Periodic Limit Cycles Research Articles

Patterns at Primary Hopf Bifurcations of a Plexus of Identical Oscillators

Mathematically, an oscillator is a system of ordinary differential equations with a periodic limit cycle, usually arising from a Hopf bifurcation for a stationary solution. A plexus is a collection of such oscillators, coupled to each other via one-way or two-way communication channels. Plexuses model a wide number of phenomena in the mathematically oriented sciences. The primary Hopf bifurcation theory of such a plexus is developed. There are features of the bifurcating solutions which depend only on the combinatorics of the plexus and not on the details of the individual oscillators. Such features are called the pattern of the bifurcation. The relation of patterns to the modes of small oscillations of mechanical systems is discussed. More detailed information is developed for weak coupling. The theory is used to analyze two particular types of plexuses; viz, rings and lines of oscillators. It is then shown that the structure of a plexus permits numerical bifurcation computations to be made more efficient, compared to straightforward bifurcation computations. A numerical algorithm is developed to this end. The numerical methods are extended to incorporate higher-order information so as to determine whether the bifurcating branch is sub- or supercritical. Finally some effects of detuning are considered.

On the existence of periodic and eventually periodic solutions of a fluid dynamic forced harmonic oscillator

For certain flow regimes, the nonlinear differential equation , Y ≥ 0, G > 0 and constant, models qualitatively the behaviour of a forced, fluid dynamic, harmonic oscillator which has been a popular department store attraction. The device consists of a ball oscillating suspended in the vertical jet from a household fan. From the postulated form of the model, we determine sets of attraction and exploit symmetry properties of the system to show that all solutions are either initially periodic, with the ball never striking the fan, or else eventually approach a periodic limit cycle, after a sufficient number of bounces away from the fan.

Resonantly Forced Rossby Waves

Abstract A shallow, rotating layer of fluid that supports Rossby waves is subjected to turbulent friction through an Ekman layer at the bottom and is driven by a wave that exerts a shear stress on the upper boundary and for which the phase approximate that of a Rossby wave. The steady-state response may be either 1) a single wave that is synchronous with the driving wave or 2) a resonant triad of waves, one of which is synchronous with the driving wave. The triadic solutions constitute a one-parameter family, of which not more than one member is stable. The evolution equations for the slowly varying complex amplitudes of the responding waves, the fixed points of which correspond to the solutions 1) and 2), are established. The stability of these fixed points, and hence the stability boundaries for 1) and 2), are determined. There are no Hopf bifurcations of the fixed-point solutions, and the evolution equations apparently do not admit periodic (limit cycle), multiply periodic or chaotic solutions.

A laboratory study of baroclinic chaos on the f-plane

Laboratory experiments on the transition to chaos in a two-layer rotating fluid are described. The topography of the upper and lower bounding surfaces, and the rotation rate of the apparatus, are chosen so that the f-plane case of uniform basic potential vorticity in each layer is reproduced. Experiments are carried out in the near-critical single-wave region of parameter space where previous theory suggests a snap-through bifurcation from steady baroclinic waves to chaos. Instead, as the friction parameter is decreased past the curve of neutral stability of the axisymmetric basic state, we find several bifurcations before chaos is observed. The transition scenario includes successive limit cycles of doubled periods between the steady-wave and chaotic regions of parameter space. There is also evidence of ageostrophic behavior, even when the system Rossby number is less than 0.1.

Existence of limit cycles for a class of autonomous systems

A proof is given for the existence of at least one stable periodic limit cycle solution for the polynomial non-linear differential equation of the formin some cases where the Levinson-Smith criteria are not directly applicable.

Elementary limit cycles with application to the testing of digital circuits

Limit cycles in recursive digital filters are normally regarded as undesirable parasitic effects that one tries to minimize. However, one can also regard limit cycles as readily observable (and predictable) patterns of behavior that can be used to verify if a digital circuit is functioning properly. In this letter patterns of simple limit cycles that occur in first-order systems subject to some form of truncation or rounding, that are excited by a constant input, are investigated and classified in accordance with elementary number theory. The simplest of these result in input-output patterns that enable rapid verification of the correct functioning of simple combinations of adders and multipliers. As such they provide a useful method for testing this type of circuit in addition to conventional methods. Necessary and sufficient conditions for the following situations are derived: (i) a unique constant solution; (ii) a limit cycle of period 2 samples and of magnitude equal to m (integer), with m = 1 the most useful case; (iii) multiple constant solutions occurring in simple clusters; (iv) multiple limit cycles of type (ii) encircling a constant solution. Of these, type (ii) provides the most useful results for testing purposes. It is shown how the occurrence or nonoccurrence of limit cycles is related to increments in the value of u. It is also shown that limit cycles of period greater than 2 samples do not exist.

Bifurcations from Stationary to Periodic Solutions in a Low-Order Model of Forced, Dissipative Barotropic Flow

A truncated spectral model of the forced, dissipative, barotropic vorticity equation on a cyclic β-plane is examined for multiple stationary and periodic solutions. External forcing on one scale of the motion provides a barotropic analog to thermal heating. For forcing of any (finite) magnitude at the maximum or minimum scale in the truncation, the truncated solution converges in the limit as t → ∞ to the known solution of the corresponding linear model. If the forcing is constant, this limit solution represents a globally attracting stationary point in phase space. These results extend the well-known spectral blocking theorem of Fjortøft (1953) to forced, dissipative flows. The main results, however, obtain from a low-order model describing two disturbance components interacting with a constant, forced, basic-flow component of intermediate scale. The zonal dependence of either the basic flow or the disturbances is flexible and determined by the choice of component wave vectors. For low-wavenumber disturbances and β ≠ 0, the basic flow represents a unique stationary solution, which becomes unstable when the forcing exceeds a critical value. An application of the Hopf bifurcation theorem in the neighborhood of critical forcing reveals the existence of a periodic solution or limit cycle, which is then derived explicitly in phase space as a closed circular orbit whose frequency is described by a linear combination of the normal-mode Rossby-wave frequencies. The limit cycle radius, which physically represents the ultimate enstrophy of the disturbances, can be depicted as a response surface on the control plane defined by the independent forcing and beta parameters. If the forcing is zonally dependent, the response surface may exhibit a pronounced fold, which arises from the existence of a snap-through bifurcation. The projection of this fold onto the parameter control plane defines a bimodal or hysteresis region in which multiple stable solutions exist for given parameters. The boundary of the hysteresis region represents parameter states at which the model can exhibit sudden flow regime transitions, analogous to those observed in the laboratory rotating annulus. This study demonstrates that the degree of nonlinearity, the scale of the forcing, and the spatial dependence of the disturbances and the forcing all crucially influence both the multiplicity and temporal nature of the stable limit solutions in a low-order, forced, dissipative model. Thus, choices in this rather complex array of physical degrees of freedom must be carefully considered in any model of the long-term evolution of large-scale atmospheric flow.

Simplified model for Belousov-Zhabotinsky reaction

Noyes' scheme of the elementary processes for the Belousov-Zhabotinsky reaction has been simplified on the basis of reaction kinetic consideration. The simplest possible analogue (model K) has been described by a set of kinetic equations, and it is solved to examine the varieties of new ordered phases, which include temporal rhythm and spatial pattern. Particular attention has been given to the onset of new phases, which is associated with an anomalous enhancement of fluctuations. A stochastic theory of fluctuations, which was developed in our previous work, has been applied to the present case. Theoretical results compare reasonably well with experimental findings, i.e. with (1) spatial periodic structure and (2) limit cycle behaviour.

Crystal size distribution dynamics in a classified crystallizer: Part I. Experimental and theoretical study of cycling in a potassium chloride crystallizer

AbstractCycles in particle size and slurry density were experimentally observed in a mixed suspension, potassium chloride crystallizer equipped with a fines dissolver and having nonrepresentative product withdrawal. Cycling behavior was achieved with a product crystal elutriator but was also observed with changes in the orientation of the product removal tube, indicating that instability was induced by product classification. Simulation of the unstable runs with a dynamic model showed crystal size distribution (CSD) limit cycles of similar period using nucleation kinetics and process conditions measured experimentally. The computer simulation showed no tendency towards cycling for the stable runs.

Limit cycles in the combinatorial implementation of digital filters

The existence of limit cycles in combinatorial filters using two's complement truncation arithmetic is investigated in this paper. Exact results for limit cycles of period one and two are presented. Some results for longer period limit cycles are obtained using an effective value linear model. Bounds on these limit cycles are also derived. The accessability of the limit cycles is briefly discussed.

An iterative procedure for determining limit cycles using Lagrangianmechanics

In many nonlinear autonomous mechanical systems, determination of the amplitudes and periods of limit cycles by direct numerical integration of the system differential equations can be very time-consuming computationally. In this investigation, an efficient iterative algorithm which converges to limit cycles of singledegree-of-freedom systems is presented. It is based on the work done by nonconservative forces in the system. Numerical results for several illustrative examples are given, including the van der Pol equation, a feedback control system with hysteresis, a system having an infinite number of stable and unstable limit cycles, and an oscillator with nonlinear dry friction.

Stable limit cycles of time dependent multispecies interactions

The standard differential system which models the interaction of n species is considered under the assumption that the coefficients (i.e., the net birth rates, the self-inhibition coefficients and the interaction coefficients) are all periodic functions of time. Conditions are given which guarantee the existence of a stable periodic limit cycle. The basic result implies, roughly, that if an n − 1 species subcommunity with a stable periodic limit cycle exists and if the interactions of the system are sufficiently weak, then the addition of the nth species will result in a stable periodic limit cycle provided its average net birth rate is larger than and close to a specified critical value; on the other hand, if this average is less than but close to the critical value, then the nth species will not survive and the system will stabilize on the limit cycle of the subcommunity. Starting with basic results concerning periodic solutions of the one species case, we apply our basic result in a “bootstrapping” manner to derive a corollary which, roughly speaking, states that if at least one species has a positive average net birth rate and if the interactions are sufficiently weak, then a multispecies community will have a stable, positive limit cycle provided that the average net birth rates of the remaining species lie in certain specified ranges.

Nonlinear Aspects of Competition Between Three Species

It is shown that for three competitors, the classic Gause–Lotka–Volterra equations possess a special class of periodic limit cycle solutions, and a general class of solutions in which the system exhibits nonperiodic population oscillations of bounded amplitude but ever increasing cycle time. Biologically, the result is interesting as a caricature of the complexities that nonlinearities can introduce even into the simplest equations of population biology ; mathematically, the model illustrates some novel tactical tricks and dynamical peculiarities for 3-dimensional nonlinear systems.

Oscillations Caused by Solid Friction : Part 3, In the Case of Maximum Static Friction Different from Kinetic Friction without Slipping

This paper concerns theoretically with self-excited oscillations caused by solid friction. A model, in which solid friction and spring force are applied to a block lying on a moving belt, is used to represent this class of mechanical problem. The solid friction force considered is assumed to vary with the relative sliding velocities between two solid bodies, namely, the friction-velocity characteristic is given by a polygon having two straight line segments and the critical value of static friction is different from the value of kinetic friction without slipping. Several types of limit cycles and the regions in which they occur are shown in figures according to the characteristics of the friction-velocity function. The appearances of amplitudes and periods of limit cycles are also described in nondimensional forms against the velocity of a moving belt for various characteristics of the friction-velocity function.

Performance cost functions for a reaction-jet controlled system during an on-off-on limit cycle

In this paper, analytical expressions are obtained for performance cost functions which describe the behavior of a second-order nonlinear control system during an on-off limit cycle. The cost functions are 1) fuel consumption per second (and the associated percent on-time and control pulse width); 2) average error; 3) peak-to-peak, limit-cycle amplitude; and 4) limit-cycle period. Each of these functions is expressed in terms of the amplitudes of the limit-cycle switch states \theta_{\delta} and \theta_{\delta} The results, which are given for a stable and an unstable plant, do not depend upon a specified control law.

Performance cost functions for a reaction-jet-controlled system during an on-off limit cycle

In this paper, analytical expressions are obtained for performance cost functions which describe the behavior of a second-order nonlinear control system during an on-off limit cycle. The cost functions are 1) fuel consumption per second (and the associated percent on-time and control pulse width); 2) average error; 3) peak-to-peak, limit-cycle amplitude; and 4) limit-cycle period. Each of these functions is expressed in terms of the amplitudes of the limit-cycle switch states <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta_{\delta}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta_{\delta}</tex> The results, which are given for a stable and an unstable plant, do not depend upon a specified control law.

Das Studium von Reibungsschwingungen mit der polygonalen Näherungsmethode

The nonlinear damping characteristicD(v) of a self-sustained system is replaced by two piece-wise linear characteristics. The integral curves calculated with their help bound the integral curve corresponding to the nonlinear damping characteristic. The method may be used for the determination of motion and velocity time history, integral curves, limit cycles and amplitude succession. The stability and period of limit cycles may also be determined. The method is used for studying oscillations due to dry friction.

Design of Nonlinear Sampled-Data System for Acceleration Measurement

The design of a sampled-data nonlinear system for three-axes acceleration measurement is discussed. The basic specific force equation is given, and the equation for the indicated velocity (integral of indicated acceleration) is derived to show the dependence of the error on the shape of the limit cycle trajectory. Emphasis is on how certain physical parameters, such as damping and dead zone width, may be chosen to fulfill the design requirement of minimum error in the indicated velocity output of the system. Expressions for design purposes relating physical parameters, such as damping, and phase plane switch line spacing to limit cycle period are derived. High limit cycle frequencies, or low periods, are shown to be less objectionable. An expression is given for the approximate dead zone width for minimum error from dynamic average and static offsets of the limit cycle trajectories. A multiscale-normalized phase plane technique is described and used to determine both the transient and final limit cycle trajectories for assumed step acceleration inputs at arbitrary points during an existing limit cycle.

An Analysis of Wind-Driven Ocean Circulation with a Limited Number of Fourier Components

The equations of motion for a square ocean basin of dimension L on the β-plane are solved approximately for the case where a wind-strew curl of the form sin (x/L) sin (y/L)[0≤x,y≤πL] is applied to the surface. The stream function is expanded in a double Fourier sine series and this representation is truncated after only four terms. The resulting set of equations contains the effects of non-linearity, time dependence, linear variation of the Coriolis parameter, friction, and wind-stress. Multiple solutions to the steady-state equations exist when the wind-stress is sufficiently strong. One of the solutions can be related to Sverdrup's (1947) solution for an ocean basin with one longitudinal boundary. A second solution is dominated by the non-linear interactions of the system. Integration of the non-linear transient equations are carried out for the case where the wind-stress starts at some initial time. In some cases the system goes through a series of oscillations of decreasing amplitude before it settles down to a steady state. For a certain range of the Rossby number (or alternatively of the strength of the wind stress) the ocean never settles down to a steady state but, after an initial transient phase, enters a periodic limit cycle. However, if initial conditions are taken sufficiently close to the known steady state solution, the system always settles down into the steady state after the initial transient phase.