Characterization Of Strange Attractors Research Articles

Irrelevance of the fractal dimension term in the fractal attrition equation

The modified Fractal Attrition Equation (mFAE) models the casualties produced by Map Aware Non-uniform Automata (MANA), an agent based combat modelling distillation, at each MANA time step. The mFAE has three important differences from the simple early twentieth century models of casualties that used only the numbers on each side to make predictions. Firstly, only those agents within range of the enemy may inflict casualties. Secondly, the detection range is assumed to be greater than the range of weapons and a fitting factor assumed to represent the gap between these two ranges, is introduced. This builds in one of the assumptions of Network Centric Warfare, that you will be able to see your enemies before they can shoot at you. Thirdly, and novelly, a fractal dimension is introduced. We postulate that the important part of the information used in calculating the fractal dimension has already been incorporated into the model through the consideration of range. We test this hypothesis by comparing the outcomes of the mFAE with and without the fractal term on three scenarios: that used by the developers of the mFAE; best practice MANA tactics from the literature; and a rout scenario. When the two models are scaled to fit the MANA casualties there is no significant difference in fit. We conclude that the fractal term in the mFAE is redundant. References J. V. Chase, Sea fights, a mathematical investigation of the effect of superiority of force in. RG 8, Box 109, XTAV(1902), Naval War College Archives. F. W. Lanchester, Aircraft in warfare: the dawn of the fourth arm---No. V, The principle of concentration, Engineering 98: 422--423, (1914). http://www.archive.org/details/aircraftinwarfar00lancrich N. J. MacKay, Lanchester combat models, Mathematics Today, 42, pp.170--173, 2006. http://arXiv.org/abs/math/0606300v1 T. W. Lucas and T. Turkes Fitting Lanchester models to the battles of Kursk and Ardennes. Nav Res Log 51: 95--116 (2004). G. C. McIntosh and M. K. Lauren, Incorporating fractal concepts into equations of attrition for military conflicts, Journal of the Operational Research Society, 2(59): 703--713, (2008). doi:10.1057/palgrave.jors.2602383 M. K. Lauren Firepower concentration in cellular automaton combat models---an alternative to Lanchester, Journal of the Operational Research Society, 53: 672--679, (2002). doi:10.1057/palgrave.jors.2601355 M. K. Lauren, J. M. Smith, J. Moffat and N. D. Perry Using the fractal attrition equation to construct a metamodel of the MANA cellular automaton combat model, The Technical Cooperation Program, Joint Systems and Analysis Group, Technical Panel 3, November 2005. G. C. McIntosh, D. P. Galligan, M. A. Anderson and M. K. Lauren, MANA (Map Aware Non-uniform Automata) Version 4 User Manual, DTA Technical Note 2007/3 NR 1465. http://arxiv.org/abs/nlin/0607051v1 J. Moffat, J. Smith and S. Witty, Emergent behaviour: Theory and experimentation using the mana model, Journal of Applied Mathematics and Decision Sciences---JAMDS , 2006, pp.1--14, 2006. doi:10.1155/JAMDS/2006/54846 D. R. Shine, An Exploratory Study of the Army-as-a-System Core Skills: Comparing the Effectiveness of Warfighting Tactics Using MANA, DSTO-TR-1663, (2005), DSTO (Australia). P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys Rev Lett 50(5): 346--349 (1983). doi:10.1103/PhysRevLett.50.346 J. G. Eisenhauer, Regression through the Origin, Teaching Statistics, 25(3): 76--80, (2003).

Characterization of strange attractors in drift wave microturbulence

A multigrid particle-in-cell algorithm for a shearless slab drift wave model with kinetic electrons is presented. The algorithm, which is based on an exact separation of adiabatic and nonadiabatic electron responses, is used to investigate the presence of strange attractors in drift wave turbulence. Although the simulation model has a large number of degrees of freedom, it is found that the strange attractor is low-dimensional and that it is strongly affected by dissipative (collisional) effects.

Characterization of chaotic dynamics—II: topological invariants and their equivalence for an autocatalytic model system and an experimental sheared polymer solution

Characterization of strange attractors exhibiting chaotic dynamics may be carried out through computation of metric, dynamical and topological invariants. The last of these are robust even under control parameter variations and hence have certain distinct advantages. In the present work, we carry out the topological analysis of the observed dynamics from a model autocatalytic reacting system and an experimental polymer solution subjected to shear. Low dimensional chaotic dynamics are observed in both these systems. The results show the global characterization and classification of the dynamics for both systems based on topological invariants, viz., linking numbers and relative rotational rates, is possible. The analyses of these invariants yield the template and the Markov transition matrix that contain in them valuable topological information about the system dynamics. The results obtained show that the two systems possess similar topological characteristics and follow the horseshoe mechanism. This information should help in developing design and control algorithms for these systems.

97/04433 Characterization of strange attractors in the self-ignition of coal stockpiles: Continillo, G. et al. Fractals Chaos Chemical Eng., Proc. Int. CFIC 96 Conf., 1996, (Pub. 1997), 385–396. Edited by Giona, M. and Biardi, G., World Scientific, Singapore

97/04433 Characterization of strange attractors in the self-ignition of coal stockpiles: Continillo, G. et al. Fractals Chaos Chemical Eng., Proc. Int. CFIC 96 Conf., 1996, (Pub. 1997), 385–396. Edited by Giona, M. and Biardi, G., World Scientific, Singapore

Simulation and characterization of strange attractors in driven diode resonator systems.

A single diode resonator system is a nonlinear circuit consisting of a series combination of a resistance, an inductance, and a p-n junction diode. Driven single diode resonator systems experimentally show the period-doubling route to chaos, while driven coupled diode resonator systems show the quasiperiodic route to chaos. We present a direct comparison of numerical simulations with experimental measurements on driven diode resonator systems. The numerical simulations are based on an extension of the Rollins and Hunt model [Phys. Rev. Lett. 49, 1295 (1982)] of the diode (ideal diode modified to include a fixed reverse capacitance and a finite reverse recovery time). The extended model allows the reverse recovery time to depend on the forward current of several previous cycles. The model is applied to both the single diode resonator system and a coupled system consisting of two parallel single diode resonators coupled through a series resistance to a sinusoidal generator. In both cases, the numerical simulations show remarkable global agreement with the experimental measurements. The universal scaling behavior of the simulated coupled diode system at the critical line for the quasiperiodic route to chaos is tested by calculating the spectrum of generalized dimensions characterized by the f(\ensuremath{\alpha}) spectrum. The results show the same f(\ensuremath{\alpha}) spectrum as obtained for the sine-circle map at the critical line.

Characterization of strange attractors as inhomogeneous fractals

We describe the structure of strange attractors by means of a generalization of ordinary fractals.

Characterization of Strange Attractors

A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and information-theoretic entropy is discussed.