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Topic: Chaotic motion

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	An Investigation of Chaotic Phenomena in a Vibratory Ball Milling System
		In this paper, the effect of the coefficient of restitution on the chaotic motion is systematically numerically investigated in terms of strange attractors and the corresponding correlation dimensions [9].
		Chaotic motion which occurred at the end of period doubling cascade for various e can be classified as two types in terms of strange attractors and their correlation dimensions are smaller than 2.
		However, the strange attractor of chaotic motion from the experiment appears to be somewhat different from that generated by the simulation due to external perturbations, such as the unstable platform velocity and coefficient of restitution.
	www.complexity.org.au /ci/vol02/han/han.html (1991 words)

	Chaos theory - Wikipedia, the free encyclopedia
		The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly.
		The first discoverer of chaos can plausibly be argued to be Jacques Hadamard, who in 1898 published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature.
		Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.
	en.wikipedia.org /wiki/Chaos_theory (2086 words)

	Chaos theory
		One way of visualizing chaotic motion, or indeed any type of motion, is to make a phase diagram of the motion.
		Such attractive motion is fittingly called an attractor for the system and is very common for forced dissipative systems.
		The Lorenz attractor is perhaps one of the best known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the eyes of an owl.
	www.ebroadcast.com.au /lookup/encyclopedia/ch/Chaos_theory.html (861 words)

	[No title]
		The equivalence of rigid body motion and billiards is of considerable value in the analysis of numerical simulations using the Poincar{\'e} surface of section technique.
		This diffusive process for chaotic orbits is not trapped by integrable KAM tori as is the case in lower-dimensional systems ({\em i.e.}, systems with two degrees of freedom) and thus, the whole of phase space is permeated by a network of stochastic trajectories.
		Evidently, positive proof of chaotic motion cannot be made numerically.} The qualitative result of this investigation is the motion becomes ``more chaotic'' as the stick grows shorter with KAM tori first evolving into resonance islands which subsequently dissolve into chaos.
	www.ma.utexas.edu /mp_arc/html/papers/96-57 (4388 words)

	Four Types of Chaotic Dynamics - Full Paper
		A representative motion of the states is depicted in five sequential frames of videotape shown in Figure lc.
		The chaotic motion of the ordered states in which ordered cells subtly change their shape and size is the most unusual and unexpected aspect of this study.
		The difficult question is whether these oscillations are chaotic and intrinsic to the structure of cellular flames, or whether they are extrinsic, due to the amplification of external noise (from nonaxial velocity components of the premixed gases) by the cellular structure.
	vip.cs.utsa.edu /flames/experiment/chaos/full.html (4404 words)

	The Chaotic Wheel
		A Lorenzian (or "chaotic") waterwheel is a physical model that perfectly corresponds to the Lorenz equations.A chaotic waterwheel is just like a normal waterwheel except for the facts that he buckets leak.
		If the parameters of the wheel are set correctly, the wheel will exhibit chaotic motion: rather than spinning in one direction at a constant speed, the wheel will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of behaviors in an unpredictable manner.
		A "C" was placed for chaotic motion, an "R" for rolling motion, and a "P" for periodic motion Please note that the graph is approximate.
	www.ace.gatech.edu /experiments2/2413/lorenz/fall02 (2824 words)

	Spectral Signatures
		The spectral signatures for periodic, quasiperiodic, and chaotic motion are illustrated in Figure 3.26.
		In periodic motion, all the peaks are rationally related to the primary peak (resonance).
		The power spectrum of a chaotic motion is easy to distinguish from periodic or quasiperiodic motion.
	cnls.lanl.gov /~nbt/Book/node92.html (413 words)

	Quantum signature of the classical chaos in the field-induced barrier crossing in a quartic potential*
		Hamilton’s equations of motion with a Hamiltonian functional properly defined for this purpose and by considering r (r, t) and (–c (r, t)) as canonically conjugate variables, where c (r, t) is the velocity potential.
		For the given set of parameter values the classical motion is regular for case (a) and chaotic for case (b) although both the cases correspond to the same value of the unperturbed energies of the double-well oscillator.
		To summarize, important insights into the quantum manifestations of the classical regular and chaotic motions of a double-well oscillator in the presence of an external field with different amplitudes have been obtained in terms of the corresponding Bohmian trajectories.
	www.ias.ac.in /currsci/may25/articles28.htm (2404 words)

	Katsirikou, Skiadas, Apostolou & Rompogiannakis Paper (Site not responding. Last check: 2007-10-24)
		A chaotic orbit is one that forever continues to experience the unstable behavior that an orbit exhibits near a source, but that is not itself fixed or periodic.
		If chaotic motion is to be observed in the motion of a physical system, it must be because the set on which the chaotic motion is occurring attracts a significant portion of initial conditions.
		The very important characteristic of the chaotic systems, that interest us, is that their behavior is never reproduced in the same way wherever is the starting point of the system, we are not in a position to describe its certain path till the end.
	www.iatul.org /conference/proceedings/vol10/papers/katsirik_full.html (5001 words)

	Ide, Kayo (1989-09-29) Regular and chaotic motion of uniform elliptical vortices in external linear time dependent ...
		The motion of a uniform vorticity of an elliptical shape (UEV) induced by an external linear flow field and a self induced effect in a two dimensional incompressible inviscid flow is investigated.
		The UEV motions in steady external linear flow fields are regular, however there are quite a few possibilities for types of motion depending on the UEV initial configuration and the external linear flow field parameters.
		A periodic motion of the UEV in a steady external linear flow field results in the generation of chaotic fluid particle mixing regions around the UEV where the fluid particles from distinct flow regions are chaotically transported.
	etd.caltech.edu /etd/available/etd-04132004-092941 (347 words)

	Chaotic Systems
		Of course the motion of the pendulum is strictly governed by Newton's Laws so we have a mathematical description of the motion.
		Certainly the motion is complex and sensitive to initial conditions.
		The differential equation describing the motion of a pendulum that is damped and driven (i.e.
	dept.physics.upenn.edu /courses/gladney/mathphys/subsection3_2_5.html (1874 words)

	Encyclopedia of the Solar System: Contents: Chapter 35
		An object in the solar system exhibits chaotic behavior in its orbit or rotation if the motion is sensitively dependent on the starting conditions, such that small changes in its initial state produce different final states.
		Possible examples of chaotic motion in the solar system include the rotation of the Saturnian satellite Hyperion, the orbital evolution of numerous asteroids and comets, and the orbit of Pluto.
		Chaotic motion has probably played an important role in determining the dynamical structure of the solar system.
	www.apnet.com /refer/solar/Contents/chap35.htm (305 words)

	Documentation on Billiards Game
		The rules of motion are as follows: a ball will move in a straight line until it hits the wall of the table.
		To help follow the motion and not get distracted by the clutter of the path on the table, mathematicians invented an associated space, called the phase space, and study the motion of the billiard in the phase space.
		Bunimovich gave a mathematical proof that the billiard motion was chaotic.
	serendip.brynmawr.edu /chaos/doc.html (1499 words)

	More on Chaos Theory
		An example of such sensitivity is the so-called "butterfly effect", whereby the flapping of a butterfly's wings is imagined to create tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a tornado to occur.
		Although chaotic planetary motion had not been observed, turbulence in fluid motion and nonperiodic oscillation in radio circuits had been encountered by experimentalists, without the benefit of a theory to explain what they were seeing.
		The key to the chaotic motion is a small pulsator (which stirs the water) that rises and falls randomly as the main pulsator rotates.
	www.artilifes.com /chaos-theory.htm (1885 words)

	Chaotic Motion of Pendulum with Support in Circular Orbit (Site not responding. Last check: 2007-10-24)
		In this paper, nonlinear plane oscillations of a pendulum, whose support is in steady, circular motion in a vertical plane, are investigated.
		Regular and chaotic motions are shown to be possible in both damped and undamped cases.
		It is shown herein that, at low speeds, both damped and undamped motion of the pendulum about a radial line become chaotic.
	www.pubs.asce.org /WWWdisplay.cgi?9100224 (161 words)

	Chaotic Compass
		Chaotic behaviour is described by the exponential deviation of nearby trajectories.
		As described in the page on the physics I have extended the equation of motion to include a damping effect.
		We have clearly observed, in a number of cases, chaotic motion dependent on the paramter λ.
	www.personal.psu.edu /students/d/j/djs479/HW2 (1038 words)

	D5 (Site not responding. Last check: 2007-10-24)
		It could be said that the period of the motion is longer than the age of the universe, and the system is chaotic.
		When the suspended magnet is given its initial displacement, the interaction with the fixed magnets is crucially dependent on the initial conditions, and chaotic motion is again observed, although this time the motion eventually stops, and the final position of the pendulum is ultimately determinable.
		Even when the system looks as if the motion is dying away, it can simply be as a result of the various spurs rotating in the same direction so as to make the structure appear as a rigid body with large moment of inertia.
	www.physics.gla.ac.uk /~kskeldon/PubSci/exhibits/D5 (403 words)

	Integrability and chaotic attractors
		We point out a link between integrability and chaotic motion by showing that the algebraic structure of the integrals of motion of a system can be used to get information on the chaotic solutions of this system or to reveal the parameters values for which there is no chaotic motion in this system.
		The chaotic attractor is stuck in the zone of phase space where there are no semi-permeable surfaces.
		Moreover with the help of the other integrals of motion of system (2), we can bound sharply its chaotic attractor.
	www.lmm.jussieu.fr /~neukirch/intechao/intechao.html (786 words)

	Brownian motion is chaotic (Site not responding. Last check: 2007-10-24)
		In case one needed any more persuasion that chaos is all around us, a Brussels-Maryland- Utah collaboration has for the first time demonstrated evidence for chaotic behavior in fluids at the microscopic level.
		A plot of the particle's position as a function of time is translated into a form which provides information about how the particle's position at one time correlates with the position at a later time.
		The chief symptom of chaos is the tendency for particles that initially follow nearby trajectories to diverge quickly from one another.
	www.phys.uni.torun.pl /~jkob/physnews98/node71.html (121 words)

	Chaotic motion of atoms in Rayleigh-Bénard conduction/convection system (Site not responding. Last check: 2007-10-24)
		The transition of temperature distributions between heat conduction and convection is observed.
		The degree of chaotic motion of atoms in the system is discussed in terms of the Lyapunov exponent.
		The degree of atomic chaos is found to increase in convection states.
	flux.aps.org /meetings/BAPSPC95/abs/SC0304.html (121 words)

	Periodic windows
		is the period of the motion) for a short interval in
		57 shows that just after the appearance of the window the attractor collapses to a period-3 attractor, and the chaotic intervals cease altogether.
		As we move away from the window, in parameter space, the intervals of periodic motion become gradually shorter and more infrequent.
	farside.ph.utexas.edu /teaching/329/lectures/node58.html (801 words)

	Wikinfo \| Chaos theory (Site not responding. Last check: 2007-10-24)
		Sensitivity on the initial conditions means that two such systems with however small a difference in their initial state eventually will end up with a finite difference between their states (however, two deterministic systems with identical initial conditions will remain identical).
		The fourth condition means that for any point in the system and any real number ε > 0 there is another point with distance d ≤ ε which is located on an periodic orbit.
		The theory has roots back to around 1900, but progressed more rapidly after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map.
	www.wikinfo.org /wiki.php?title=Chaos_theory (1103 words)

	REGULAR AND CHAOTIC DYNAMICS Home Page (Site not responding. Last check: 2007-10-24)
		This has been broadened to include a thorough introduction to chaotic motion in dissipative systems in the final two chapters.
		We present practical methods for describing the dynamics, for determining transitions from regular to stochastic behavior and routes to chaos, and for characterizing Hamiltonian chaos and dissipative chaotic attractors.
		Regular and Chaotic Dynamics is available through the ASUC Bookstore (510) 642-0770 and Cody's (510) 845-7852.
	www.eecs.berkeley.edu /~lieber/Textbook1.html (194 words)

	Chaotic motion in the Solar System (Site not responding. Last check: 2007-10-24)
		This page is designed to accompany the article by Carl Murray entitled, "Chaotic motion in the Solar System" in the Encyclopedia of the Solar System (Academic Press, 1998).
		In the article reference was made to the technique of "surface of section" whereby the regular or chaotic nature of an orbit can be recognised by plotting successive values of the x-component of the test particle's position and velocity whenever y=0 with the y-component of the velocity being positive.
		These quantities all refer to motion in a rotating reference frame.
	www.maths.qmw.ac.uk /Astronomy/SSD/chaos.html (434 words)

	Batteries' Technology: Chaotic motion to ordered motion (Site not responding. Last check: 2007-10-24)
		Everybody knows that all things are made of tiny particles that move in completely chaotic manner - it's called Brownian motion (Einstein explained the deal).
		Could it be possible to turn Brownian motion into ordered motion?
		It claims that scientists covered surface with some material, which was sensitive to light and had something to do with Brownian motion.
	battech.blogspot.com /2005/08/chaotic-motion-to-ordered-motion.html (185 words)

	Chaotic Motion and the Lorenzian water wheel
		Author: matthew r tokle I need help researching a topic on Chaotic Motion and the Lorenzian water- wheel.
		See page 27 for the water wheel and nearby find more examples of chaotic motion.
		The latter poses some nice computer experiments to illustrate theory and the weaknesses of computers in demonstrating it in some circum- stances.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH113.HTM (181 words)

	Direct Observations Of Hyperion's Chaotic Motion (Site not responding. Last check: 2007-10-24)
		Hyperion's irregular shape and the gravitational pull of Titan, a larger satellite of Saturn, make it a prime candidate for chaotic motion.
		The laws of motion and the largest computers are helpless here; although computer simulation can identify situations where chaos might develop.
		Contemplate what might happen -- or might have happened already -- if any of the other planets moved chaotically.
	www.science-frontiers.com /sf067/sf067a05.htm (140 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		We study the conditions under which the nucleons inside a deformed nucleus can undergo chaotic motion.
		To do this we perform self-consistent calculations in semiclassical approximation utilizing a multipole-multipole interaction of the Bohr-Mottelson type for quadrupole and octupole deformations.
		For the case of a harmonic static potential, we find that neither deformation leads to chaotic motion, whereas we observe chaotic damping of both multipole motions for an r{sup 6} static potential.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=386976 (198 words)

	Find in a Library: Chaotic motion of an elasto-plastic beam
		Find in a Library: Chaotic motion of an elasto-plastic beam
		Sorry, no libraries with the specified item were found.
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/9248275d1968febba19afeb4da09e526.html (54 words)

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