... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal C... more ... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal Cutting Emily Stone Department of Mathematical Sciences, The University of Montana-Missoula Missoula, MT 59812-0864, USA ... Mechanical engineers Page 2. 60 Emily Stone et. al. ...
We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, ... more We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, Dyn. Syst. 17 (2002), pp. 65-85] for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the ...
A rather general 'explanation' is outlined of intermittent turbulence production in flo... more A rather general 'explanation' is outlined of intermittent turbulence production in flow dominated, at the larger scales, by coherent structures. Symmetries in a 'perfect' underlying flow lead to bundles of recurrent solutions which circulate near a heteroclinic attractor in the ...
The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensio... more 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.
The application of archetypal analysis to high-dimensional data arising from video-taped images i... more 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.
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particul... more 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 ...
A comparison is made between the principal component or Karhunen-Loève decomposition of two sets ... more A comparison is made between the principal component or Karhunen-Loève 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.
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a ... more The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a particular stoma is open, CO 2 is absorbed through its aperture, but H 2 O is also lost due to evaporation. Thus a plant will seek a stomatal aperture that balances its need for CO 2 ...
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal C... more ... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal Cutting Emily Stone Department of Mathematical Sciences, The University of Montana-Missoula Missoula, MT 59812-0864, USA ... Mechanical engineers Page 2. 60 Emily Stone et. al. ...
We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, ... more We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, Dyn. Syst. 17 (2002), pp. 65-85] for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the ...
A rather general 'explanation' is outlined of intermittent turbulence production in flo... more A rather general 'explanation' is outlined of intermittent turbulence production in flow dominated, at the larger scales, by coherent structures. Symmetries in a 'perfect' underlying flow lead to bundles of recurrent solutions which circulate near a heteroclinic attractor in the ...
The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensio... more 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.
The application of archetypal analysis to high-dimensional data arising from video-taped images i... more 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.
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particul... more 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 ...
A comparison is made between the principal component or Karhunen-Loève decomposition of two sets ... more A comparison is made between the principal component or Karhunen-Loève 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.
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a ... more The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a particular stoma is open, CO 2 is absorbed through its aperture, but H 2 O is also lost due to evaporation. Thus a plant will seek a stomatal aperture that balances its need for CO 2 ...
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
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