Jim Swift - Academia.edu (original) (raw)
Papers by Jim Swift
Journal of Nonlinear Science
Physica D: Nonlinear Phenomena, 1998
We show how the symmetry of attractors of equivariant dynamical systems can be observed by equiva... more We show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space. Equivariant projections have long been used, but they can give misleading results if used improperly and have been considered untrustworthy. We find conditions under which an equivariant projection generically shows the correct symmetry of the attractor.
Physica D: Nonlinear Phenomena, 1984
This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-l~na... more This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-l~nard convection with reflectional symmetry in the horizontal midplane. This symmetry is a consequence of the Boussinesq approximation, provided the boundary conditions are the same on the top and bottom plates. All possible local bifurcation diagrams (assuming certain non-degeneracy conditions) are found using only group theory. The results are therefore applicable to other systems with the same symmetries. Rolls, hexagons, or a new solution, regular triangles, can be stable depending on the physical system. Rolls are stable in ordinary Rayleigh-B~nard convection. The results are compared to those of Buzano and Golubitsky [1] without the midplane reflection symmetry. The bifurcation behavior of the two cases is quite different, and a connection between them is established by considering the effects of breaking the reflectional symmetry. Finally, the relevant experimental results are described.
J. Ineq. Pure Appl. Math, 2004
for helpful discussions during this research.
We define a graph network to be a coupled cell network where there are only one type of cell and ... more We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and anti-synchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and anti-synchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and anti-synchrony subspaces for several graph networks. We also apply o...
We numerically find solutions to the vector Ginzburg-Landau equation with a triple-well potential... more We numerically find solutions to the vector Ginzburg-Landau equation with a triple-well potential (as studied by Flores, Padilla, and Tonegawa). We use the Galerkin Newton Gradient Algorithm (by Neuberger and Swift) and bifurcation techniques to find solutions to this problem. With a small parameter, we find a Morse index 2 solution which approximates a pattern formation with triple junction structure whose nodal set is of minimal length and intersects the boundary at right angles. ii Acknowledgements I would like to thank Prof. John M. Neuberger for his encouragement and countless hours of explanation he gave while this work was in progress. Dr. Neuberger’s confidence in me was unwavering and often motivated me to work harder to try to live up to his expectations. I would also like to thank Prof. Pablo Padilla of UNAM for partially funding a trip to Mexico so that we could gain valuable insight into this problem.
We study a two-parameter family of so-called Hamiltonian systems defined on a region Ω in R with ... more We study a two-parameter family of so-called Hamiltonian systems defined on a region Ω in R with the bifurcation parameters λ and μ of the form: ∆u+ ∂ ∂v Hλ,μ(u, v) = 0 ∆v + ∂ ∂u Hλ,μ(u, v) = 0 on Ω, taking Hλ,μ to be a function of two variables that satisfies certain conditions. We use numerical methods adapted from Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs(Inter. J. Bif. Chaos, 2009) to approximate solution pairs. After providing a symmetry analysis of the solution space of pairs of functions defined on the unit square, we numerically approximate bifurcation surfaces over the two dimensional parameter space. A cusp catastrophe is found on the diagonal in the parameter space where λ = μ and is explained in terms of symmetry breaking bifurcation. Finally, we suggest a more theoretical direction for our future work on this topic.
Nature Computational Science, 2021
SIAM Journal on Applied Dynamical Systems, 2020
SIAM Journal on Applied Dynamical Systems, 2019
Physical Review E, 1995
We derive the averaged equations describing a series array of Josephson junctions shunted by a pa... more We derive the averaged equations describing a series array of Josephson junctions shunted by a parallel inductor-resistor-capacitor load. We assume that the junctions have negligable capacitance (β = 0), and derive averaged equations which turn out to be completely tractable: in particular the stability of both in-phase and splay states depends on a single parameter, δ. We find an explicit expression for δ in terms of the load parameters and the bias current. We recover (and refine) a common claim found in the technical literature, that the in-phase state is stable for inductive loads and unstable for capacitive loads.
Physical Review Letters, 1984
ABSTRACT
Thesis University of California Berkeley 1985 Source Dissertation Abstracts International Volume 46 09 Section B Page 3100, 1985
International Journal of Parallel Programming, 2012
We present a simple and easy to apply methodology for using high-level self-submitting parallel j... more We present a simple and easy to apply methodology for using high-level self-submitting parallel job queues in an MPI environment. Using C++, we implemented a library of functions, MPQueue, both for testing our concepts and for use in real applications. In particular, we have applied our ideas toward solving computational combinatorics problems and for finding bifurcation diagrams of solutions of partial differential equations (PDE). Our method is general and can be applied in many situations without a lot of programming effort. The key idea is that workers themselves can easily submit new jobs to the currently running job queue. Our applications involve complicated data structures, so we employ serialization to allow data to be effortlessly passed between nodes. Using our library, one can solve large problems in parallel without being an expert in MPI. We demonstrate our methodology and the features of the library with several example programs, and give some results from our current PDE research. We show that our techniques are efficient and effective via overhead and scaling experiments.
SIAM Journal on Applied Dynamical Systems, 2006
We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift to find solutions to ... more We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and post-process the basis, rendering it suitable for input to the GNGA. The GNGA uses Newton's method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods, and follows branches created at symmetry-breaking bifurcations, so the human user does not need to supply initial guesses for Newton's method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.
Contemporary Mathematics, 1986
Electronic Journal of Differential Equations
Contemporary Mathematics, 1984
Journal of Nonlinear Science
Physica D: Nonlinear Phenomena, 1998
We show how the symmetry of attractors of equivariant dynamical systems can be observed by equiva... more We show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space. Equivariant projections have long been used, but they can give misleading results if used improperly and have been considered untrustworthy. We find conditions under which an equivariant projection generically shows the correct symmetry of the attractor.
Physica D: Nonlinear Phenomena, 1984
This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-l~na... more This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-l~nard convection with reflectional symmetry in the horizontal midplane. This symmetry is a consequence of the Boussinesq approximation, provided the boundary conditions are the same on the top and bottom plates. All possible local bifurcation diagrams (assuming certain non-degeneracy conditions) are found using only group theory. The results are therefore applicable to other systems with the same symmetries. Rolls, hexagons, or a new solution, regular triangles, can be stable depending on the physical system. Rolls are stable in ordinary Rayleigh-B~nard convection. The results are compared to those of Buzano and Golubitsky [1] without the midplane reflection symmetry. The bifurcation behavior of the two cases is quite different, and a connection between them is established by considering the effects of breaking the reflectional symmetry. Finally, the relevant experimental results are described.
J. Ineq. Pure Appl. Math, 2004
for helpful discussions during this research.
We define a graph network to be a coupled cell network where there are only one type of cell and ... more We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and anti-synchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and anti-synchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and anti-synchrony subspaces for several graph networks. We also apply o...
We numerically find solutions to the vector Ginzburg-Landau equation with a triple-well potential... more We numerically find solutions to the vector Ginzburg-Landau equation with a triple-well potential (as studied by Flores, Padilla, and Tonegawa). We use the Galerkin Newton Gradient Algorithm (by Neuberger and Swift) and bifurcation techniques to find solutions to this problem. With a small parameter, we find a Morse index 2 solution which approximates a pattern formation with triple junction structure whose nodal set is of minimal length and intersects the boundary at right angles. ii Acknowledgements I would like to thank Prof. John M. Neuberger for his encouragement and countless hours of explanation he gave while this work was in progress. Dr. Neuberger’s confidence in me was unwavering and often motivated me to work harder to try to live up to his expectations. I would also like to thank Prof. Pablo Padilla of UNAM for partially funding a trip to Mexico so that we could gain valuable insight into this problem.
We study a two-parameter family of so-called Hamiltonian systems defined on a region Ω in R with ... more We study a two-parameter family of so-called Hamiltonian systems defined on a region Ω in R with the bifurcation parameters λ and μ of the form: ∆u+ ∂ ∂v Hλ,μ(u, v) = 0 ∆v + ∂ ∂u Hλ,μ(u, v) = 0 on Ω, taking Hλ,μ to be a function of two variables that satisfies certain conditions. We use numerical methods adapted from Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs(Inter. J. Bif. Chaos, 2009) to approximate solution pairs. After providing a symmetry analysis of the solution space of pairs of functions defined on the unit square, we numerically approximate bifurcation surfaces over the two dimensional parameter space. A cusp catastrophe is found on the diagonal in the parameter space where λ = μ and is explained in terms of symmetry breaking bifurcation. Finally, we suggest a more theoretical direction for our future work on this topic.
Nature Computational Science, 2021
SIAM Journal on Applied Dynamical Systems, 2020
SIAM Journal on Applied Dynamical Systems, 2019
Physical Review E, 1995
We derive the averaged equations describing a series array of Josephson junctions shunted by a pa... more We derive the averaged equations describing a series array of Josephson junctions shunted by a parallel inductor-resistor-capacitor load. We assume that the junctions have negligable capacitance (β = 0), and derive averaged equations which turn out to be completely tractable: in particular the stability of both in-phase and splay states depends on a single parameter, δ. We find an explicit expression for δ in terms of the load parameters and the bias current. We recover (and refine) a common claim found in the technical literature, that the in-phase state is stable for inductive loads and unstable for capacitive loads.
Physical Review Letters, 1984
ABSTRACT
Thesis University of California Berkeley 1985 Source Dissertation Abstracts International Volume 46 09 Section B Page 3100, 1985
International Journal of Parallel Programming, 2012
We present a simple and easy to apply methodology for using high-level self-submitting parallel j... more We present a simple and easy to apply methodology for using high-level self-submitting parallel job queues in an MPI environment. Using C++, we implemented a library of functions, MPQueue, both for testing our concepts and for use in real applications. In particular, we have applied our ideas toward solving computational combinatorics problems and for finding bifurcation diagrams of solutions of partial differential equations (PDE). Our method is general and can be applied in many situations without a lot of programming effort. The key idea is that workers themselves can easily submit new jobs to the currently running job queue. Our applications involve complicated data structures, so we employ serialization to allow data to be effortlessly passed between nodes. Using our library, one can solve large problems in parallel without being an expert in MPI. We demonstrate our methodology and the features of the library with several example programs, and give some results from our current PDE research. We show that our techniques are efficient and effective via overhead and scaling experiments.
SIAM Journal on Applied Dynamical Systems, 2006
We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift to find solutions to ... more We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and post-process the basis, rendering it suitable for input to the GNGA. The GNGA uses Newton's method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods, and follows branches created at symmetry-breaking bifurcations, so the human user does not need to supply initial guesses for Newton's method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.
Contemporary Mathematics, 1986
Electronic Journal of Differential Equations
Contemporary Mathematics, 1984