In recent years heterogeneous catalytic reaction systems have been shown to produce a complex dynamic behavior. For example the heterogeneous catalytic oxidation of ethanol on palladium was found to exhibit multistability as well as relaxation oscillations (Jaeger., 1986, Ber. Bunsenges. phys. Chem.90, 1075–1079; 1990, Chem. Engng Sci.45, 947–953). The reaction provides the clear experimental identification of a global Hopf bifurcation and so-called overlapping states near the transition point. In this paper a dynamic model will be developed which is based on these experimental facts. The model describes all instabilities observed and predicts the corresponding transitions in agreement with the observations.

Experimental bifurcation diagrams for the heterogeneous catalytic oxidation of ethanol on a supported palladium catalyst are presented, depending on the oxygen or the ethanol content in the feed as control parameters. The identification of the bifurcations leads to a proposal for the formal construction of a mathematical model following Sheintuch and Luss (1985, Chem. Engng Sci. 40 , 1653–1664; 1987a, Chem. Engng Sci. 42 , 41–52; 1987b, Chem. Engng Sci. 42 , 233–243). The identification of a global Hopf bifurcation in the experimental system requires an extension of their model.

The concepts of entropy and lowering of entropy in comparison to the equilibrium state are introduced. It is shown that entropy lowering is connected with the contraction of the occupied part of the phase space due to the formation of attractors. As concrete examples the excitation of oscillations in solids and turbulence in liquid flows are discussed.

The entropy lowering is calculated in the laminar as well as in the turbulent regime. For the Poiseuille flow it is shown that the entropy lowering increases monotonically if while increasing the Reynolds number the energy of the flow is kept fixed. The transition from laminar to turbulent motion turns out to be a nonequilibrium transition to a more ordered state.

We investigate the effect of additive colored noise on Hopf-bifurcating systems in the limit of small correlation times. It is shown that it results in an advancement of the oscillating regime. Several examples are studied.

The Brownian motion of nonlinear oscillators imbedded in a heat bath is studied. For the excitation of sustained oscillations, realized by van der Pol oscillators, the entropy is calculated in the limit of weak dissipation and weak noise. At fixed average energy the entropy decreases monotonically if the feedback strength increases.

The behaviour of the entropy is studied during a nonequilibrium phase transition. It is shown that for the generation of sustained oscillations the entropy decreases monotonously if the average oscillator energy remains fixed. This result is valid both for hard and soft excitation of oscillations.

Thermodynamics is a rather old discipline of physics, however, it is not oldfashioned. On the contrary,such modern topics as the hot big bang model, the theory of black holes, as well as the theory of biological systems /1,2/, show that thermodynamics goes through a renaissance. Thermodynamics is also intimately related to information theory, a key discipline for the study of selforganization and evolution /3,4/. The very origin of this discipline is closely connected with thermodynamical reasoning, as shown in the fundamental papers of SZILARD (1929), SHANNON (1948) and BRILLOUIN (1956). Thus STRATONOVICH, one of the pioneers of several branches of modern information theory, writes that thermodynamics and statistical physics are the cement which hold together the disciplines forming modern information theory /5/. Besides the informational aspects also the direct consideration of thermodynamic functions and of the entropy production is of much interest for the study of selforganization processes /6,7/. The structures created in the process of selforgánization are often called “dissipative structures” /6-8/. Besides this term, which underlines the aspect of dissipation, we shall also use the term “autostructures”, which underlines the aspect autonomy /9/. The term autostructure is a generalization of well-known terms as “autooscillations” and “autowaves” /10/.

It is well known that under appropriate conditions nonlinear physical, chemical or biological systems show various instabilities such as, for example, transitions leading to multistability, limit cycle oscillations or chaotic motion [1–3]. These instabilities correspond to bifurcations of the underlying dynamic equations. A realistic treatment requires one to take into account fluctuations, which appear as thermal fluctuations or are due to the discrete change of particle numbers in nonlinear chemical reactions. Another origin of noise may be the influence of the environment—which in general varies more or less randomly—on the thermodynamically open systems. Usually the description of complex systems, e.g. biochemical oscillations, is restricted to simplifying models which represent the observed behaviour only qualitatively. Then fluctuations may appear as a result of “hidden” reactions neglected in the model.

Several entropy concepts (Clausius-, Shannon-, Boltzmann-, Kolmogorov-entropies) are discussed. The lowering of thermodynamic entropy with respect to an equilibrium system at the same energy is introduced as an operational measure of order. It reflects the formation of attractors (selforganization) on the energy shell. The conditions of selforganization in the Universe are discussed.

The influence of external fluctuations on the bifurcational behavior of two-dimensional dynamical systems exhibiting limit cycles is investigated. Studying both exactly and approximately solvable examples it is shown that the variances of the external fluctuations occur as additional bifurcation parameters. The threshold values for soft as well as for hard self-excitation of oscillations are affected by the external fluctuations. To classify bifurcations of dynamical systems in the presence of fluctuations some aspects of catastrophe theory are applied to the corresponding stationary probability distributions.