Asymmetric Limit Cycles Research Articles

GLUING BIFURCATIONS IN CHUA OSCILLATOR

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

Accordion waves in Myxococcus xanthus.

Myxococcus xanthus are Gram-negative bacteria that glide on solid surfaces, periodically reversing their direction of movement. When starved, M. xanthus cells organize their movements into waves of cell density that sweep over the colony surface. These waves are unique: Although they appear to interpenetrate, they actually reflect off one another when they collide, so that each wave crest oscillates back and forth with no net displacement. Because the waves reflect the coordinated back and forth oscillations of the individual bacteria, we call them "accordion" waves. The spatial oscillations of individuals are a manifestation of an internal biochemical oscillator, probably involving the Frz chemosensory system. These internal "clocks," each of which is quite variable, are synchronized by collisions between individual cells using a contact-mediated signal-transduction system. The result of collision signaling is that the collective spatial behavior is much less variable than the individual oscillators. In this work, we present experimental observations in which individual cells marked with GFP can be followed in groups of unlabeled cells in monolayer cultures. These data, together with an agent-based computational model demonstrate that the only properties required to explain the ripple patterns are an asymmetric biochemical limit cycle that controls direction reversals and asymmetric contact-induced signaling between cells: Head-to-head signaling is stronger than head-to-tail signaling. Together, the experimental and computational data provide new insights into how populations of interacting oscillators can synchronize and organize spatially to produce morphogenetic patterns that may have parallels in higher organisms.

PERIODIC-IMPACT MOTIONS AND BIFURCATIONS OF THE VIBRATORY SYSTEM WITH A CLEARANCE

A multi-degree-of-freedom vibratory system with a clearance is considered. The system consists of linear components, but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Such models play an important role in the studies of mechanical systems with clearances or gaps. Period one double-impact symmetrical motions are derived analytically according to the set of periodicity and matching conditions, and associated Poincare map is established. Stability and local bifurcations of the fixed point of double-impact symmetrical motion is analyzed by using the Poincare map. A two-degree-of-freedom vibratory system with a clearance is used as an example to demonstrate the validity of the analysis. Stability of periodic-impact motions, bifurcations, grazing singularities and routes to chaos are analyzed for the two-degree-of-freedom vibratory system with a clearance, in turn. Dynamics of the fundamental element in vehicle dynamics, a suspended, rolling wheelset is described. The diversity of dynamical behavior in this rolling wheelset with vibro-impact is demonstrated. Interesting features like both symmetric and asymmetric limit cycles, pitchfork bifurcation, period-doubling bifurcation, grazing boundary singularity and chaos, etc., are found.

Spontaneous phase symmetry breaking in a four-frequency ring gas laser

Lasing regimes of a single-mode four-frequency class-A ring gas laser with elliptical polarization of the emitted waves are studied numerically. Stationary regimes typical of both standing-and traveling-wave lasing are discovered. Self-oscillations exhibiting the properties of asymmetric and symmetric limit cycles are also found. It is shown that transition between cycles with different symmetry may result in the spontaneous phase symmetry breaking and the appearance of chaos arising due to the period doubling bifurcation cascade of the asymmetric limit cycle.

Parameter estimation from relay autotuning with asymmetric limit cycle data

Obtaining the parameters for PID controllers based on limit cycle information for the process in a relay controlled feedback loop has become an accepted practical procedure. Often the only information used is the frequency and amplitude of the limit cycle, which then using the describing function analysis provides estimates for the ultimate gain and frequency of the process. Alternatively, limit cycle measurements may be used to estimate the parameters of an assumed transfer function, usually again using describing function analysis. Both these methods can work well but may lead to problems in cases where the describing function analysis is inaccurate. In this paper, it is shown how the parameters of first and second order plant transfer functions, stable or unstable, with time delay can be found exactly, assuming no measurement errors, from measurements of the parameters of a single asymmetrical limit cycle in a relay controlled feedback loop. Examples are given to compare the approach with results using the describing function method and also to show how good controller design based on a first or second order time delay model can be achieved for plants of higher order.

Dynamic phase transition, universality, and finite-size scaling in the two-dimensional kinetic Ising model in an oscillating field.

We study the two-dimensional kinetic Ising model below its equilibrium critical temperature, subject to a square-wave oscillating external field. We focus on the multidroplet regime, where the metastable phase decays through nucleation and growth of many droplets of the stable phase. At a critical frequency, the system undergoes a genuine nonequilibrium phase transition, in which the symmetry-broken phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. We investigate the universal aspects of this dynamic phase transition at various temperatures and field amplitudes via large-scale Monte Carlo simulations, employing finite-size scaling techniques adopted from equilibrium critical phenomena. The critical exponents, the fixed-point value of the fourth-order cumulant, and the critical order-parameter distribution all are consistent with the universality class of the two-dimensional equilibrium Ising model. We also study the cross-over from the multidroplet regime to the strong-field regime, where the transition disappears.

Testing and Modeling Threshold Autoregressive Processes

Abstract The threshold autoregressive model is one of the nonlinear time series models available in the literature. It was first proposed by Tong (1978) and discussed in detail by Tong and Lim (1980) and Tong (1983). The major features of this class of models are limit cycles, amplitude dependent frequencies, and jump phenomena. Much of the original motivation of the model is concerned with limit cycles of a cyclical time series, and indeed the model is capable of producing asymmetric limit cycles. The threshold autoregressive model, however, has not received much attention in application. This is due to (a) the lack of a suitable modeling procedure and (b) the inability to identify the threshold variable and estimate the threshold values. The primary goal of this article, therefore, is to suggest a simple yet widely applicable model-building procedure for threshold autoregressive models. Based on some predictive residuals, a simple statistic is proposed to test for threshold nonlinearity and specify the ...

Testing and Modeling Threshold Autoregressive Processes

Abstract The threshold autoregressive model is one of the nonlinear time series models available in the literature. It was first proposed by Tong (1978) and discussed in detail by Tong and Lim (1980) and Tong (1983). The major features of this class of models are limit cycles, amplitude dependent frequencies, and jump phenomena. Much of the original motivation of the model is concerned with limit cycles of a cyclical time series, and indeed the model is capable of producing asymmetric limit cycles. The threshold autoregressive model, however, has not received much attention in application. This is due to (a) the lack of a suitable modeling procedure and (b) the inability to identify the threshold variable and estimate the threshold values. The primary goal of this article, therefore, is to suggest a simple yet widely applicable model-building procedure for threshold autoregressive models. Based on some predictive residuals, a simple statistic is proposed to test for threshold nonlinearity and specify the threshold variable. Some supplementary graphic devices are then used to identify the number and locations of the potential thresholds. Finally, these statistics are used to build a threshold model. The test statistic and its properties are derived by simple linear regression. Its performance in the finite-sample case is evaluated by simulation and real-world data analysis. The statistic performs well as compared with an alternative test available in the literature. Further applications of threshold autoregressive models are also suggested, including handling heterogeneous time series and modeling random processes with periodic variances whose periodicity is not fixed. The latter phenomenon is commonly encountered in practice, especially in econometrics and biological sciences.