1,720,976 research outputs found
Numerical computation and continuation of invariant manifolds connecting fixed points
No full textFriedman, Mark J.; Doedel, Eusebius J.. (1989). Numerical computation and continuation of invariant manifolds connecting fixed points. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/5110
Numerical methods for bifurcation problems and large-scale dynamical systems
No full textNumerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems is based on the combined proceedings of two workshops devoted to computational issues.
The workshops were an integral part of the 1997-98 IMA program on "EMERGING APPLICATIONS OF DYNAMICAL SYSTEMS."
I would like to thank Donald G. Aronson, University of Minnesota (Mathematics); Wolf-Juergen Beyn, Universitaet Bielefeld (Fakultaet fuer Mathematik); Eusebius Doedel, California Institute of Technology (Applied Mathematics); Bernold Fiedler, Free University of Berlin (Mathematics); H.B. Keller, Caltech (Applied Mathematics); Yannis Kevrekidis, Princeton University (Chemical Enginering); Jens Lorenz, University of New Mexico (Mathematics and Statistics); Edriss S. Titi, University of California (Mathematics); Laurette S. Tuckerman, Laboratoire d'Informatique pour la Mecanique et les Sciences de l'Ingenieur (LIMSI) for their excellent work as organizers of the meeting.
Special appreciation to Eusebius Doedel and Laurette S. Tuckerman for serving as editors of the proceedings. I also take this opportunity to thank the National Science Foundation (NSF), and the National Security Agency (NSA), whose financial support made the workshop possible
Computational methods for global analysis of homoclinic and hetero-clinic orbits: a case study
No full textFriedman, Mark J.; Doedel, Eusebius J.. (1991). Computational methods for global analysis of homoclinic and hetero-clinic orbits: a case study. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/1619
Numerical solution of linear second order parabolic partial differential equations by the methods of collacation with cubic splines
No full textCollocation with cubic splines is used as a method for solving Linear second order parabolic partial differential equations. The collocation method is shown to be equivalent to a finite difference method that is consistent with the differential equation and stable in the sense of Von Neumann. Results of numerical computations are given, as well as an application of the method to a moving boundary problem for the heat equation.Science, Faculty ofMathematics, Department ofGraduat
The Construction of Finite Difference Approximations to Ordinary Differential Equations
Get PDFFinite difference approximations of the form Σ^(si)_(i=-rj)d_(j,i)u_(j+i)=Σ^(mj)_(i=1) e_(j,if)(z_(j,i)) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r_j + s_j + m_j - n, and higher for certain special choices of the points Z_(j,i). Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated
Investigating torus bifurcations in the forced Van der Pol oscillator
No full textWe demonstrate a new algorithm for computing one-dimensional stable and unstable manifolds of (Poincaré) maps that we have implemented in DsTool [1]. As an example we investigate the most complicated sequence of bifurcations in the forced Van der Pol oscillator as the amplitude of the forcing is increased. This bifurcation sequence has recently been used to test algorithms for the computation of invariant tori
A Lin's method approach to heteroclinic connections involving periodic orbits - Analysis and numerics
Get PDFThe topic of the thesis is the bifurcation analysis of heteroclinic cycles
connecting a hyperbolic equilibrium and a hyperbolic periodic orbit. An
extension of Lin's method that is based on the coupling of the global
continuous system and the discrete system that describes the dynamics near
the periodic orbit is developed in the first part. The method allows to
formulate bifurcation equations for the given scenario. The bifurcation
equations for homoclinic orbits to the equilibrium and for homoclinic
orbits to the periodic orbit are qualitatively solved and discussed, the
case of a quadratic tangency is also considered. In the second part a
numerical method based on the theoretical results is developed that allows
to find and continue in parameters a heteroclinic connection between an
equilibrium and a periodic orbit. The method is demonstrated on three
selected examples and the theoretical results from the first part are
verified. An extension of this numerical method to orbits that connect two
periodic orbits is also given.Die Dissertationsschrift beschäftigt sich mit der Bifurkationsanalyse von
heteroklinen Zyklen, die eine hyperbolische Gleichgewichtslage und einen
hyperbolischen periodischen Orbit miteinander verbinden. Im ersten Teil der
Arbeit wird eine Erweiterung von Lins Methode entwickelt, die auf einer
Kopplung des globalen kontinuierlichen Systems und des diskreten Systems,
das die Dynamik in der Umgebung des periodischen Orbits beschreibt, beruht.
Die Methode erlaubt es, Bifurkationsgleichungen für das gegebene Szenario
zu formulieren. Die Bifurkationsgleichungen für homokline Orbits an die
Gleichgewichtslage und homokline Orbits an den periodischen Orbit werden
qualitativ gelöst und diskutiert, ebenso wird der Fall einer quadratischen
Berührung behandelt. Im zweiten Teil der Dissertation wird auf Basis der
theoretischen Ergebnisse eine numerische Methode entwickelt, die es
erlaubt, verbindende Orbits zwischen Gleichgewichtslagen und periodischen
Orbits zu finden und im Parameterraum zu verfolgen. Die Methode wird an
drei ausgewählten Beispielen demonstriert, dabei werden die theoretischen
Ergebnisse aus dem ersten Teil der Arbeit bestätigt. Eine Erweiterung der
numerischen Methode auf verbindende Orbits zwischen zwei hyperbolischen
periodischen Orbits wird abgeleitet
Difference methods for ordinary differential equations with applications to parabolic equations
No full textThe first chapter of the thesis is concerned with the construction of finite difference approximations to boundary value problems in linear nth order ordinary differential equations. This construction is based upon a local collocation procedure with polynomials, which is equivalent to a method of undetermined coefficients. It is shown that the coefficients of these finite difference approximations can be expressed as the determinants of matrices of relatively small dimension. A basic theorem states that these approximations are consistent, provided only that a certain normalization factor does not vanish. This is the case for compact difference equations and for difference equations with only one collocation point. The order of consistency may be improved by suitable choice of the collocation points. Several examples of known, as well as new difference approximations are given. Approximations to boundary conditions are also treated in detail. The stability theory of H. O. Kreiss is applied to investigate the stability of finite difference schemes based upon these approximations. A number of numerical examples are also given.
In the second chapter it is shown how the construction method of the first chapter can be extended to initial value problems for systems of linear first order ordinary differential equations. Specific examples are 'included and the well-known stability theory for these difference equations is summarized.
It is then shown how these difference methods may be applied to linear parabolic partial differential equations in one space variable after first discretizing in space by a suitable method from the first chapter. The stability of such difference schemes for parabolic equations is investigated using an eigenvalue-eigenvector analysis. In particular, the effect of various approximations to the boundary conditions is considered. The relation of this analysis to the stability theory of J. M. Varah is indicated. Numerical examples are also included.Science, Faculty ofMathematics, Department ofGraduat
Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations
Get PDFBeyn W-J, Doedel E. Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations. SIAM Journal on Scientific and Statistical Computing. 1981;2(1):107-120.A large class of consistent and unconditionally stable discretizations of nonlinear boundary value problems is defined. The number of solutions to the discretizations is compared to the number of solutions of the continuous problem. We state conditions under which these numbers must agree for all sufficiently small mesh sizes. Various examples, including bifurcation problems, illustrate our theoretical results
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