Research for Dynamics

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Land Warfare and Complexity, Part II: An Assessment of the Applicability of Nonlinear Dynanics and Complex Systems Theory to the Study of Land Warfare
July 1, 1996
The Commanding General, Marine Corps Combat Development Command (MCCDC) asked the Center for Naval Analyses to assess the general applicability of the new science to land warfare. 'New Sciences' is a catch-all phrase that refers to the tools and methodologies used in nonlinear dynamics and complex systems theory to study physical dynamical systems exhibiting a 'complicated dynamics.' This report concludes that the concepts, ideas, theories, tools and general methodologies of nonlinear dynamics and complex systems theory show enormous, almost unlimited, potential for not just providing better solutions for certain existing problems of land combat, but for fundamentally altering our general understanding of the basic processes of war, at all levels. Indeed, the new sciences' greatest legacy may, in the end, prove to be not just a set of creative answers to old questions but and entirely new set of questions to be asked of what really happens on the battlefield. The central thesis of this paper is that land combat is a complex adaptive system. That is to say, that land combat is essentially a nonlinear dynamical system composed of many interacting semi-autonomous and hierarchically organized agents continuously adapting to a changing environment. See also CIM 461.10.
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Land Warfare and Complexity, Part I: Mathematical Background and Technical Sourcebook
July 1, 1996
The purpose of this paper is to provide the theoretical framework and mathematical background necessary to understand and discuss the various ideas of nonlinear dynamics and complex system theory to plant seeds for a later, more detailed discussion (provided in Part II of this report) of how these ideas might apply to land warfare issues. This paper is also intended to be a general technical sourcebook of information. The main idea put forth in this paper is that significant new insights into the fundamental processes of land warfare can be obtained by viewing land warfare as a complex adaptive system. That is to say, by viewing a military 'conflict' as a nonlinear dynamical system composed of many interacting semi-autonomous and hierarchically organized agents continuously adapting to a changing environment. See also CRM 96-68.
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OSCILLATIONS, FLUCTUATIONS, AND THE HOPF BIFURCATION
February 1, 1981

This paper considers the effects of small random perturbations on deterministic systems of differential equations. The deterministic systems of interest have oscillatory dynamics and may undergo a bifurcation (the Hopf bifurcation). A first exit problem for experiments beginning near stable and unstable limit cycles is formulated. The unstable limit cycle is surrounded by an annulus. Of interest is the probability of first exit from the annulus through a specified boundary, conditioned on initial position. The diffusion approximation is used, so that the conditional probability satisfies a backward diffusion equation. Appropriate solutions on the backward equation are constructed by an asymptotic method. The behavior of the stochastic system in the vicinity of stable and unstable limit cycles is compared. When the deterministic system exhibits the Hopf bifurcation, the above analysis must be modified. Uniform solutions of the backward equation are constructed. The solutions are analogous to Hadamard's solution of the point source problem for the wave equation. Numerical examples are used to compare the theory with Monte Carlo experiments.

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UNIFORM TREATMENT OF FLUCTUATIONS AT CRITICAL POINTS
February 1, 1981

A generalized critical point is characterized by the vanishing of certain linear relationships. In particular, the dynamics near such a point are completely non-linear. This paper analyzes fluctuations at such points of spatially homogeneous systems. Thermodynamic critical points as a special case are discussed, but the main emphasis is on stochastic kinetic equations. It is shown that fluctuations at a critical point cannot be characterized by a Gaussian density, but more sophisticated densities yield reasonable results. The theory is applied to the critical harmonic oscillator.

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FRANCK-CONDON THEORY OF CHEMICAL DYNAMICS. VI. ANGULAR DISTRIBUTIONS OF REACTION PRODUCTS
February 1, 1981
The Franck-Condon model and additional approximations are applied in this paper to determine product angular distributions of the type A + BC -> AB + C. The interest is in the relation of the anular distribution to certain features of the potential energy surface, the possibility of finding reduced variables, that is, scaling factors, with which to compare chemical reactions and the issue of the independence of angular from energy distributions of reaction products.
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