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NASA NTRS2005

Stochastic Representation of Chaos using Terminal Attractors

A nonlinear version of the Liouville equation based upon terminal attractors is proposed for describing post-instability motions of dynamical systems with exponential divergence of trajectories such as those leading to chaos and turbulence. As a result, the post-instability motions are represented by expectations, variances, and higher moments of the state variables as functions of time. The proposed approach can be applied to conservative chaos, and in particular, to n-bodies problem, as well as to dissipative systems, and in particular, to chaotic attractors and turbulence.

stochastic processes↗

NASA NTRS1991

Stability and bifurcation analysis of oscillators with piecewise-linear characteristics - A general approach

A general approach is developed for determining the periodic solutions and their stability of nonlinear oscillators with piecewise-smooth characteristics. A modified harmonic balance/Fourier transform procedure is devised for the analysis. The procedure avoids certain numerical differentiation employed previously in determining the periodic solutions, therefore enhancing the reliability and efficiency of the method. Stability of the solutions is determined via perturbations of their state variables. The method is applied to a forced oscillator interacting with a stop of finite stiffness. Flip and fold bifurcations are found to occur. This led to the identification of parameter ranges in which chaotic response occurred.

Noah, S. T.↗

NASA NTRS1987

Deterministic representation of chaos with application to turbulence

Chaotic motions of nonlinear dynamical systems are decomposed into mean components and fluctuations. The approach is based upon the concept that the fluctuations driven by the instability of the original (unperturbed) motion grow until a new stable state is approached. The Reynolds-type equations written for continuous as well as for finite-degrees-of-freedom dynamical systems are closed by using this stabilization principle. The theory is applied to conservative systems, to strange attractors and to turbulent motions.

Zak, M.↗

NASA NTRS2000

On The Computational Capabilities of Physical Systems: Relationship With Conventional Computer Science - Part 2

In the first of this pair of papers, it was proven that there cannot be a physical computer to which one can properly pose any and all computational tasks concerning the physical universe. It was then further proven that no physical computer C can correctly carry out all computational tasks that can be posed to C. As a particular example, this result means that no physical computer that can, for any physical system external to that computer, take the specification of that external system's state as input and then correctly predict its future state before that future state actually occurs; one cannot build a physical computer that can be assured of correctly "processing information faster than the universe does". These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine. This generality is a direct consequence of the fact that a novel definition of computation - "physical computation" - is needed to address the issues considered in these papers, which concern real physical computers. While this novel definition does not fit into the traditional Chomsky hierarchy, the mathematical structure and impossibility results associated with it have parallels in the mathematics of the Chomsky hierarchy. This second paper of the pair presents a preliminary exploration of some of this mathematical structure. Analogues of Chomskian results concerning universal Turing Machines and the Halting theorem are derived, as are results concerning the (im)possibility of certain kinds of error-correcting codes. In addition, an analogue of algorithmic information complexity, "prediction complexity", is elaborated. A task-independent bound is derived on how much the prediction complexity of a computational task can differ for two different reference universal physical computers used to solve that task, a bound similar to the "encoding" bound governing how much the algorithm information complexity of a Turing machine calculation can differ for two reference universal Turing machines. Finally, it is proven that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity), in that there is one and only version of it that can be applicable throughout our universe.

Wolpert, David H.↗

NASA NTRS1997

Detection of chaotic determinism in time series from randomly forced maps

Time series from biological system often display fluctuations in the measured variables. Much effort has been directed at determining whether this variability reflects deterministic chaos, or whether it is merely "noise". Despite this effort, it has been difficult to establish the presence of chaos in time series from biological sytems. The output from a biological system is probably the result of both its internal dynamics, and the input to the system from the surroundings. This implies that the system should be viewed as a mixed system with both stochastic and deterministic components. We present a method that appears to be useful in deciding whether determinism is present in a time series, and if this determinism has chaotic attributes, i.e., a positive characteristic exponent that leads to sensitivity to initial conditions. The method relies on fitting a nonlinear autoregressive model to the time series followed by an estimation of the characteristic exponents of the model over the observed probability distribution of states for the system. The method is tested by computer simulations, and applied to heart rate variability data.

Non-NASA Center↗

NASA NTRS1991

Lyapunov exponent of magnetospheric activity from AL time series

A correlation dimension analysis of the AE index indicates that the magnetosphere behaves as a low-dimensional chaotic system with a dimension close to 4. Similar techniques are used to determine if the system's behavior is due to an intrinsic sensitivity to initial conditions and thus is truly chaotic. The quantity used to measure the sensitivity to initial conditions is the Liapunov exponent. Its calculation for AL shows that it is nonzero (0.11 + or - 0.05/min). This gives the exponential rate at which initially similar configurations of the magnetosphere evolve into completely different states. Also, predictions of deterministic nonlinear models are expected to deviate from the observed behavior at the same rate.

Vassiliadis, D.↗

NASA NTRS1990

On spurious steady-state solutions of explicit Runge-Kutta schemes

The bifurcation diagram associated with the logistic equation v sup n+1 = av sup n (1-v sup n) is by now well known, as is its equivalence to solving the ordinary differential equation u prime = alpha u (1-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. Runge-Kutta schemes applied to both the equation u prime = alpha u (1-u) and the cubic equation u prime = alpha u (1-u)(b-u) were studied computationally and analytically and their behavior was contrasted with the explicit Euler scheme. Their spurious fixed points and periodic orbits were noted. In particular, it was observed that these may appear below the linearized stability limits of the scheme and, consequently, computation may lead to erroneous results.

Sweby, P. K.↗

NASA NTRS2013

Probability of US Heat Waves Affected by a Subseasonal Planetary Wave Pattern

Heat waves are thought to result from subseasonal atmospheric variability. Atmospheric phenomena driven by tropical convection, such as the Asian monsoon, have been considered potential sources of predictability on subseasonal timescales. Mid-latitude atmospheric dynamics have been considered too chaotic to allow significant prediction skill of lead times beyond the typical 10-day range of weather forecasts. Here we use a 12,000-year integration of an atmospheric general circulation model to identify a pattern of subseasonal atmospheric variability that can help improve forecast skill for heat waves in the United States. We find that heat waves tend to be preceded by 15-20 days by a pattern of anomalous atmospheric planetary waves with a wavenumber of 5. This circulation pattern can arise as a result of internal atmospheric dynamics and is not necessarily linked to tropical heating.We conclude that some mid-latitude circulation anomalies that increase the probability of heat waves are predictable beyond the typical weather forecast range.

Mid-Latitude Circulation↗

NASA NTRS1998

Upper Limits of Predictability in Long-Range Climate/Hydrologic Forecasts

The accurate forecasting of el nino or la nina conditions in the tropical Pacific can potentially lead to valuable predictions of hydrological anomalies over land at seasonal to interannual timescales. Even with highly accurate earth system models, though, our ability to generate these continental forecasts will always be limited by the chaotic nature of the atmospheric circulation. The nature of this fundamental limitation is explored through the use of 16-member ensembles of multi-decade GCM simulations. In each simulation of the first ensemble, sea surface temperatures (SSTs) are given the same realistic interannual variations over a 45-year period, and land surface state is allowed to evolve with that of the atmosphere. Analysis of the results shows that the SSTs control the temporal organization of continental precipitation anomalies to a significant extent in the tropics and to a much smaller extent in midlatitudes. In each simulation of the second ensemble, we prescribe SSTs as before, but we also prescribe interannual variations in the low frequency component of evaporation efficiency over land. Thus, in the second ensemble, we effectively make the extreme assumption that surface boundary conditions across the globe are perfectly predictable, and we quantify the consistency with which the atmosphere (particularly precipitation) responds to these boundary conditions. The resulting "absolute upper limit" on the predictability of precipitation is found to be quite high in the tropics yet only moderate in many midlatitude regions.

Koster, R. D.↗

NASA NTRS2005

Problems in Microgravity Fluid Mechanics: G-Jitter Convection

This is the final report on our NASA grant, Problems in Microgravity Fluid Mechanics NAG3-2513: 12/14/2000 - 11/30/2003, extended through 11/30/2004. This grant was made to Stanford University and then transferred to the University of California at Santa Barbara when the PI relocated there in January 2001. Our main activity has been to conduct both experimental and theoretical studies of instabilities in fluids that are relevant to the microgravity environment, i.e. those that do not involve the action of buoyancy due to a steady gravitational field. Full details of the work accomplished under this grant are given below. Our work has focused on: (i) Theoretical and computational studies of the effect of g-jitter on instabilities of convective states where the convection is driven by forces other than buoyancy (ii) Experimental studies of instabilities during displacements of miscible fluid pairs in tubes, with a focus on the degree to which these mimic those found in immiscible fluids. (iii) Theoretical and experimental studies of the effect of time dependent electrohydrodynamic forces on chaotic advection in drops immersed in a second dielectric liquid. Our objectives are to acquire insight and understanding into microgravity fluid mechanics problems that bear on either fundamental issues or applications in fluid physics. We are interested in the response of fluids to either a fluctuating acceleration environment or to forces other than gravity that cause fluid mixing and convection. We have been active in several general areas.

Homsy, G. M.↗

NASA NTRS1990

Are earthquakes an example of deterministic chaos?

A simple mass-spring model is used to systematically examine the dynamical behavior introduced by fault zone heterogeneities. The model consists of two sliding blocks coupled to each other and to a constant velocity driver by elastic springs. The state of this system can be characterized by the positions of the two blocks relative to the driver. A simple static/dynamic friction law is used. When the system is symmetric, cyclic behavior is observed. For an asymmetric system, where the frictional forces for the two blocks are not equal, the solutions exhibit deterministic chaos. Chaotic windows occur repeatedly between regions of limit cycles on bifurcation diagrams. The model behavior is similar to that of the one-dimensional logistic map. The results provide substantial evidence that earthquakes are an example of deterministic chaos.

Huang, Jie↗

NASA NTRS2003

A Time Integration Algorithm Based on the State Transition Matrix for Structures with Time Varying and Nonlinear Properties

A variable order method of integrating the structural dynamics equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. When the time variation of the system can be modeled exactly by a polynomial it produces nearly exact solutions for a wide range of time step sizes. Solutions of a model nonlinear dynamic response exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with solutions obtained by established methods.

Bartels, Robert E.↗

NASA NTRS1984

High-resolution IUE observations of interstellar absorption lines in the Vela supernova remnant

Ultraviolet spectra of 45 stars in the vicinity of the Vela supernova remnant were recorded by the short-wavelength echelle spectrograph aboard the International Ultraviolet Explorer (IUE). Over one-third of the stars show interstellar absorption lines at large radial velocities (greater than 60 km/s). The mapping of these high-velocity components in the sky suggests the motions are chaotic, rather than from a coherent expansion of the remnant material. In accord with earlier conclusions from Copernicus data, the gas at high velocity exhibits higher than normal ionization and shows substantially less depletion of nonvolatile elements than normal interstellar material at low velocities. Relatively strong lines from neutral carbon in the two excited fine-structure states indicate that the neutral clouds within the remnant have had their pressures enhanced by the passage of the blast wave from the supernova. Also, the remnant seems to show a significant enhancement in the abundances of low-velocity Si IV, C IV, and N V over those found in the general interstellar medium.

Jenkins, E. B.↗

NASA NTRS2002

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

Bartels, Robert E.↗

NASA NTRS1995

Steady-state dynamic behavior of an auxiliary bearing supported rotor system

This paper investigates the steady-state responses of a rotor system supported by auxiliary bearings in which there is a clearance between the rotor and the inner race of the bearing. A simulation model based upon the rotor of a production jet engine is developed and its steady-state behavior is explored over a wide range of operating conditions for various parametric configurations. Specifically, the influence of rotor imbalance, support stiffness, and damping is studied. It is found that imbalance may change the rotor responses dramatically in terms of frequency contents at certain operating speeds. Subharmonic responses of 2nd order through 10th order are all observed except the 9th order. Chaotic phenomenon is also observed. Jump phenomena (or double-valued responses) of both hard-spring type and soft-spring type are shown to occur at low operating speeds for systems with low auxiliary bearing damping or large clearance even with relatively small imbalance. The effect of friction between the shaft and the inner race of the bearing is also discussed.

Xie, Huajun↗

NASA NTRS2006

Designing capture trajectories to unstable periodic orbits around Europa

The hostile environment of third body perturbations restricts a mission designer's ability to find well-behaved reproducible capture trajectories when dealing with limited control authority as is typical with low-thrust missions. The approach outlined in this paper confronts this shortcoming by utilizing dynamical systems theory and an extensive preexisting database of Restricted Three Body Problem (RTBP) periodic orbits. The stable manifolds of unstable periodic orbits are utilized to attract a spacecraft towards Europa. By selecting an appropriate periodic orbit, a mission designer can control important characteristics of the captured state including stability, minimum altitudes, characteristic inclinations, and characteristic radii among others. Several free parameters are optimized in the non-trivial mapping from the RTBP to a more realistic model. Although the ephemeris capture orbit is ballistic by design, low-thrust is used to target the state that leads to the capture orbit, control the spacecraft after arriving on the unstable quasi-periodic orbit, and begin the spiral down towards the science orbit. The approach allows a mission designer to directly target fuel efficient captures at Europa in an ephemeris model. Furthermore, it provides structure and controllability to the design of capture trajectories that reside in a chaotic environment.

low-thrust↗

NASA NTRS2002

Nonlinear Dynamic Models in Advanced Life Support

To facilitate analysis, ALS systems are often assumed to be linear and time invariant, but they usually have important nonlinear and dynamic aspects. Nonlinear dynamic behavior can be caused by time varying inputs, changes in system parameters, nonlinear system functions, closed loop feedback delays, and limits on buffer storage or processing rates. Dynamic models are usually cataloged according to the number of state variables. The simplest dynamic models are linear, using only integration, multiplication, addition, and subtraction of the state variables. A general linear model with only two state variables can produce all the possible dynamic behavior of linear systems with many state variables, including stability, oscillation, or exponential growth and decay. Linear systems can be described using mathematical analysis. Nonlinear dynamics can be fully explored only by computer simulations of models. Unexpected behavior is produced by simple models having only two or three state variables with simple mathematical relations between them. Closed loop feedback delays are a major source of system instability. Exceeding limits on buffer storage or processing rates forces systems to change operating mode. Different equilibrium points may be reached from different initial conditions. Instead of one stable equilibrium point, the system may have several equilibrium points, oscillate at different frequencies, or even behave chaotically, depending on the system inputs and initial conditions. The frequency spectrum of an output oscillation may contain harmonics and the sums and differences of input frequencies, but it may also contain a stable limit cycle oscillation not related to input frequencies. We must investigate the nonlinear dynamic aspects of advanced life support systems to understand and counter undesirable behavior.

Jones, Harry↗

NASA NTRS2000

On the Computational Capabilities of Physical Systems: The Impossibility of Infallible Computation - Part 1

In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical universe. Next it is proven that no physical computer C can correctly carry out any computational task in the subset of such tasks that can be posed to C. This result holds whether the computational tasks concern a system that is physically isolated from C, or instead concern a system that is coupled to C. As a particular example, this result means that there cannot be a physical computer that can, for any physical system external to that computer, take the specification of that external system's state as input and then correctly predict its future state before that future state actually occurs; one cannot build a physical computer that can be assured of correctly 'processing information faster than the universe does'. The results also mean that there cannot exist an infallible, general-purpose observation apparatus, and that there cannot be an infallible, general-purpose control apparatus. These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine. This generality is a direct consequence of the fact that a novel definition of computation - a definition of 'physical computation' - is needed to address the issues considered in these papers. While this definition does not fit into the traditional Chomsky hierarchy, the mathematical structure and impossibility results associated with it have parallels in the mathematics of the Chomsky hierarchy. The second in this pair of papers presents a preliminary exploration of some of this mathematical structure, including in particular that of prediction complexity, which is a 'physical computation analogue' of algorithmic information complexity. It is proven in that second paper that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity), in that there is one and only version of it that can be applicable throughout our universe.

Wolpert, David H.↗