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15 Result(s)

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  1. No Access

    Chapter and Conference Paper

    Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

    We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqu...

    Hermen Jan Hupkes, Leonardo Morelli… in Difference Equations and Discrete Dynamica… (2020)

  2. No Access

    Article

    A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

    Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential eq...

    Andrew J. Steyer, Erik S. Van Vleck in BIT Numerical Mathematics (2018)

  3. No Access

    Article

    Neutral Mixed Type Functional Differential Equations

    We extend the existing Fredholm theory for mixed type functional differential equations developed by Mallet-Paret (J Dyn Differ Equ 11:1–47, 1999) to the case of implicitly defined mixed type functional different...

    Charles Lamb, Erik S. Van Vleck in Journal of Dynamics and Differential Equations (2016)

  4. No Access

    Reference Work Entry In depth

    Lyapunov Exponents: Computation

    Luca Dieci, Erik S. Van Vleck in Encyclopedia of Applied and Computational Mathematics (2015)

  5. No Access

    Chapter

    Continuous Matrix Factorizations

    Continuous matrix factorizations show great promise in a number of contexts. In this chapter we survey results on continuous matrix factorizations paying particular attention to smooth matrix factorizations of...

    Erik S. Van Vleck in Numerical Algebra, Matrix Theory, Differen… (2015)

  6. No Access

    Article

    QR methods and error analysis for computing Lyapunov and Sacker–Sell spectral intervals for linear differential-algebraic equations

    In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker–Sell spectral intervals for linear differential-algebraic equations. Furthermore,...

    Vu Hoang Linh, Volker Mehrmann, Erik S. Van Vleck in Advances in Computational Mathematics (2011)

  7. No Access

    Article

    Exponential Dichotomy for Asymptotically Hyperbolic Two-Dimensional Linear Systems

    We consider the problem of determining the existence of exponential dichotomy for a class of linear nonautonomous ODEs. An approach is explored that combines numerical techniques with rigorous perturbation the...

    Weishi Liu, Erik S. Van Vleck in Journal of Dynamics and Differential Equations (2010)

  8. No Access

    Article

    Lyapunov and Sacker–Sell Spectral Intervals

    In this work, we show that for linear upper triangular systems of differential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also –under some restrictions...

    Luca Dieci, Erik S. Van Vleck in Journal of Dynamics and Differential Equations (2007)

  9. No Access

    Article

    Perturbation Theory for Approximation of Lyapunov Exponents by QR Methods

    Motivated by a recently developed backward error analysis for QR methods, we consider the error in the Lyapunov exponents of perturbed triangular systems. We consider the case of stable and distinct Lyapunov expo...

    Luca Dieci, Erik S. Van Vleck in Journal of Dynamics and Differential Equations (2006)

  10. No Access

    Article

    On the error in computing Lyapunov exponents by QR Methods

    We consider the error introduced using QR methods to approximate Lyapunov exponents. We give a backward error statement for linear non-autonomous systems, and further discuss nonlinear autonomous problems. In ...

    Luca Dieci, Erik S. Van Vleck in Numerische Mathematik (2005)

  11. No Access

    Article

    A Variant of Newton's Method for the Computation of Traveling Waves of Bistable Differential-Difference Equations

    We consider a variant of Newton's method for solving nonlinear differential-difference equations arising from the traveling wave equations of a large class of nonlinear evolution equations. Building on the Fre...

    Christopher E. Elmer, Erik S. Van Vleck in Journal of Dynamics and Differential Equations (2002)

  12. No Access

    Chapter and Conference Paper

    Continuous Orthonormalization for Linear Two-Point Boundary Value Problems Revisited

    In this work we revisit the continuous orthonormalization technique for solving linear two-point boundary value problems and discuss a specific implementation of the method. We infer stability of the method by...

    Luca Dieci, Erik S. van Vleck in Dynamics of Algorithms (2000)

  13. No Access

    Article

    Computation of orthonormal factors for fundamental solution matrices

    In this work, we introduce and analyze two new techniques for obtaining the Q factor in the QR factorization of some (or all) columns of a fundamental solution matrix Y of a linear differential system. These tec...

    Luca Dieci, Erik S. Van Vleck in Numerische Mathematik (1999)

  14. No Access

    Article

    Orthosymplectic integration of linear Hamiltonian systems

    The authors describe a continuous, orthogonal and symplectic factorization procedure for integrating unstable linear Hamiltonian systems. The method relies on the development of an orthogonal, symplectic chan...

    Benedict J. Leimkuhler, Erik S. Van Vleck in Numerische Mathematik (1997)

  15. No Access

    Chapter and Conference Paper

    Piecewise Smooth Orthonormal Factors for Fundamental Solution Matrices

    The purpose of this note is to report on some recent results concerning QR factorizations of smooth full rank matrices. The focus of this note is on the computation of the Q factor. In particular, unlike previous...

    Luca Dieci, Erik S. Van Vleck in Foundations of Computational Mathematics (1997)