit cycle. This behavior is preserved under a "contraction" of the system to 3 variables. A second bichaotic mode (2 spiral-type chaotic regimes separated by a symmetrical steady state) is also possible. A preliminary result on a third of chaos is also presented. Thus, "turbulence" (or chaos) may be a general behavioral possibility of interaction-type morphogenetic systems.
Different types of chaotic flow are possible in the 3-dimensional state spaces of two simple non-linear differential equations. The first equation consists of a 2-variable, double-focus subsystem complemented by a linearly coupled third variable. It produces at least three types of chaos: Lorenzian chaos, "sandwich" chaos, and "horseshoe" chaos. Two figure 8-shaped chaotic regimes of the latter type are possible simultaneously, running through each other like 2 links of a chain. In the second equation, a transition between two different types of horseshoe chaos (spiral chaos and screw chaos) is possible. While sandwich chaos allows for a genuine strange attractor, the same has not yet been demonstrated for horseshoe chaos. Unlike the situation in the analogous 1-dimensional case, an emergent period-3 solution is not necessarily stable in the horseshoe. Since chaos is a "super-oscillation" (emergent with the third dimension), the existence of "super-chaos" is postulated for the nect level.
A six-minute, super-8 sound film, demonstrating the different behavioral modes and their bifurcations in the 2 equations, has been prepared. Chaos sounds as musical as a snore.