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Controlling chaos in a high dimensional system with periodic parametric perturbations

Description: The effect of applying a periodic perturbation to an accessible parameter of a high-dimensional (coupled-Lorenz) chaotic system is examined. Numerical results indicate that perturbation frequencies near the natural frequencies of the unstable periodic orbits of the chaotic system can result in limit cycles or significantly reduced dimension for relatively small perturbations.
Date: October 1, 1998
Creator: Mirus, K. A. & Sprott, J. C.
Partner: UNT Libraries Government Documents Department

(Some basic research problems related to energy: Annual progress report)

Description: A new simple relation has been found, between the transport coefficients of a fluid, in particular the viscosity, and the maximum and minimum Lyapunov exponents of the fluid in a non-equilibrium stationary state. This relation holds arbitrarily far from equilibrium, as long as the transport coefficients themselves make sense. It is hoped that this new relation will provide new insight in the nature and properties of transport coefficients or, in general, in fluid behavior far from equilibrium, i.e., in the strongly non-linear regime.
Date: January 1, 1991
Partner: UNT Libraries Government Documents Department

Adaptive external torque estimation by means of tracking a Lyapunov function

Description: A real-time method is presented to adoptively estimate three-dimensional unmodeled external torques acting on a spacecraft. This is accomplished by forcing the tracking error dynamics to follow the Lyapunov function underlying the feedback control law. For the case where the external torque is constant, the tracking error dynamics are shown to converge asypmtotically. The methodology applies not only to the control law used in this paper, but can also be applied to most Lyapunov derived feedback control laws. The adaptive external torque estimation is very robust in the presence of measurement noise, since a numerical integration is used instead of a numerical differentiation. Spacecraft modeling errors, such as in the inertia matrix, are also compensated for by this method. Several examples illustrate the practical significance of these ideas.
Date: March 1, 1996
Creator: Schaub, H.; Junkins, J.L. & Robinett, R.D.
Partner: UNT Libraries Government Documents Department

Anisotropies in magnetic field evolution and local Lyapunov exponents

Description: The natural occurrence of small scale structures and the extreme anisotropy in the evolution of a magnetic field embedded in a conducting flow is interpreted in terms of the properties of the local Lyapunov exponents along the various local characteristic (un)stable directions for the Lagrangian flow trajectories. The local Lyapunov exponents and the characteristic directions are functions of Lagrangian coordinates and time, which are completely determined once the flow field is specified. The characteristic directions that are associated with the spatial anisotropy of the problem, are prescribed in both Lagrangian and Eulerian frames. Coordinate transformation techniques are employed to relate the spatial distributions of the magnetic field, the induced current density, and the Lorentz force, which are usually followed in Eulerian frame, to those of the local Lyapunov exponents, which are naturally defined in Lagrangian coordinates.
Date: January 13, 2000
Creator: Tang, X.Z. & Boozer, A.H.
Partner: UNT Libraries Government Documents Department

A theory of state space reconstruction in the presence of noise

Description: Takens' theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a single time series. This theorem does not treat the effect of noise, however, and so gives no guidance about practical considerations for reconstructing a good state space. We study the problem of reconstructing a state space with observational noise, examining the likelihood for a particular state given a series of noisy observations. We define a quantity called distortion, which is proportional to the covariance of the likelihood function in a reconstructed state space. This is related to the noise amplification, which corresponds to the root-mean-square errors for time series prediction with an ideal model. We prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification. We derive several asymptotic scaling laws for distortion and noise amplification. They depend on properties of the state space reconstruction, such as the sampling time and the reconstruction dimension, and properties of the dynamical system, such as the dimension and Lyapunov exponents. When the dimension and Lyapunov exponents are sufficiently large these scaling laws show that, no matter how the state space is reconstructed, there is an explosion in the noise amplification -- from a practical point of view all determinism is lost, even for short times, so that the time series is effectively a random process. In the low noise, large data limit we show that the technique of local principal value decomposition (PVD) is an optimal method of state space reconstruction, in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension. 20 refs., 12 figs.
Date: January 1, 1990
Creator: Casdagli, M.; Eubank, S.; Farmer, J.D. & Gibson, J. (Los Alamos National Lab., NM (USA) Santa Fe Inst., NM (USA))
Partner: UNT Libraries Government Documents Department

Nonequilibrium molecular dynamics: The first 25 years

Description: Equilibrium Molecular Dynamics has been generalized to simulate Nonequilibrium systems by adding sources of thermodynamic heat and work. This generalization incorporates microscopic mechanical definitions of macroscopic thermodynamic and hydrodynamic variables, such as temperature and stress, and augments atomistic forces with special boundary, constraint, and driving forces capable of doing work on, and exchanging heat with, an otherwise Newtonian system. The underlying Lyapunov instability of these nonequilibrium equations of motion links microscopic time-reversible deterministic trajectories to macroscopic time-irreversible hydrodynamic behavior as described by the Second Law of Thermodynamics. Green-Kubo linear-response theory has been checked. Nonlinear plastic deformation, intense heat conduction, shockwave propagation, and nonequilibrium phase transformation have all been simulated. The nonequilibrium techniques, coupled with qualitative improvements in parallel computer hardware, are enabling simulations to approximate real-world microscale and nanoscale experiments.
Date: August 1, 1992
Creator: Hoover, W.G.
Partner: UNT Libraries Government Documents Department

Finite dimensionality in the complex Ginzburg-Landau equation

Description: Finite dimensionality is shown to exist in the complex Ginzburg-Landau equation periodic on the interval (0,1). A cone condition is derived and explained which gives upper bounds on the number of Fourier modes required to span the universal attractor and hence upper bounds on the attractor dimension itself. In terms of the parameter R these bounds are not large. For instance, when vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, the Fourier spanning dimension is 0(R/sup 3/2/). Lower bounds are estimated from the number of unstable side-bands using ideas from work on the Eckhaus instability. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, computing exactly) the Lyapunov dimension and invoking a recent theorem of Constantin and Foias, which asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula, is an upper bound on the Hausdorff dimension. 39 refs., 7 figs.
Date: January 1, 1987
Creator: Doering, C.R.; Gibbon, J.D.; Holm, D.D. & Nicolaenko, B.
Partner: UNT Libraries Government Documents Department

Predictability of Rayleigh-Taylor instability

Description: Numerical experiments modeling the Rayleigh Taylor instability are carried out using a two-dimensional incompressible Eulerian hydrodynamic code VFTS. The method of integrating the Navier-Stokes equations including the viscous terms is similar to that described in Kim and Moin, except that Lagrange particles have been added and provision for body forces is given. The Eulerian method is 2nd order accurate in both space and time, and the Poisson equation for the effective pressure field is solved exactly at each time step using a cyclic reduction method. 3 refs., 3 figs.
Date: March 27, 1986
Creator: Viecelli, J.A.
Partner: UNT Libraries Government Documents Department

Chaotic dynamics in dense fluids

Description: We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential both in equilibrium and nonequilibrium steady states. Lyapunov characteristic exponents lambda/sub n/ describe the mean exponential rates of divergence and convergence of neighbouring trajectories in phase-space. They are useful in characterizing the stochastic properties of a dynamical system. A new algorithm for their calculation is presented which incorporates ideas from control theory and constraint nonequilibrium molecular dynamics. 4 refs., 1 fig.
Date: September 1, 1987
Creator: Posch, H.A. & Hoover, W.G.
Partner: UNT Libraries Government Documents Department

Construction of the Courant-Snyder invariants for the non-linear equations of motion and criterion for the long-term stability of the beam in a storage ring

Description: The Courant-Snyder invariants become Lyapunov functions when the [beta]-functions admit non-zero lower, and finite upper bounds. The long-term stability of motion then follows. This alternative criterion for the long-term stability of motion can be generalized to the nonlinear case. A single particle subjected to an arbitrary static magnetic field is considered in some detail, as an example.
Date: April 26, 1993
Creator: Garczynski, V.
Partner: UNT Libraries Government Documents Department