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NASA NTRS1986

The onset of time-dependent convection in spherical shells as a clue to chaotic convection in the earth's mantle

This work presents a detailed numerical study of the dynamical behavior of convection in a spherical shell, as applied to mantle convection. From both two-dimensional (120 radial and 360 tangential points) and three-dimensional (60 radial levels and spherical harmonics up to order and degree L = 33, m = 33), it is shown that for a spherical shell (with inner-to-outer radii ratio eta = 0.62) convection becomes time-dependent, with l = 2 dominating, at a Rayleigh number of about 31 times supercritical for a constant-viscosity, base-heated configuration. This secondary instability is characterized by oscillatory time dependence, with higher frequencies involved, at slightly higher Rayleigh numbers. In illustrating the onset of time dependence, the analysis is extended to show that the onset of weak turbulence in spherical-shell convection takes place at about 60 times the critical Rayleigh number via a quasi-periodic mode.

Machetel, Philippe↗

NASA NTRS1992

Generation of magnetic fields by chaotic fluid convection - The fast dynamo problem

In the kinematic fast dynamo problem, the underlying nonlinear dynamics of the flow play a critical role in the behavior of a dynamo field. It is presently noted that the two important facets of the problem are the approximately lognormal distribution of vector lengths, and the presence of partial cancellation. It is suggested that these features may be reflected in the magnetic fields observed on the sun.

Finn, John M.↗

NASA NTRS1987

Application of non-linear dynamics to the characterization of cardiac electrical instability

Beat-to-beat alternation in the morphology of the ECG has been previously observed in hearts susceptible to fibrillation. In addition, fibrillation has been characterized by some as a chaotic state. Period doubling phenomena, such as alternation, and the onset of chaos have been connected by non-linear dynamical systems theory. In this paper, we describe the use of a technique from nonlinear dynamics theory, the construction of a first return nap, to assess the susceptibility to fibrillation threshhold in canine experiments.

NASA Discipline Cardiopulmonary↗

NASA NTRS1982

The applicability of Lyapunov characteristic numbers in the study of the stability of satellite orbits

It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.

Broucke, R.↗

NASA NTRS1990

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Yee, H. C.↗

NASA NTRS1991

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics

Yee, H. C.↗

NASA NTRS1980

Nonlinear dynamics; Proceedings of the International Conference, New York, NY, December 17-21, 1979

Papers were presented on turbulence, ergodic and integrable behavior, chaotic maps and flows, chemical and fully developed turbulence, and strange attractors. Specific attention was given to measures describing a turbulent flow, stochastization and collapse of vortex systems, a subharmonic route to turbulent convection, and weakly nonlinear turbulence in a rotating convection layer. The Korteweg-de Vries and Hill equations, plasma transport in three dimensions, a horseshoe in the dynamics of a forced beam, and the explosion of strange attractors exhibited by Duffing's equation were also considered.

Helleman, R. H. G.↗

NASA NTRS2007

Fuel-Optimal Trajectories in a Planet-Moon Environment Using Multiple Gravity Assists

For low energy spacecraft trajectories such as multi-moon orbiters for the Jupiter system, multiple gravity assists by moons could be used in conjunction with ballistic capture to drastically decrease fuel usage. In this paper, we outline a procedure to obtain a family of zero-fuel multi-moon orbiter trajectories, using a family of Keplerian maps derived by the first author previously. The maps capture well the dynamics of the full equations of motion; the phase space contains a connected chaotic zone where intersections between unstable resonant orbit manifolds provide the template for lanes of fast migration between orbits of different semimajor axes. Patched three body approach is used and the four body problem is broken down into two three-body problems, and the search space is considerably reduced by the use of properties of the Keplerian maps. We also introduce the notion of Switching Region where the perturbations due to the two perturbing moons are of comparable strength, and which separates the domains of applicability of the corresponding two Keplerian maps.

Ross, Shane D.↗

NASA NTRS1988

Thermal attractor in chaotic convection with high-Prandtl-number fluids

The onset of chaotic convection in large-Prandtl-number fluids has been found to occur in the form of elongated cells with aperiodic boundary-layer instabilities. The dynamics of this phenomenon has been monitored in the physical domain, and also in the spectral and phase spaces. In order to describe more precisely the underlying mechanisms of chaotic thermal convection, the notion of 'thermal attractor' is introduced in the physical domain and a systematic study is conducted of the evolution of the spectral energy surfaces. A relationship between this thermal attractor and the presence of a strange attractor in the phase space is proposed.

Vincent, Alain P.↗

NASA NTRS1987

Solar system dynamics

The rotational dynamics of irregularly shaped satellites and the origin of Kirkwood Gaps are discussed. The chaotic tumbling of Hyperion and the anomalously low eccentricity of Deimos are examined. The Digital Orrery is used to explore the phase space of the ellipic restricted three body problem near the principal commensurabilities (2/1, 5/2, 3/1, and 3/2). The results for the 3/1 commensurability are in close agreement with those found earlier with the algebraic mapping method. Large chaotic zones are associated with the 3/1, 2/1 and 5/2 resonances, where there are gaps in the distribution of asteroids. The region near the 3/2 resonance, where the Hilda group of asteroids is located, is largely devoid of chaotic behavior. Thus, there is a qualitative agreement between the character of the motion and the distribution of asteroids.

Wisdom, Jack↗

NASA NTRS1993

Convective Effects During the Physical Vapor Transport Process. II - Thermosolutal Convection

The effect of an inert gas on the diffusive-convective physical vapor transport process is investigated for the case when the temperature gradient is stabilizing, and the concentration gradient destabilizing, for a wide parametric range. When an inert gas is present, the thermal and solutal convection oppose each other. The solutal field is destabilizing while the thermal field and the advective-diffusive flux stabilize the flow field. When the pressure of the inert component is increased, the stabilizing effect of the advective-diffusive flux is decreased; thus, convection becomes more vigorous. The nonlinear dynamics of the flow field here show a transition from quasi-periodic to chaotic state. When both stabilizing mechanisms are present, the flow field shows a transition to a steady state. Toward steady state, growth and amalgamation of rolls occur, which result in an overturning motion. This leads to a superposed flow consisting of one roll and a unidirectional flow. However, when the pressure is increased, the advective-diffusive stability mechanism is decreased. and oscillations of the flow field occur. The low gravity environment is effective at eliminating oscillatory behavior of the flow field and results in uniform temperature and concentration gradients.

Duval, Walter M. B.↗

NASA NTRS1994

Some recent results of nonlinear acoustics in combustion chambers

The chief purpose of this paper is to provide a brief review of some recent work on nonlinear acoustics applicable to combustion instabilities, largely in the framework of an approximate analysis. Some connections will be made with the modern theory of nonlinear dynamical systems, including very recent and incomplete attempts by others to assess the possible chaotic behavior observed in laboratory tests.

Culick, F. E. C.↗

NASA NTRS1991

Indications of low dimensionality in magnetospheric dynamics

Using three separate but related approaches, the question of whether the dynamic response of the magnetosphere to the solar wind input may be described by a low-order system of equations is examined. First, it is determined that the dimension of the subset (the attractor) in the high-dimensional magnetospheric phase space associated with the westward auroral electrojet (AL) index for some of the data sets compiled by Bargatze et al. (1985) is 4.0 + or - 0.2, seemingly independent of activity level. Second, direct modeling of the magnetosphere, considering the bulk properties of the tail plasma, leads to a system of equations that is similar to those previously reported as a dripping faucet model; here, the focus is specifically on the prediction of a natural frequency in this model. Finally, a peak is identified with the predicted frequency in power spectra of AL computed for intervals with both low and high activity. Peaks at other frequencies also appear in the spectra, and such resonances would be expected for a chaotic nonlinear oscillator. Combining these approaches it is concluded that at least some aspects of magnetospheric dynamics may be meaningfully modeled by low-dimensional sets of equations.

Roberts, D. A.↗

NASA NTRS2003

Probable Disastrous Consequences of Collision Between Unknown Small (100 m) Asteroids with Known (Approximately 1 km) Near Earth Orbiting (NEO) Asteroids

The long-term stability of the Solar System is not well understood. Ironically its stability is taken for granted even though our knowledge of all the constituents [comets, asteroids. (The Asteroid Belt between Mars and Jupiter, Trojan Asteroids, Kuiper belt, Ort Cloud), planetoids, planets, moons, etc], and its long-term dynamics cannot be easily computed. At best one might say that the solar system is chaotic, but much of the time it seems to exists near a quasi-stationary state. An asteroid that passes near the Earth regularly returns with clock-like precision. Taking into account every known detail of its path through the solar system, its orbit is calculated forward thousands of years with no untoward calamity on the horizon. And then one day, this passive visitor slams into the Earth during a sunny afternoon picnic! Can this happen? Unfortunately, this is a real possibility in the ordinary history of the solar system. In fact our knowledge of the solar system in the small is sketchy, as will be pointed out. Events, which lie outside our awareness, can precipitate disasters that we may perceive when it's too late to launch effective counter measures. In this work, one such scenario is described and the direct consequences for the Earth are calculated.

Smalley, Larry↗

NASA NTRS1989

Do steady fast magnetic dynamos exist?

This paper considers the question of the existense of a steady fast kinematic magnetic dynamo for a conducting fluid with a steady velocity field and vanishingly small electrical resistivity. The analysis of examples of steady dynamos, found by considering the zero-resistivity dynamics, indicated that, for sufficiently small resistivity, dynamo action can indeed occur in steady smooth three-dimensional chaotic fluid flows and that fast dynamos should consequently be a typical occurrence for such flows.

Finn, John M.↗

NASA NTRS1992

The kinematics of buoyancy induced mixing

A model problem to study mixing driven by buoyancy flow fields due to a time dependent body force which consists of a steady and an oscillatory component is considered. Flow fields generated by a time dependent body force, which consists of a steady and an oscillatory component from vibrations or g-jitter, mix two fluids inside a cavity by stretching and folding the interface into various morphological patterns. These patterns show two basic structures depending on the dominant components of the body force. These structures are whirls and tendrils, corresponding to the dominance of either the steady or oscillatory component, respectively. A combination of these structures also occurs in the interaction region. In this region the effect of the steady component opposes the oscillatory component and the result is to smooth the interface, and prevent breakup of the interface. For the parametric range considered, two basic flow regimes occur: convective and chaotic. In the convective regime, the morphological patterns are topological. In the chaotic regime, mixing occurs by a repetitive sequence of bubble formation, necking, and breakup which spreads the concentration field throughout the cavity. The length stretch of the interface increases exponentially with time. The dynamical system characteristics show that in the convective regime, at a point in space, the flow field is oscillatory with elliptical phase space trajectories, and its power spectrum indicates that the flow field responds to the corresponding input frequency. Whereas, in the chaotic regime, the flow field is aperiodic, the phase space trajectories show irregular patterns, and the power spectrum shows a broadband distribution.

Duval, Walter M. B.↗

NASA NTRS1990

Non-linear dynamic analysis of geared systems, part 2

A good understanding of the steady state dynamic behavior of a geared system is required in order to design reliable and quiet transmissions. This study focuses on a system containing a spur gear pair with backlash and periodically time-varying mesh stiffness, and rolling element bearings with clearance type non-linearities. A dynamic finite element model of the linear time-invariant (LTI) system is developed. Effects of several system parameters, such as torsional and transverse flexibilities of the shafts and prime mover/load inertias, on free and force vibration characteristics are investigated. Several reduced order LTI models are developed and validated by comparing their eigen solution with the finite element model results. Several key system parameters such as mean load and damping ratio are identified and their effects on the non-linear frequency response are evaluated quantitatively. Other fundamental issues such as the dynamic coupling between non-linear modes, dynamic interactions between component non-linearities and time-varying mesh stiffness, and the existence of subharmonic and chaotic solutions including routes to chaos have also been examined in depth.

Singh, Rajendra↗

NASA NTRS1990

The relation between ferroelasticity and superconductivity

The high-temperature superconductivity is explained widely by the layered crystal structure. The one- and two-dimensional subsystems and their interaction are investigated here. It is assumed that the high-T(sub c) superconductivity takes place in the two-dimensional subsystem and the increase of the phase transition temperature from 60 K up to 90 K is the consequence of turning on the influence of one-dimensional chains. The interaction between the two subsystems is transferred along the c axis by the phonons of breathing mode, which causes the hybridization of the electronic bonds between these subsystems. The experimental works indicate that the existence of both the chains Cu(1)-O and their interaction with the superconducting plane of Cu(2)-O modify the temperature of the transition to the superconducting state. It is seen from the neutron scattering data that the rates of the interatomic distance dependencies on temperature are changed around 140 K and 90 K. The 'zig-zag' order in Cu(1)-O chains has been postulated but, on the other hand, the vibrations with a large amplitude only were reported. The bi-stabilized situation of the oxygen ions can be caused by the change of distance between these ions and the Ba ions. It leads to the appearance of a two-well potential. Its parameters depend on temperature and the dynamics of the oxygen ions' movement. They can induce the antipolar order, which can be, however, more or less chaotic. The investigation of the ferroelastic properties of Y-Ba-Cu-O samples lead to the conclusion that they are related to jumps of ions inside the given chain and not to a diffusion between different sites in the ab plane. Researchers deduce thus that the fluctuating oxygen ions from these chains create dipoles in the ab plane. They can be described with the pseudo-spin formalism/ - Pauli matrices/. The system can be described with the Ising model. The pseudo-spins interact with phonons and influence the superconductivity in the second subsystem.

Molak, A.↗