Nonlinear Dynamical Systems Theory Research Articles

Discrete analytical solution of a smooth non-linear optimal control problem

With certain smoothness assumptions, conditions are established under which solutions of an optimal control problem are also solutions of the system of differential controls of the object and the control law, and vice versa. This equivalence is used to formulate and prove theorems that define, in explicit analytical form, discrete solutions of non-linear optimal control problems and problems in the qualitative theory of non-linear dynamic systems.

Neurobiological approach to computing devices

According to the old metaphor of classical cybernetics the brain can be considered as a computer. Newer theoretical endeavours reverse the question and ask: what could neurobiology offer to engineers of near-future generation computer systems? Three not completely disjoint abstract functions of the nervous system, namely pattern formation, pattern recognition and action, can be treated in a unified conceptual framework. Storage and retrieval mechanisms of information are connected to fault-tolerant, adaptive parallel structures. “Learning” and “plastic behaviour” are interpreted in terms of the theory of non-linear dynamic systems. As neural development and plasticity can be approached by deterministic models superimposed by random influence, noise might also have a positive role to play during the operation of technical computing devices. Molecular computation is discussed in relation to eventual hardware realization of “neurobiology-based” computers.

Non-linearities in laser diodes

Non-linear operating characteristics of semiconductor lasers are described. It is shown that understanding of such features is deepened when use is made of techniques and results from the theory of non-linear dynamical systems. The potential for practical exploitation of the non-linear properties of laser diodes is also examined.

Chaotic dynamics in brain activity

The aim of this paper is to report on a new attempt at characterizing the electroencephalogram (EEG), which is based on recent progress in the theory of nonlinear dynamical systems (Brandstater et al. 1983; Nicolis and Nicolis 1984, 1986; Babloyantz et al. 1985). The method is independent of any modeling of brain activity. It relies solely on the analysis of data obtained from a single-variable time series. From such a “one-dimensional” view of the system, one reconstructs the X k (where k = 1,…, n) variables necessary for the description of systems dynamics. With the help of these variables, phase-space trajectories are drawn. Provided that the dynamics of the system can be reduced to a set of deterministic laws, the system reaches in time a state of permanent regime. This fact is reflected by the convergence of families of phase trajectories toward a subset of the phase space. This invariant subset is called an “attractor.”

Complex quasiperiodic and chaotic states observed in thermally induced oscillations of gas columns.

Highly nonlinear phenomena are observed in Taconis oscillations which are spontaneous oscillations of gas columns thermally induced in a tube with steep temperature gradients. Near the overlapped region and the intersection of the stability curves for two different modes with incommensurate frequencies, we find that both modes can be excited simultaneously and competition between them leads to complex quasiperiodic and chaotic states. Experimental data are anlayzed by use of recent methods from the theory of nonlinear dynamical systems.

CORTISIM: A complex dynamical model of the cortisol regulation system

The concept of complex dynamical systems is an outgrowth of the mathematical theory of nonlinear dynamical systems, devel oped in the context of physiological modeling. This paper de scribes an exemplary application of this new modeling strategy to the cortisol regulation system, along with some preliminary results of its simulation. The eventual intent of this project is the creation of a research tool for endocrine physiology.

Non-linear dynamical systems theory and locational analysis

Recent developments in the application of dynamical systems theory to locational problems are reviewed. Application areas include retailing, agricultural and industrial location, housing and an integrated land-use package. Illustrations of numerical simulations are presented. A number of issues that arise from the use of the associated models are also discussed.

Can a solar pulsation event be characterized by a low-dimensional chaotic attractor?

Using the theory of nonlinear dynamical systems a time-series analysis of a pulsation event in solar radio emission suggests that there exists a low-dimensional attractor. The power spectrum cannot be interpreted as a superposition of periodic components. Estimates of the maximum Lyapunov exponent and the Kolmogorov entropy give some hints to deterministic chaos. Consequences for the physical modelling of the event are discussed.

Is there a climatic attractor?

Much of our information on climatic evolution during the past million years comes from the time series describing the isotope record of deep-sea cores. A major task of climatology is to identify, from this apparently limited amount of information, the essential features of climate viewed as a dynamic system. Using the theory of nonlinear dynamic systems we show how certain key properties of climate can be determined solely from time series data.

A Probabilistic Theory of Nonlinear Dynamical Systems Based on the Cell State Space Concept

Developed in the paper is a probabilistic theory for nonlinear dynamical systems. The theory is based on discretizing the state space into a cell structure and using the cell probability functions to describe the state of a system. Although the dynamical system may be highly nonlinear the probabilistic formulation always leads to a set of linear ordinary differential equations. The evolution of the probability distribution among the cells can then be studied by applying the theory of Markov processes to this set of equations. It is believed that this development possibly offers a new approach to the global analysis of nonlinear systems.