Numerical Methods for the Continuation of Invariant Tori

To Amanda and our child iii ACKNOWLEDGEMENTS First and foremost, the author thanks his advisor for being available for frequent discussions about hyperbolicity, Fréchet derivatives, and other minutiae. Thanks also go to the School of Mathematics at Georgia Tech for supporting the author financially with a teaching assistantship for two years, and to the National Science

[1]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[2]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[3]  W. Langford.  Periodic and Steady-State Mode Interactions Lead to Tori , 1979 .

[4]  U. Ascher,et al.  Reformulation of Boundary Value Problems into “Standard” Form , 1981 .

[5]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[6]  Gene H. Golub,et al.  Matrix computations , 1983 .

[7]  W. Langford.  Numerical Studies of Torus Bifurcations , 1984 .

[8]  Hans G. Othmer,et al.  Synchronization, Phase-Locking and Other Phenomena in Coupled Cells , 1985 .

[9]  H. Othmer.  Nonlinear Oscillations in Biology and Chemistry , 1986 .

[10]  M. van Veldhuizen.  A new algorithm for the numerical approximation of an invariant curve , 1987 .

[11]  Hans G. Othmer,et al.  An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators , 1987 .

[12]  L. Chua,et al.  Chaos via torus breakdown , 1987 .

[13]  H. Othmer,et al.  On the collapse of the resonance structure in a three-parameter family of coupled oscillators , 1988 .

[14]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Connecting Orbits in Dynamical Systems , 1990 .

[15]  Robert D. Russell,et al.  Numerical Calculation of Invariant Tori , 1991, SIAM J. Sci. Comput..

[16]  K. Schmitt,et al.  Persistence of invariant tori in systems of coupled oscillators I: Regular and singular problems , 1991 .

[17]  H. Othmer,et al.  Persistence of invariant tori in systems of coupled oscillators: II. degenerate problems , 1991 .

[18]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[19]  Luca Dieci,et al.  Solution of the Systems Associated with Invariant Tori Approximation. II: Multigrid Methods , 1994, SIAM J. Sci. Comput..

[20]  S. Wiggins.  Normally Hyperbolic Invariant Manifolds in Dynamical Systems , 1994 .

[21]  G. Moore,et al.  Geometric methods for computing invariant manifolds , 1995 .

[22]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[23]  Luca Dieci,et al.  Computation of invariant tori by the method of characteristics , 1995 .

[24]  L. Dieci,et al.  Block iterations and compactification for periodic block dominant systems associated to invariant tori approximation , 1995 .

[25]  G. Moore,et al.  Computation and parametrization of periodic and connecting orbits , 1995 .

[26]  G. Moore,et al.  Computation and Parametrisation of Invariant Curves and Tori , 1996 .

[27]  Hinke M. Osinga,et al.  Computing invariant manifolds , 1996 .

[28]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[29]  G. Vegter,et al.  Algorithms for computing normally hyperbolic invariant manifolds , 1997 .

[30]  Lixin Liu,et al.  Computation and Continuation of Homoclinic and Heteroclinic Orbits with Arclength Parameterization , 1997, SIAM J. Sci. Comput..

[31]  Tassilo Küpper,et al.  Computation of Invariant Tori by the Fourier Methods , 1997, SIAM J. Sci. Comput..

[32]  Andrew Y. T. Leung,et al.  Construction of Invariant Torus Using Toeplitz Jacobian Matrices/Fast Fourier Transform Approach , 1998 .

[33]  G. Moore,et al.  Algorithms for constructing stable manifolds of stationary solutions , 1999 .

[34]  Volker Reichelt,et al.  Computing Invariant Tori and Circles in Dynamical Systems , 2000 .

[35]  Bernd Krauskopf,et al.  Investigating torus bifurcations in the forced Van der Pol oscillator , 2000 .

[36]  Manfred R. Trummer.  Spectral methods in computing invariant tori , 2000 .

[37]  Nicholas J. Higham,et al.  A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra , 2000, SIAM J. Matrix Anal. Appl..

[38]  T. Valkering,et al.  Dynamics of two capacitively coupled Josephson junctions in the overdamped limit , 2000 .

[39]  Luca Dieci,et al.  Lyapunov-type numbers and torus breakdown: numerical aspects and a case study , 1997, Numerical Algorithms.