Optimization of nonlinear characteristics
Optimal characteristics for single-input single- output memoryless time invariant nonlinear dynamic systems
Desoer, C. A.↗
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Results for “nonlinear dynamical system”
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Estimation of nonlinear pilot model parameters including time delay.
Investigation of the feasibility of using a Kalman filter estimator for the identification of unknown parameters in nonlinear dynamic systems with a time delay. The problem considered is the application of estimation theory to determine the parameters of a family of pilot models containing delayed states. In particular, the pilot-plant dynamics are described by differential-difference equations of the retarded type. The pilot delay, included as one of the unknown parameters to be determined, is kept in pure form as opposed to the Pade approximations generally used for these systems. Problem areas associated with processing real pilot response data are included in the discussion.
Schiess, J. R.↗
A method for the analysis of nonlinearities in aircraft dynamic response to atmospheric turbulence
An analytical method is developed which combines the equivalent linearization technique for the analysis of the response of nonlinear dynamic systems with the amplitude modulated random process (Press model) for atmospheric turbulence. The method is initially applied to a bilinear spring system. The analysis of the response shows good agreement with exact results obtained by the Fokker-Planck equation. The method is then applied to an example of control-surface displacement limiting in an aircraft with a pitch-hold autopilot.
Sidwell, K.↗
Relative motion characteristics of 2 near-Earth Satellites
The stability of the nonlinear dynamical system of two GRAVSAT - type satellites was investigated by performing several numerical experiments which provide the simulations of the relative motion characteristics between the two satellites for various specified time intervals. The simulations included the relative range, range-rate, and relative acceleration magnitude. These simulations were generated with respect to appropriate initial orbital elements which were obtained such that the instantaneous separation distance between the two satellites has small fluctuations from a specified constant separation distance. The simulation results indicate that the behavior of the relative motions is very sensitive to the initial orbital elements of the satellites and that for a specified time interval of interest. A stable behavior is possible only with the use of an appropriate set of initial orbital elements compatible with the gravity field used to derive them.
Schutz, B. E.↗
Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics
This paper examines the robustness properties of existing adaptive control algorithms to unmodeled plant high-frequency dynamics and unmeasurable output disturbances. It is demonstrated that there exist two infinite-gain operators in the nonlinear dynamic system which determines the time-evolution of output and parameter errors. The pragmatic implications of the existence of such infinite-gain operators is that: (1) sinusoidal reference inputs at specific frequencies and/or (2) sinusoidal output disturbances at any frequency (including dc), can cause the loop gain to increase without bound, thereby exciting the unmodeled high-frequency dynamics, and yielding an unstable control system. Hence, it is concluded that existing adaptive control algorithms as they are presented in the literature referenced in this paper, cannot be used with confidence in practical designs where the plant contains unmodeled dynamics because instability is likely to result. Further understanding is required to ascertain how the currently implemented adaptive systems differ from the theoretical systems studied here and how further theoretical development can improve the robustness of adaptive controllers.
Rohrs, C. E.↗
Dynamic Identification for Control of Large Space Structures
This is a compilation of reports by the one author on one subject. It consists of the following five journal articles: (1) A Parametric Study of the Ibrahim Time Domain Modal Identification Algorithm; (2) Large Modal Survey Testing Using the Ibrahim Time Domain Identification Technique; (3) Computation of Normal Modes from Identified Complex Modes; (4) Dynamic Modeling of Structural from Measured Complex Modes; and (5) Time Domain Quasi-Linear Identification of Nonlinear Dynamic Systems.
Ibrahim, S. R.↗
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.↗
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications.
Crouch, Peter E.↗
Chaos in the sunspot cycle - Analysis and prediction
The variability of solar activity over long time scales, given semiquantitatively by measurements of sunspot numbers, is examined as a nonlinear dynamical system. First, a discussion of the data set used and the techniques utilized to reduce the noise and capture the long-term dynamics inherent in the data is presented. Subsequently, an attractor is reconstructed from the data set using the method of time delays. The reconstructed attractor is then used to determine both the dimension of the underlying system and also the largest Lyapunov exponent, which together indicate that the sunspot cycle is indeed chaotic and also low dimensional. In addition, recent techniques of exploiting chaotic dynamics to provide accurate, short-term predictions are utilized in order to improve upon current forecasting methods and also to place theoretical limits on predictability extent. The results are compared to chaotic solar-dynamo models as a possible physically motivated source of this chaotic behavior.
Mundt, Michael D.↗
Connecting coherent structures and strange attractors
A concept of turbulence derived from nonlinear dynamical systems theory suggests that turbulent solutions to the Navier-Stokes equations are restricted to strange attractors, and, by implication, that turbulent phenomenology must find some expression or source in the structure of these mathematical objects. Examples and discussions are presented to link coherent structures to some of the commonly known characteristics of strange attractors. Basic to this link is a geometric interpretation of conditional sampling techniques employed to educe coherent structures that offers an explanation for their appearance in measurements as well as their size.
Keefe, Laurence R.↗
The mathematical bases for qualitative reasoning
The practices of researchers in many fields who use qualitative reasoning are summarized and explained. The goal is to gain an understanding of the formal assumptions and mechanisms that underlie this kind of analysis. The explanations given are based on standard mathematical formalisms, particularly on ordinal properties, continuous differentiable functions, and the mathematics of nonlinear dynamic systems.
Kalagnanam, Jayant↗
Examination of time-variable input effects in a nonlinear analogue magnetosphere model
The plasma physical analog model (an extension of the damped, harmonic-oscillator dripping faucet model) is employed to consider explicitly the effect of time-varying the inputs. This work is equivalent to considering the effects of northward and southward turnings of the interplanetary magnetic field for various periods of time. It is found that relatively extended episodes (not less than 2 hours) of turned-on input with shorter (about 1 hour) periods of turned-off input lead to model behavior much like the continuously driven case. Going to short input intervals with longer periods of zero input leads to highly irregular and dramatically fluctuating episodes of magnetotail unloading. These results give an insight into the diversity of apparent magnetospheric responses during relatively isolated substorm conditions. This work shows the absolutely critical interdependence (in a nonlinear dynamical system) of input phasing and internal magnetospheric response cycles.
Baker, D. N.↗
The attractor dimension of solar decimetric radio pulsations
The temporal characteristics of decimetric pulsations and related radio emissions during solar flares are analyzed using statistical methods recently developed for nonlinear dynamic systems. The results of the analysis is consistent with earlier reports on low-dimensional attractors of such events and yield a quantitative description of their temporal characteristics and hidden order. The estimated dimensions of typical decimetric pulsations are generally in the range of 3.0 + or - 0.5. Quasi-periodic oscillations and sudden reductions may have dimensions as low as 2. Pulsations of decimetric type IV continua have typically a dimension of about 4.
Kurths, J.↗
The numerical dynamic for highly nonlinear partial differential equations
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
Lafon, A.↗
Applications of CFD and visualization techniques
In this paper, three applications are presented to illustrate current techniques for flow calculation and visualization. The first two applications use a commercial computational fluid dynamics (CFD) code, FLUENT, performed on a Cray Y-MP. The results are animated with the aid of data visualization software, apE. The third application simulates a particulate deposition pattern using techniques inspired by developments in nonlinear dynamical systems. These computations were performed on personal computers.
Saunders, James H.↗
Lyapunov exponents from CHUA's circuit time series using artificial neural networks
In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks.
Gonzalez, J. Jesus↗
Multiscale low-frequency circulation modes in the global atmosphere
In this paper, fundamental multiscale circulation modes in the global atmosphere are identified with the objective of providing better understanding of atmospheric low-frequency variabilities over a wide range of spatial and temporal scales. With the use of a combination of rotated principal component technique, singular spectrum analysis, and phase space portraits, three categories of basic multiscale modes in the atmosphere are found. The first is the interannual-mode (IAM), which is dominated by time scales longer than a year and can be attributed to heating and circulation anomalies associated with the coupled tropical ocean-atmosphere, in particular the El Nino-Southern Oscillation. The second is a set of tropical intraseasonal modes consisting of three separate multiscale patterns (ISO-1, -2, -3) related to tropical heating that can be identified with the different phases of the Madden-Julian Oscillation (MJO), including its teleconnection to the extratropics. The ISO spatial and temporal patterns suggest that the extratropical wave train in the North Pacific and North America is related to heating over the Maritime Continent and that the evolution of the MJO around the equator may require forcing from the extratropics spawning convection over the Indian Ocean. The third category represents extratropical intraseasonal oscillations arising from internal dynamics of the basic-state circulation. In the Northern Hemisphere, there are two distinct circulation modes with multiple frequencies in this category: the Pacific/North America (PNA) and the North Atlantic/Eurasia (NAE). In the Southern Hemisphere, two phase-locked modes (PSA-1 and PSA-2) are found depicting an eastward propagating wave train from eastern Australia, via the Pacific South America to the South Atlantic. The extratropical modes exhibit temporal characteristics such as phase locking and harmonic oscillations possibly associated with quadratically nonlinear dynamical systems. Additionally, the observed monthly and seasonal anomalies arise from a complex interplay of the various multiscale low-frequency modes. The relative dominance of the different modes varies widely from month to month and from year to year. On the monthly time scale, while one or two mechanisms may dominate in one year, no single mechanism seems to dominate for all years. There are indications that when the IAM, that is, ENSO heating patterns are strong, the extratropical modes may be suppressed and vice versa. For the seasonal mean, the interannual mode tends to dominate and the contribution from the PNA remains quite significant.
Lau, K.-M.↗
Wavelets, period-doubling, and time-frequency localization with application to organization of convection over the tropical western Pacific
In this paper, preliminary results in using orthogonal and continuous wavelet transform (WT) to identify period doubling and time-frequency localization in both synthetic and real data are presented. First, the Haar WT is applied to synthetic time series derived from a simple nonlinear dynamical system- a first-order quadratic difference equation. Second, the complex Morlet WT is used to study the time-frequency localization of tropical convection based on a high-resolution Japanese Geostationary Meteorological Satellite infrared (IR) radiance dataset. The Haar WT of the synthetic time series indicates the presence and distinct separation of multiple frequencies in a period-doubling sequence. The period-doubling process generates a multiplicity of intermediate frequencies, which are manifested in the nonuniformity in time with respect to the phase of oscillations in the lower frequencies. Wavelet transform also enables the detection of extremely weak signals in high-order subharmonics resulting from the period-doubling bifurcations. These signals are either undetected or considered statistically insignificant by traditional Fourier analysis. The Morlet WT of the IR radiance dataset indicates the presence of multiple timescales, which are localized in both frequency and time. There are two regimes in the variation of IR radiance, corresponding to the wet and dry periods. Multiple timescales, ranging from semidiurnal, diurnal, synoptic, to intraseasonal with embedding structures, are active in the wet regime. In particular, synoptic variability is more prominent during the wet phase of an intensive intraseasonal cycle. These are not only consistent with, but also show more details than, previous findings by using other techniques. The phase-locking relationships among the oscillations with different time-scales suggest that both synoptic and intraseasonal variations may be mixed oscillations due to the interaction of self-excited oscillations in the tropical atmosphere and external forcings such as annual and diurnal solar radiation variations. Both examples show that WT is a powerful tool for analysis of phenomena involving multiscale interactions that exhibit localization in both frequency and time. A discussion on the caveats in the use of WT in geophysical data analysis is also presented.
Weng, Hengyi↗