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Topic: Limit cycles

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Chaos (mythology)

	Wavelet Analysis To Predict Limit Cycles
		In a typical application that involves aeroelasticity, for the purpose of the present method, limit cycles are behaviors that basically consist of steady oscillations of parts of an aircraft.
		This limit cycle occurs at an airspeed of 9.8 m/s.
		A prediction function for the limit cycle can be computed as a ratio between (1) the magnitude of the maximum change in scale and (2) the time when this change occurs.
	www.nasatech.com /Briefs/Mar01/DRC0110.html (683 words)

	The effects of static friction and backlash on extended physiological proprioception control of a powered prosthesis (Site not responding. Last check: 2007-10-14)
		Previous attempts to understand the nature of the observed limit cycles demonstrated that static friction was a factor in the development of limit cycles in an EPP-controlled Michigan Feeder Arm [16].
		Systems that develop a limit cycle contain variable damping that will cause the amplitude of the oscillations of the system to grow or decay until a fixed amplitude is reached [17].
		To eliminate the asymmetric effects of gravity and reduce the number of variables that could be affecting the development of limit cycles, we mounted the elbow on its side so that the forearm moved in a plane parallel to the ground as the elbow flexed and extended.
	www.vard.org /jour/05/42/3/farrell.html (7123 words)

	Limit Cycles and Chaos (Site not responding. Last check: 2007-10-14)
		Non-equilibrium dynamics may be of 2 types: a limit cycle when the trajectory repeats itself, and chaotic when the trajectory does not repeat itself.
		The confidence interval for parameter values is usually large enough to cover both chaotic and non-chaotic (limit cycle) model dynamics.
		I suspect that in these cases chaotic dynamics was induced by the seasonal cycle in population numbers.
	www.ento.vt.edu /~sharov/PopEcol/lec9/chaos.html (467 words)

	limitcycle (Filter Design Toolbox)
		an integer indicating the repeat period of the limit cycle (-1 if the filter converged and the last state is zero, 0 if the last state is not zero and no limit cycle was detected).
		In this example, there is a region of initial conditions in which no limit cycles occur, and a region where they do.
		If no limit cycles are detected before the Monte Carlo trials are over, the state sequence spirals to zero.
	www.weizmann.ac.il /matlab/toolbox/filterdesign/limitcycle.html (478 words)

	Lab 4C (Site not responding. Last check: 2007-10-14)
		The van der Pol equation gives an example of a system of differential equations exhibiting a limit cycle, i.e., there is a closed curve in the phase plane corresponding to periodic time series (see Figure 4.1 on page 348 of our textbook).
		A limit cycle is called attracting, if solutions nearby approach the cycle (the van der Pol equation's limit cycle is attracting!), it is called repelling if nearby solutions move away from the cycle.
		Last not least, a limit cycle is called semi-stable, if it is attractive for solutions from the outside and repelling for solutions from the inside, or vice versa.
	www.math.utep.edu /classes/3226/lab4c/lab4c.html (278 words)

	Bifurcations in the modified Hodgkin-Huxley neuron model
		The motion on the saddle limit cycle slows down approaching the vicinity of the equilibrium state along its stable manifold.
		The gray dots represent the chaotic attractor; the fl filled circle is the saddle-focus equilibrium state; the unstable limit cycles of periods 1, 2 and 3 are shown by blue, red and green bold lines, respectively.
		Moreover, they are extremely close to each other, that is, the period-1 cycle is a part of the period-2 cycle, the period-2 cycle is a part of the period-3 cycle and so on.
	www.phy.ohiou.edu /~neiman/CR7/node3.html (1253 words)

	WWU Math Department - Colloquium
		Limit cycles in dynamical systems represent a specific periodic solution in a phase plane diagram.
		In applications limit cycles mark periodic phenomena, such as in the modeling of the human heartbeat.
		In general finding the number of limit cycles for a specific dynamical system can be quite complicated and has connections to the unsolved Hilbert's 16th problem.
	www.wwu.edu /depts/math/colloquium/c_053105.htm (196 words)

	The origin of power dropouts
		Therefore the stabity criteria for the mode with a frequency close to zero may be obtained from the stability analysis of the equation for a laser with phase conjugated feedback.
		As further decreases, the limit cycle may coalesce with a saddle point (antimode) through a saddle node bifurcation.
		then the limit cycle is destroyed and the trajectory falls onto the next limit cycle with a lower frequency.
	www.ucc.ie /ucc/depts/physics/opto/feedback/node2.html (745 words)

	Second-Order Digital Filters Done Right
		Another phenomenon is called a limit cycle, which in general is an oscillation, with a frequency that's often not too far from the filter frequency setting.
		There are two types of these oscillations: one is a full-scale limit cycle that occurs when excessive output values are not clipped, but instead "wrap around." This refers to a characteristic of a signed number encoding (called twos complement), where increasing a value beyond its range causes it to suddenly change sign.
		Limit cycles, overflow oscillations, and the jump phenomenon are less likely but may occur.
	www.rane.com /note157.html (4956 words)

	Egwald Mathematics — Nonlinear Dynamics: Limit Cycles in Two Dimensional Flows
		Limit cycles and fixed points structurally determine the dynamics of solutions to differential equation systems.
		If this system has a limit cycle surrounding the origin, it will be an attracting limit cycle, as the following diagrams confirm.
		For the van der Pol equation, the shape and size of the limit cycle depends on the parameter μ.
	www.egwald.com /nonlineardynamics/limitcycles.php (1432 words)

	The Phase Plane
		Limit cycles in nonlinear systems are different than what you find in linear systems.
		In many nonlinear systems limit cycles may not have variable amplitude, and you may end up at the exact same limit cycle from many different starting points.
		You should be able to see that the system goes to the same limit cycle behavior no matter where the system starts.
	www.facstaff.bucknell.edu /mastascu/eControlHTML/Nonlinear/NonLinear1PhasePlane.htm (1252 words)

	Limit Cycles near Stationary Points in the Lorenz System (Site not responding. Last check: 2007-10-14)
		The limit cycles in the Lorenz system near the stationary points are analysed numerically.
		The stable limit cycle is smaller than the unstable one and has not been previously reported in the literature.
		In addition, all the limit cycles in the Lorenz system are theoretically proven not to be planar.
	stacks.iop.org /0256-307X/22/2780 (246 words)

	The Magnificent Seven - Print Version
		Mathematically, multiples integrals of pi (i.e., pi, 2pi, 3pi, and 4pi) are the "eigenvalues" of "stable limit* cycles" for a chaotic "predator-prey" (nonlinear) system of equations of economics in which the predator is the wage output and the prey is the rate of employment and the time rate of change of debt.
		The two-dimensional "limit cycles" or "weekly long-wave cycles" presented in the Magnificent Seven are actually n-dimensional waves of an "expanding vortex of money in circulation around the world".
		That is, when the 176-week cycle tops and heads downward (downtrend in the major market indices), the Fed Funds Rate starts significantly upward (this major interest-rate cycle is now starting again with the 176-week cycle topping in the S&P 500 in Nov 2004!).
	www.gold-eagle.com /editorials_04/rinehart080504pv.html (1553 words)

	A simplified single mode model
		In the model, the LFF attractor is born as a consequence of the merging of many limit cycles.
		As further decreases, the limit cycle may coalesce with a saddle point (anti-mode) through a saddle node bifurcation.
		This gluing of the two limit cycles yields a large limit cycle.
	www.ucc.ie /ucc/depts/physics/opto/hegarty/node42.html (1026 words)

	Limit Cycles
		When overflow occurs, even otherwise stable filters may get stuck in a large-scale limit cycle, which is a short-period, almost full-scale persistent filter output caused by overflow.
		Saturation arithmetic has been proved to prevent zero-input limit cycles, which is one reason why all DSP microprocessors support this feature.
		Poles close to the unit circle increase the magnitude and likelihood of small-scale limit cycles.
	cnx.org /content/m11928/1.2 (327 words)

	Limit Cycle, Mu-Ency at MROB (Site not responding. Last check: 2007-10-14)
		A limit cycle is also called an attractor.
		When iteration diverges or is chaotic (failing to diverge or converge) there is no attractor and "limit cycle" is undefined for that iteration.
		The number of points in a limit cycle is called the period.
	www.mrob.com /pub/muency/limitcycle.html (91 words)

	CJM - Bifurcations of Limit Cycles From Infinity in Quadratic Systems
		We investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic differential systems in the plane having a degenerate critical point at infinity.
		It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from infinity together with eventual limit cycles around it.
		We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center.
	journals.cms.math.ca /cgi-bin/vault/view/gavrilov1074 (214 words)

	Financial Sense Univ. ~ The Magnificent Seven - Part 4 by Stephen Rinehart 07.25.2004
		The historic 45-year interest rate lows were to reinitialize the economic cycles to the same baseline for all world central banks to possibly prevent a deflationary response while increasing M3.
		These appear to be the “source” for the amplitudes of the cycles in all major world indexes (it feeds the increasing amplitudes in world indices).
		That is, when the 176-week cycle tops and heads downward (downtrend in the major market indices), the Fed Funds Rate starts significantly upward (this major interest-rate cycle is now starting again with the 176-week cycle topping in the SandP 500 in Nov 2004!).
	www.financialsense.com /fsu/editorials/rinehart/2004/0725.html (1711 words)

	Cartan's Corner : The Hopf Index and Limit Cycles (Site not responding. Last check: 2007-10-14)
		A limit cycle of the Van der Pohl type is a strange robust thing.
		It is not a cycle in the usual Hamiltonian sense of a harmonic oscillator, for its total mechanical energy is not a constant of the motion.
		The limit cycle, unlike the oscillator, evidently interacts with its surroundings similar to a "breathing system", taking in energy and exhuding energy twice for each complete phase space cycle.
	www22.pair.com /csdc/car/carfre61.htm (211 words)

	XPP TUTORIAL IV
		When a system of differential equations has a stable limit cycle solution, one is often interested in the behavior when several such systems are coupled together.
		The phase shift that results will depend on both the component of the limit cycle that was perturbed and the time in the cycle at which the perturbation occured.
		In the limit that the perturbations become infinitesimally small, this phase shift function tends to a computable vector quantity called the adjoint or response function.
	mrb.niddk.nih.gov /xpp/xpptut4.html (2259 words)

	February 15 (Site not responding. Last check: 2007-10-14)
		Wilson suggested already in 1971 that RG solutions could display a limit cycle behavior.
		After an introduction into this subject, I will discuss an example of an effective field theory involving a limit cycle with applications in atomic and nuclear physics.
		I will illustrate the manifestation of limit cycles in observables and address the possibility of an infrared limit cycle in the fundamental theory of the strong interaction, Quantum Chromodynamics.
	www.physics.arizona.edu /~shufang/coll_spring05/feb15.html (131 words)

	PlanetMath: limit cycle
		In simpler terms a limit cycle is an isolated periodic solution of the system.
		This is version 5 of limit cycle, born on 2005-02-06, modified 2006-07-21.
		(Ordinary differential equations :: Qualitative theory :: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramif)
	planetmath.org /encyclopedia/UnstableLimitCycle.html (142 words)

	Integrated Composite Analyzer (ICAN/JAVA) Load Input
		Basically, the type of input needed is the lower and upper limits of the cyclic load, the number of cycles, and the cyclic degradation coefficient beta
		Entries one and two are the upper and the lower cyclic load limits.
		The third entry is the total number of cycles to be applied.
	www.grc.nasa.gov /WWW/StructMech/ICAN/html/ican.doc.input.loads.html (2573 words)

	Population Cycles in Mammals
		A number of mathematical models have shown that prey-predator models can produce limit cycles, but they have not demonstrated that the cyclic fluctuations observed in natural populations can be explained by the mechanisms they incorporate, and the parameters they define.
		The parameters of the model are estimated from studies on the biology of cyclic lynx and hare populations, rather than by fitting time-series data to the model.
		Parameters of the model give rise to limit cycles when they are changed in the way they are expected to change from south to north, which is consistent with the observations on the latitudinal patterns in cyclicity.
	www.ramas.com /cycles.htm (292 words)

	On the number of limit cycles of the Li'enard equation - Giacomini, Neukirch (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system.
		We obtain also a sequence of polynomials R n (x) whose roots of odd multiplicity are related to the number and location of the limit cycles of the system.
		Giacomini and S. Neukirch, "The number of limit cycles of the Li'enard equation ", Physical Review E 56 3809 (1997).
	citeseer.ist.psu.edu /24348.html (252 words)

	Limit Cycles of xy (mod x+y)
		Of course, it follows that {7,5,11} is also a limit cycle.
		There is an interesting class of cycles of period 9, in which the cycle consists of just four distinct numbers in the pattern A BC A DB A CD.
		In this cycle the greatest common divisor of the cycle is one of the terms of the cycle.
	www.mathpages.com /home/kmath193.htm (198 words)

	AMCA: Exact number of limit cycles for a family of rigid systems by Armengol Gasull (Site not responding. Last check: 2007-10-14)
		For a given family of planar differential equations it is a very difficult problem to get an upper bound for its number of limit cycles.
		Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle.
		As we will see, the condition that allows to separate the existence and the non existence limit cycles can be described but it is very intricate.
	at.yorku.ca /c/a/l/o/87.htm (194 words)

	Relative positions of limit cycles in a Kolmogorov-type system (Site not responding. Last check: 2007-10-14)
		To estimate the relative position of limit cycles for a system is always useful in the qualitative analysis of the system.
		If the system has only one limit cycle, determining the position of the limit cycle is more important.
		Furthermore, the approach to the location of limit cycles is very applicable because the region is easy to compute.
	stacks.iop.org /0305-4470/22/L317 (248 words)

	Limit cycles and stochastic resonance in a Langevin equation subject to white noise (Site not responding. Last check: 2007-10-14)
		Limit cycles and stochastic resonance in a Langevin equation subject to white noise
		Limit cycles and stochastic resonance in a periodically driven Langevin equation subject to white noise
		Use of this service is subject to compliance with the terms and conditions of use.
	stacks.iop.org /0305-4470/37/7473 (283 words)

	Detection of slow-fast limit cycles in a model for electrical activity in the pancreatic {beta}-cell -- LENBURY et al. ...
		Detection of slow-fast limit cycles in a model for electrical activity in the pancreatic {beta}-cell -- LENBURY et al.
		limit cycles composed of alternate slow and fast transitions.
		Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues.
	imammb.oxfordjournals.org /cgi/content/abstract/13/1/1 (260 words)

	SSRN-The Birth of Limit Cycles in Cournot Oligopoly Models with Time Delays by Carl Chiarella
		We discuss types of nonlinearities which may be present to bound the motion and introduce time lags in production and information which may serve as bifurcation parameters.
		In the case of identical firms we apply the Hopf bifurcation theorem to determine conditions under which limit cycle motion is born.
		Chiarella, Carl, "The Birth of Limit Cycles in Cournot Oligopoly Models with Time Delays" (December 1991).
	papers.ssrn.com /sol3/papers.cfm?abstract_id=883572 (239 words)

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