Gottfried Mayer-Kress

A Nonlinear Dynamical Model of the Impact of SDI on the Arms Race

We present numerical results from a nonlinear dynamical model with discrete time that simulates the implications of ballistic missile defense systems (SDI) on the arms race between the two superpowers. As dynamical variables we introduce the number of intercontinental ballistic missiles (ICBMs), antiballistic missile systems (ABMs) and anti-ABM systems such as antisatellite weapons (ASAT) of each of the two sides. The time evolution of these systems (arms race) is simulated numerically under various parameter assumptions (scenarios). The a priori unpredictability of human decisions is simulated through random fluctuations of the buildup parameters. The results of our idealized model indicate that for most parameter combinations, the introduction of SDI systems leads to an extension of the offensive arms race rather than a transition to a defense-dominated strategic configuration. A reduction in the number of offensive weapons, that is, an approach to a defense-dominated strategy, was observed if either the number of reentry vehicles per ICBM (MIRV) is limited to much smaller values than presently realized or if the accuracy of offensive weapons is significantly reduced. For the case of a strongly accelerated arms buildup (either offensive or defensive), we observe a loss of stability of the solutions that we interpret as a transition to unpredictable chaos. We also incorporate a discussion of economic and risk parameters, both of which also tend to increase with the introduction of SDI systems.

Dimensional analysis of heart rate variability in heart transplant recipients

Abstract Periodicities in the heart rate have been known for some time. We discuss these periodicities in normal and transplanted hearts. We then consider the possibility of dimensional analysis of these periodicities in transplanted hearts and problems associated with the record.

Bifurcations and Intrinsic Chaotic and 1/f Dynamics in an Isolated Perfused Rat Heart

The application of the theory of chaotic dynamical systems has gradually evolved from computer simulations to assessment of erratic behavior of physical, chemical, and biological systems. Whereas physical and chemical systems lend themselves to fairly good experimental control, biologic systems, because of their inherent complexity, are limited in this respect. This has not, however, prevented a number of investigators from attempting to understand many biologic periodicities. This has been especially true regarding cardiac dynamics: the spontaneous beating of coupled and non-coupled cardiac pacemakers provides a convenient comparison to the dynamics of oscillating systems of the physical sciences. One potentially important hypothesis regarding cardiac dynamics put forth by Goldberger and colleagues, is that normal heart beat fluctuations are chaotic, and are characterized by a 1/f-like power spectrum. To evaluate these conjectures, we studied the heart beat intervals (R wave toR wave of the electocardiogram) of isolated, perfused rat hearts and their response to a variety of external perturbations. The results indicate bifurcations between complex patterns, states with positive dynamical entropies, and low values of fractal dimensions frequently seen in physical, chemical and cellular systems, as well as power law scaling of the spectrum. Additionally, these dynamics can be modeled by a simple, discrete map, which has been used to describe the dynamics of the Belousov-Zhabotinsky reaction.

Time Evolution of Local Complexity Measures and Aperiodic Perturbations of Nonlinear Dynamical Systems

We discuss numerical algorithms for estimating dimensional complexity of observed time-series with special emphasis on biological and medical applications. Factors which enter the procedure are discussed and applied to local estimates of pointwise dimensions or crowding indices. We illustrate the concepts with the help of experimental time-series obtained from speech signals. The temporal evolution of the crowding index shows oscillations which can be correlated with properties of the time-series. We compare the time evolution of the dimensional complexity parameter with the original time-series and also with recurrence plots of the embedded time series.

Observations of the fractal dimension of deep-and shallow-water ocean surface gravity waves

Abstract The fractal dimension of field measurements of time series of sea surface elevation (ocean waves) was calculated. For a range of oceanic conditions, the fractal dimension estimates of deep water waves did not show low values. Although the mutual information content of waves approaching a beach increased with decreasing water depth, the corresponding fractal dimension was practically independent of depth. The observed dimensions are not statistically different from the dimension of a linear process with the same power spectrum as the field data, although the shallow water data are known to be nonlinear.

An explicit construction of a class of suspensions and autonomous differential equations for diffeomorphisms in the plane

From a large class of diffeomorphisms in the plane, which are known to produce chaotic dynamics, we explicitly construct their continuous suspension on a three dimensional cylinder. This suspension is smooth (C1) and can be characterized by the choice of two smooth functions on the unit interval, which have to fulfill certain boundary conditions. For the case of entire Cremona transformations, we are able to construct the corresponding autonomous differential equations of the flow explicitly. Thus it is possible to relate properties of discrete maps to those of ordinary differential equations in a quantitative manner. Furthermore, our construction makes it possible to study the exact solutions of chaotic differential-equations directly.

Chaos versus predictability in formulating national strategic security policy

The relation between chaos and predictability in deterministic theories, as developed and used in physical models, is also useful in studying important nonphysical problems such as that of international security. In the latter case, the well‐known transition from laminar to turbulent flow is a heuristic analogy to the transition from cold to hot war. A simplified procurement model for the Strategic Defense Initiative (SDI) is presented, which can be used to determine the outcome of various deployment modes. A specific outcome, even if desirable from the point of view of a transition from an offensive to a defensive strategy, may represent a dangerous crisis instability—conducive to the outbreak of war—if it occurs in the model’s chaotic regime. Some numerical examples of the various possibilities are given.

Chaotic attractors of a locally conservative hyperbolic map with overlap

Abstract We introduce a class of hyperbolic maps which are locally conservative but globally dissipative. The dissipation stems not from local contraction of phase space but from the overlap that arises from the noninvertibility of the map. The dynamics of these maps are chaotic, and the attractors are strange, although not necessarily fractal. Though several examples of this class are provided, our analysis concentrates on a single example which is obtained as a modification of the bakers transform. We discuss the morphology of the attractors as the map parameter is varied. We find that some attractors are “thin” fractals and others are “fat” sets of positive measure. We are able to characterize the parameter values that lead to these two kinds of attractors, and for certain parameter values, we can derive analytically exact values for the fractal dimension or for the fat fractal scaling exponent. Taken together, these results lead us to conjecture that the generic attractor of this map has zero measure and full dimension.

Chaotic Heart Rate Dynamics in Isolated Perfused Rat Hearts

The application of the theory of chaotic dynamical systems has gradually evolved from computer simulations to assessment of erratic behavior of physical, chemical, and biological systems (Rossler, 1976; Swinney, 1983; Glass, et al., 1983; West, et al., 1986, Zbilut, et al., 1988). Whereas physical and chemical systems lend themselves to fairly good experimental control, biological systems are limited in this respect. Indeed, investigations in this area are often restricted by constrained experimental designs necessary for laboratory control, or ethical concerns regarding human subjects (Koslow, et al., 1987). Moreover the transitions from the behavior of discrete units such as cardiac cells to synergistic cooperation and self-organization as in a heart have not yet been fully explored (Haken, 1982). To overcome some of these lacunae, we studied the heart beat intervals (R wave to R wave of the electrocardiogram) of isolated, perfused rat hearts and their response to a variety of external perturbations. We have chosen the interbeat interval as a primary observable because we are interested in the long time dynamics of the heart as opposed to the temporal pattern of individual heart beats, which have been the subject of clinical studies for some time. The results indicate bifurcations between complex patterns, states with positive dynamical entropies and low values of fractal dimensions frequently seen in physical, chemical and cellular systems. The fact that these chaotic dynamical states can be elicited by global forcing situations and external perturbations suggest a hierarchical self-similarity in the organization of the heart.

Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations

Abstract The relation between the parameters of a differential equation and corresponding discrete maps is becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. An iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODEs with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is a polynomial in the state variables and if the ODE has a fixed point at the origin. For several nonlinear systems approximations of different orders are investigated.