Emily Stone

Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

We study a model for chatter in twist drills derived by Stone and Askari [Dynam. Sys., 17, 1 (2002), 65–85], in which a linear vibration mode interacts with nonlinear cutting forces. This results in a delay differential equation describing an oscillator that is nonlinear in damping and with cross-terms in the damping and the delay. Linear stability analysis of the model with significant nonlinear terms is performed, and an analysis of the nonlinear stability of the primary Hopf bifurcation is observed. The latter is done via the construction, using symbolic algebra, of a two-dimensional centre manifold in the infinite dimensional space employing an algorithm developed by Campbell and Bélair [SIAM J. Appl. Math., 54, 5 (1994), 1402–1424; and Can. Appl. Math. Quart., 3, 2 (1995), 137–154]. Our analysis shows that the stability of the Hopf bifurcation depends on the type of vibration in question and on the cutting speed. These results are confirmed numerically and further bifurcations in the high-speed limit are also explored numerically, with tantalizing results that could be the basis of much future work.

Noisy heteroclinic networks

The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random switching between the heteroclinic cycles determined by the linear dynamics near one of the saddle points, noise induced stability of a cycle, and intermittent switching between cycles. All three responses are explained by examining the size of the stable and unstable eigenvalues at the equilibria.

Introduction to archetypal analysis of spatio-temporal dynamics

Abstract A comparison is made between the principal component or Karhunen-Loeve (KL) decomposition of spatio-temporal data and a new procedure called archetypal analysis (Cutler and Breiman, 1994). Archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values. We show that archetypes may be more appropriate than KL when the data do not have elliptical distributions, and they are often well-suited to studying regimes in which the system spends a long time near a “steady” state, punctuated with quick excursions to outliers in the data set, which may represent intermittency. We also introduce a variation of archetypal analysis that is designed to track moving structures, such as traveling waves or solitions. By using this method the traveling part of the motion is separated from the stationary (or semi-stationary) pattern. Advantages and disadvantages of each method are discussed.

Archetypal analysis of spatio-temporal dynamics

Abstract A comparison is made between the principal component or Karhunen-Loeve decomposition of two sets of spatio-temporal data (one numerical, the other experimental) and a new procedure called archetypal analysis (Cutler and Breiman, 1994). Archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values. Archetypes may be more appropriate than KL when the data do not have elliptical distributions, and are often well-suited to studying regimes in which the system spends a long time near a “steady” state, punctuated with quick excursions to outliers in the data set, which may represent intermittency. Other advantages and disadvantages of each method are discussed.

Noise and O(1) amplitude effects on heteroclinic cycles.

The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained. (c) 1999 American Institute of Physics.

Exploring archetypal dynamics of pattern formation in cellular flames

The application of archetypal analysis to high-dimensional data arising from video-taped images is presented. Included in the analysis are intermittent regimes which have not been analyzed previously by other statistical methods such as principal component analysis (PCA). A hybrid PCA/archetypes technique has been developed to overcome the difficulties of applying archetypes to data sets with points living in a space of dimension higher than about 500. The advantages of the method lie in the creation of patterns typical of the set as a whole, and an expression of the dynamics in terms of these patterns. Archetypes are particularly useful in identifying intermittent regimes, where low energy events that might be missed by a severe principal component truncation are none-the-less crucial to understanding the dynamics. They are part of a suite of data analysis techniques that can be used on dynamic data sets (such as FFT, PCA and other spectral decompositions). This hybrid method extends the application of archetypes to spatio-temporal dynamics in two-dimensional patterns.

Effect of a refractory period on the entrainment of pulse-coupled integrate-and-fire oscillators

Abstract It was shown by Mirollo and Strogatz [SIAM J. Appl. Math. 50 (1990) 1645] that a population of globally coupled, identical, integrate-and-fire oscillators will almost always become entrained. We find that the inclusion of a refractory period in the cycle of each oscillator can result in open sets of initial configurations evolving to asynchronous states.

Analyzing the dynamics of cellular flames

Abstract Video data from experiments on the dynamics of two-dimensional flames are analyzed. The Karhumen-Loeve (KL) analysis is used to identify the dominant spatial structures and their temporal evolution fot several dynamical regimes of the flames. A data analysis procedure to extract and process the boundaries of flame cells is described. It is shown how certain spatial structures are associated with certain temporal events. The existence of small scale, high frequency, turbulent background motion in almost all regimes is revealed.