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Theoretical developments enable suppression of chaotic structual motions. Theory enables prediction, avoidance, and suppression of chaotic vibrations in structures, especially using dynamic feedback stabilization. In new formulation, motion both repeatable and predictable.
This paper provides insights about the behavior of chaotic complex systems, and the sensitive dependence of the system on the initial starting conditions. How much does a small change in the initial conditions of a complex system affect it in the long term? Do complex systems exhibit what is called the "Butterfly Effect"? This paper uses an agent-based modeling approach to address these questions. An existing model from NetLogo library was extended in order to compare chaotic complex systems with near-identical initial conditions. Results show that small changes in initial starting conditions can have a huge impact on the behavior of chaotic complex systems. The term the "butterfly effect" is attributed to the work of Edward Lorenz [1]. It is used to describe the sensitive dependence of the behavior of chaotic complex systems on the initial conditions of these systems. The metaphor refers to the notion that a butterfly flapping its wings somewhere may cause extreme changes in the ecological system's behavior in the future, such as a hurricane.
Attention is given to solar system cases in which chaotic solutions of Newton's equations are important, as in chaotic rotation and orbital evolution. Hyperion is noted to be tumbling chaotically; chaotic orbital evolution is suggested to be of fundamental importance to an accounting for the Kirkwood gaps in asteroid distribution and for the phase space boundary of the chaotic zone at the 3/1 mean-motion commensurability with Jupiter. In addition, chaotic trajectories in the 2/1 chaotic zone reach very high eccentricities by a route that carries them to high inclinations temporarily.
The evolution of the entire planetary system has been numerically integrated for a time span of nearly 100 million years. This calculation confirms that the evolution of the solar system as a whole is chaotic, with a time scale of exponential divergence of about 4 million years. Additional numerical experiments indicate that the Jovian planet subsystem is chaotic, although some small variations in the model can yield quasi-periodic motion. The motion of Pluto is independently and robustly chaotic.
The discovery (by Laskar, 1989, 1990) that the evolution of the solar system is chaotic, made in a numerical integration of the averaged secular approximation of the equations of motions for the planets, was confirmed by Sussman and Wisdom (1992) by direct numerical integration of the whole solar system. This paper presents results of direct integrations of the rotation of Mars in the chaotically evolved planetary system, made using the same model as that used by Sussman and Wisdom. The numerical integration shows that the obliquity of Mars undergoes large chaotic variations, which occur as the system evolves in the chaotic zone associated with a secular spin-orbit resonance.
In this project, we looked at the application of harmonic balancing as a tool for identifying parameters (HBID) in a nonlinear dynamical systems with chaotic responses. The main idea is to balance the harmonics of periodic orbits extracted from measurements of each coordinate during a chaotic response. The periodic orbits are taken to be approximate solutions to the differential equations that model the system, the form of the differential equations being known, but with unknown parameters to be identified. Below we summarize the main points addressed in this work. The details of the work are attached as drafts of papers, and a thesis, in the appendix. Our study involved the following three parts: (1) Application of the harmonic balance to a simulation case in which the differential equation model has known form for its nonlinear terms, in contrast to a differential equation model which has either power series or interpolating functions to represent the nonlinear terms. We chose a pendulum, which has sinusoidal nonlinearities; (2) Application of the harmonic balance to an experimental system with known nonlinear forms. We chose a double pendulum, for which chaotic response were easily generated. Thus we confronted a two-degree-of-freedom system, which brought forth challenging issues; (3) A study of alternative reconstruction methods. The reconstruction of the phase space is necessary for the extraction of periodic orbits from the chaotic responses, which is needed in this work. Also, characterization of a nonlinear system is done in the reconstructed phase space. Such characterizations are needed to compare models with experiments. Finally, some nonlinear prediction methods can be applied in the reconstructed phase space. We developed two reconstruction methods that may be considered if the common method (method of delays) is not applicable.
Three concepts proposed for system management and control. Concepts are: internal system for autonomous management and control, separate system for environment monitoring, and optical intelligent system required in especially chaotic environments. Intelligent system responds by making decisions and solving problems in novel situations preprogramed.
The motion of planetary bodies is the archetypal clockwork system. Indeed, clocks and calendars were developed to keep track of the relative motions of the Earth, the Sun and the Moon. However, studies over the past few decades imply that this predictable regularity does not extend to small bodies, nor does it apply to the precise trajectories of the planets themselves over long timescale.s. Various examples of chaotic motion within our Solar System and, extrasolar planetary systems will be discussed.
Topics discussed in these proceedings include nonlinear processing and communications; feature extraction and recognition; image gathering, interpolation, and restoration; image coding; and wavelet transform. Papers are presented on noise reduction for signals from nonlinear systems; driving nonlinear systems with chaotic signals; edge detection and image segmentation of space scenes using fractal analyses; a vision system for telerobotic operation; a fidelity analysis of image gathering, interpolation, and restoration; restoration of images degraded by motion; and information, entropy, and fidelity in visual communication. Attention is also given to image coding methods and their assessment, hybrid JPEG/recursive block coding of images, modified wavelets that accommodate causality, modified wavelet transform for unbiased frequency representation, and continuous wavelet transform of one-dimensional signals by Fourier filtering.
The chaos in a one-dimensional system, which would be nonlinear stationary Alfven waves in the absence of an external driver, is characterized. The evolution equations are numerically integrated for the transverse wave magnetic field amplitude and phase using the derivative nonlinear Schroedinger equation (DNLS), including resistive wave damping and a long-wavelength monochromatic, circularly polarized driver. A Poincare map analysis shows that, for the nondissipative (Hamiltonian) case, the solutions near the phase space (soliton) separatrices of this system become chaotic as the driver amplitude increases, and 'strong' chaos appears when the driver amplitude is large. The dissipative system exhibits a wealth of dynamical behavior, including quasiperiodic orbits, period-doubling bifurcations leading to chaos, sudden transitions to chaos, and several types of strange attractors.
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.
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.
Researchers with NASA's Season-to-Interannual Prediction Project (NSIPP) refer to different types of memory when running models on NCCS computers: the computer memory required for their models and the memory of the atmosphere or the ocean. Because of the atmosphere's chaotic nature, its memory is short. For weather predictions, the initial information taken from atmospheric observations has a limited useful life. Currently, there is no way to take observations, initialize an atmosphere model, integrate ahead in time, and make an accurate weather forecast beyond about 2 weeks. After that, the system becomes chaotic. What conditions could be used to make predictions beyond 2 weeks? If not conditions in the atmosphere, then the memory must be found somewhere else. That place is in the oceans. Although most changes in the atmosphere vary on a short timescale, the weather being a prime example, some important large atmospheric climate variations occur over much longer timescales-month s, years, or decades. NSIPP is interested specifically in those phenomena that occur over timescales of several months to a few years, and the El Nino Southern Oscillation (ENSO) is the most significant of these.
Synthetic aperture radar data was used to construct an estimation algorithm for development of information on long waves. The evolution of chaotic dynamic systems was also explored.
Problems in the design of feedback controllers for chaotic dynamical systems are considered theoretically, focusing on two cases where chaos arises only when a nonchaotic continuous-time system is discretized into a simpler discrete-time systems (exponential discretization and pseudo-Euler integration applied to Lotka-Volterra competition and prey-predator systems). Numerical simulation results are presented in extensive graphs and discussed in detail. It is concluded that care must be taken in applying standard dynamical-systems methods to control systems that may be discontinuous or nondifferentiable.
Magnetic field signatures associated with a substorm onset event are examined by making use of simultaneous observations from the Active Magnetosphere Particle Tracer Explorer (AMPTE)/Charge Composition Explorer (CCE) and Spacecraft Charging AT High Altitude (SCATHA). The observations of the two satellites are discussed in relation to their differences and the relative positions of the satellites. Despite the small separation between the satellites, AMPTE/CCE observed the start of irregular magnetic field fluctuations a few tens of seconds earlier than SCATHA, indicating that the CCE was within, or closer to, the onset region. It was found that the amplitude of the fluctuations was largest in the north-south component. The results indicate that the magnetic field fluctuations were excited locally and the coherence length was less than a multiple of Larmor radius of thermal protons. It is suggested that the tail current disruption is described as a system of chaotic filamentary electric currents which flow in various directions, but preferentially anti-parallel to the cross-tail current, and that ions play an important role in the triggering of the tail current disruption.
The orbits around the Lagrange points L1 and L2 of the Sun-Earth-Moon system are chaotic and unstable by nature. In this work, we describe a simple method to control Halo Orbits with a single maneuver at the XZ-plane crossing for each revolution of the orbit in the Bicircular Problem. Examples of up to 500 revolutions of the Halo Orbit are controlled using this method. With optimization and adjustment of the maneuver design, the 𝚫V can be significantly reduced. The simplicity of the method and the infrequency of the maneuvers make this a good candidate for future autonomous control of libration orbits.
Due to the unavailability and, later, prohibitive cost of the computational power required, many phenomena in nonlinear dynamic systems have in the past been addressed in terms of linear systems. Linear systems respond to periodic inputs with periodic outputs, and may be characterized in the time domain or in the frequency domain as convenient. Reduction to the frequency domain is frequently desireable to reduce the amount of computation required for solution. Nonlinear systems are only soluble in the time domain, and may exhibit a time history which is extremely sensitive to initial conditions. Such systems are termed chaotic. Dynamic buckling, aeroelasticity, fatigue analysis, control systems and electromechanical actuators are among the areas where chaotic vibrations have been observed. Direct transient analysis over a long time period presents a ready means of simulating the behavior of self-excited or externally excited nonlinear systems for a range of experimental parameters, either to characterize chaotic behavior for development of load spectra, or to define its envelope and preclude its occurrence.