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Lecture Notes on Numerical Analysis of Nonlinear Equations. By E. J. Doedel. Last Modified: Winter 2007.
Global Bifurcations of the Lorenz Model. E. J. Doedel, B. Krauskopf, H. M. Osinga. Nonlinearity. Volume 19. 2006. Pages 2947-2972.
A Survey of Methods for Computing (Un)Stable Manifolds of Vector Fields. B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. M. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. Volume 15. Number 3. 2005. Pages 763-791. Abstract. The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example --- the two-dimensional stable manifold of the origin in the Lorenz system. [Alternate URL: Bristol Centre for Applied Nonlinear Mathematics Preprint 2004.14]
Elemental Periodic Orbits of the CR3BP: A Brief Selection of Computational Results. E. J. Doedel, D. J. Dichmann, J. Gal?-Vioque, H. B. Keller, R. C. Paffenroth, A. Vanderbauwhede. Proceedings of the EQUADIFF 2003, Hasselt, Belgium, Pages 163-168. World Scientific, Singapore. 2005. Abstract. We illustrate how numerical boundary value techniques can be used to obtain a rather complete classification of certain types of periodic solutions of the Circular Restricted 3-Body Problem (CR3BP), for all values of the mass-ratio parameter.
Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles. Y. A. Kuznetsov, W. Govaerts, E. J. Doedel, and A. Dhooge. SIAM Journal on Numerical Analysis. Volume 43. Number 4. 2005. Page 1407-1435. Abstract. Explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous ODEs are derived. They include second-order coefficients for the fold (limit point) bifurcation, as well as third-order coefficients for the flip (period-doubling) and Neimark--Sacker (torus) bifurcations. The formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0,T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the right-hand sides near the cycle. The formulas allow us to distinguish between sub- and supercritical bifurcations, in agreement with earlier asymptotic expansions of the bifurcating solutions. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The actual implementation is described in detail. We include three numerical examples, in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients.
Apparent Discontinuities in the Phase-Resetting Response of Cardiac Pacemakers. T. Krogh-Madsen, L. Glass, E. J. Doedel and M. R. Guevara. Journal of Theoretical Biology. Volume 230. Issue 4. October 21, 2004. Pages 499-519. Special Issue in honour of Arthur T. Winfree. Abstract. Injection of a brief stimulus pulse resets the spontaneous periodic activity of a sinoatrial node cell: a stimulus delivered early in the cycle generally delays the time of occurrence of the next action potential, while the same stimulus delivered later causes an advance. We investigate resetting in two models, one with a slow upstroke velocity and the other with a fast upstroke velocity, representing central and peripheral nodal cells, respectively. We first formulate each of these models as a classic Hodgkin-Huxley type of model and then as a model representing a population of single channels. In the Hodgkin-Huxley-type model of the slow-upstroke cell the transition from delay to advance is steep but continuous. In the corresponding single-channel model, due to the channel noise then present, repeated resetting runs at a fixed stimulus timing within the transitional range of coupling intervals lead to responses that span a range of advances and delays. In contrast, in the fast-upstroke model the transition from advance to delay is very abrupt in both classes of model, as it is in experiments on some cardiac preparations ("all-or-none" depolarization). We reduce the fast-upstroke model from the original seven-dimensional system to a three-dimensional system. The abrupt transition occurs in this reduced model when a stimulus transports the state point to one side or the other of the stable manifold of the trajectory corresponding to the eigendirection associated with the smaller of two positive eigenvalues. This stable manifold is close to the slow manifold, and so canard trajectories are seen. Our results demonstrate that the resetting response is fundamentally continuous, but extremely delicate, and thus suggest one way in which one can account for experimental discontinuities in the resetting response of a nonlinear oscillator.
Continuation of Periodic Orbits in Conservative and Hamiltonian Systems. F. J. Mu?z-Almaraz, E. Freire, J. Gal?, E. Doedel and A. Vanderbauwhede. Physica D: Nonlinear Phenomena. Volume 181. Issues 1-2. July 1, 2003. Pages 1-38. Abstract. We introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals, and in particular in Hamiltonian systems having several independent symmetries. Our method is based on a generalization of the concept of a normal periodic orbit as introduced by Sepulchre and MacKay [Nonlinearity 10 (1997) 679]. We illustrate the continuation method on some integrable Hamiltonian systems with two degrees of freedom and briefly discuss some further applications.
Computation of Periodic Solutions of Conservative Systems with Application to the 3-Body Problem. E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. J. Dichmann, J. Gal?-Vioque, A. Vanderbauwhede. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. Volume 13. Number 6. 2003. Pages 1353-1381. Abstract. We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Sim? as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.
Computation of Periodic Solution Bifurcations in ODEs Using Bordered Systems. E. J. Doedel, W. Govaerts, and Y. A. Kuznetsov. SIAM Journal on Numerical Analysis. Volume 41. Number 2. 2003. Page 401-435. Abstract. We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ODEs, namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark--Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic BVP that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization ofthe periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS.
The Computation of Periodic Solutions of the 3-Body Problem using the Numerical Continuation Software AUTO. D. J. Dichmann, E. J. Doedel, R. C. Paffenroth. In: "Libration Point Orbits and Applications". G. G?ez. M. W. Lo, J. J. Masdemont (Editors). World Scientific. 2003. Pages 489-528.
Stability of Piecewise Polynomial Collocation for Computing Periodic Solutions of Delay Differential Equations. K. Engelborghs, E. J. Doedel. Numerische Mathematik. Volume 91. Number 4. 2002. Pages 627-648. Abstract. Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation $\dot y(t) = a(t)y(t)+b(t)y(t-\tau) + f(t)$ by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are "unconditional", that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. [Alternate URL: Stability of Piecewise Polynomial Collocation for Computing Periodic Solutions of Delay Differential Equations.]
An Analytical and Numerical Study of a Modified Van der Pol Oscillator. E. J. Doedel, E. Freire, E. Gamero and A. J. Rodr?uez-Luis. Journal of Sound and Vibration. Volume 256. Issue 4. September 26, 2002. Pages 755-771. Abstract. A three-dimensional system of differential equations that models an electronic oscillator is considered. The equations allow a variety of periodic orbits that originate from a degenerate Hopf bifurcation, which is analytically studied. Numerical results are presented that show the existence of saddle-node cusps of periodic orbits, as well as period-doubling bifurcations, that result in the coexistence of multiple "canard" orbits if one of the parameters is small. The presence of chaotic attractors is also detected.
Numerical Continuation and Computation of Normal Forms. W. J. Beyn, A. Champneys, E. J. Doedel, W. Govaerts, Y. A. Kuznetsov, B. Sandstede. In: Handbook of Dynamical Systems. Volume 2. B. Fiedler (Editor). Elsevier Science. 2002. Pages 149-219. ISBN: 0-444-50168-1.
Convergence of a Boundary Value Difference Equation for Computing Periodic Solutions of Neutral Delay Differential Equations. K. Engelborghs, E. J. Doedel. Journal of Difference Equations and Applications. Volume 7. 2001. Pages 927-940. [Alternate URL: Convergence of a Boundary Value Difference Equation for Computing Periodic Solutions of Neutral Delay Differential Equations.]
Continuaci? de ?bitas peri?icas en Hamiltonianos con Simetr?: Aplicaci? a Mec?ica Celeste. J. Gal?, F. J. Mu?z Almaraz, E. Freire, E. J. Doedel, A. Vanderbauwhede. Actas XVII Cedya Congreso de Ecuaciones Diferenciales y Aplicaciones. Salamanca, Espa?. Septiembre 2001. Resumen: Presentamos un esquema de continuaci? de ?bitas peri?icas para sistemas hamiltonianos con simetr?. Justificamos, desde un punto de vista te?ico la validez del m?odo y lo aplicamos al problema de los tres cuerpos en forma de ocho, recientemente descubierta por Chenciner y Montgomery. [Alternate URL: Continuaci? de ?bitas peri?icas en Hamiltonianos con Simetr?: Aplicaci? a Mec?ica Celeste.]
Continuation of Periodic Orbits around Lagrange Points and AUTO2000. R. C. Paffenroth, E. J. Doedel, D. J. Dichmann. Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Session on "Lagrange Point Missions". M. Lo and K. Howell (Organizers). Quebec City, Canada. July 2001. [Related: Talk. R. C. Paffenroth.]
The AUTO2000 Command Line User Interface. R. C. Paffenroth, E. J. Doedel. Ninth International Python Conference. Long Beach, California, USA. March 5-8, 2001. Pages 233-241.
Lecture Notes on "Numerical Analysis of Bifurcation Problems". E. J. Doedel. International Course on Bifurcations and Stability in Structural Engineering. Alain L?er (Coordinator). EDF/CNRS LMA Marseille, Maison Europ?nne des Technologies, Universit?Pierre et Marie Curie (Paris VI). November 2000. 43 Pages.
Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. E. J. Doedel, L. S. Tuckerman (Editors). Series: The IMA Volumes in Mathematics and its Applications. Volume 119. Springer-Verlag. 2000. ISBN: 0-387-98970-6. Description: The Institute for Mathematics and its Applications (IMA) devoted its 1997-1998 program to Emerging Applications of Dynamical Systems. Dynamical systems theory and related numerical algorithms provide powerful tools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been developed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating more complicated objects, such as higher- codimension bifurcations of fixed points, periodic orbits, and connecting orbits, as well as the calcuation of invariant manifolds. Another challenge is to extend the applicability of algorithms to the very large systems that result from discretizing partial differential equations. Even the calculation of steady states and their linear stability can be prohibitively expensive for large systems (e.g. 10_3- -10_6 equations) if attempted by simple direct methods. Several of the papers in this volume treat computational methods for low and high dimensional systems and, in some cases, their incorporation into software packages. A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self -organized criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering.
Collocation Methods for Continuation Problems in Nonlinear Elliptic PDEs. E. J. Doedel, H. Sharifi. Issue on Continuation Methods in Fluid Mechanics. D. Henry and A. Bergeon (Editors). Notes on Numerical Fluid Mechanics. Volume 74. Vieweg. 2000. Pages 105-118. [Alternate URL: Collocation Methods for Continuation Problems in Nonlinear Elliptic PDEs.]
Lecture Notes on "Numerical Analysis of Bifurcation Problems". E. J. Doedel. Short Course on Numerical Bifurcation Analysis. Summer School on Nonlinear Dynamics in Physiology and Medicine. McGill University Centre for Nonlinear Dynamics. May 2000. 23 Pages.
Continuaci? Num?ica de ?bitas Peri?icas en Sistemas Hamiltonianos con Simetr?. F. J. Mu?z-Almaraz, J. Gal?, E. Freire, E. J. Doedel. Actas XVI Cedya Congreso de Ecuaciones Diferenciales y Aplicaciones. Las Palmas de Gran Canaria. Espa?. 1999. R. Montenegro, G. Montero, G. Winter (Editores). P?inas 395-402. Resumen: Presentamos un procedimiento para continuar num?icamente soluciones peri?icas en sistemas hamiltonianos. El objetivo es la detecci? de situaciones de degeneraci? y el estudio de la correspondiente conducta de bifurcaci? partiendo de los modos normales del sistema o de alguna otra familia de soluciones peri?icas. Exponemos m?odos de continuaci? aplicados a un caso integrable de dos grados de libertad con simetr? a giros. [Alternate URL: Continuaci? Num?ica de ?bitas Peri?icas en Sistemas Hamiltonianos con Simetr?.]
On the Construction of Discretizations of Elliptic Partial Differential Equations. E. J. Doedel. Journal of Difference Equations and Applications. Volume 3. 1997. Pages 389-416. [Alternate URL: On the Construction of Discretizations of Elliptic Partial Differential Equations.]
Numerical Methods for Boundary Value Problems with Application to Bifurcation Problems. E. J. Doedel. Journal of Shanghai Jiaotong University. E-3 Sup (I). 1998. Pages 27-36.
Nonlinear Numerics. E. J. Doedel. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Volume 7. Number 9. 1997. Pages 2127-2143. Abstract. The objectives and some basic methods of numerical bifurcation analysis are described. Several computational examples are used to illustrate the power as well as the limitations of these techniques. Directions in which further algorithmic and software development is desirable are also discussed. [Alternate URL: Nonlinear Numerics. E. J. Doedel. Journal of the Franklin Institute. Volume 334. Issues 5-6. September-November 1997. Pages 1049-1073. Visions of Nonlinear Mechanics in the 21st Century. Abstract. The objectives and some basic methods of numerical bifurcation analysis are described. Several computational examples are used to illustrate the power as well as the limitations of these techniques. Future directions of algorithmic and software development are also discussed.]
Successive Continuation for Locating Connecting Orbits. E. J. Doedel, M. J. Friedman, B. Kunin. Numerical Algorithms. Volume 14. Issue 1-3. 1997. Pages 103-124. Abstract. A successive continuation method for locating connecting orbits in parametrized systems of autonomous ODEs is considered. A local convergence analysis is presented and several illustrative numerical examples are given.
A Codimension-Two Point associated with Coupled Josephson Junctions. D. G. Aronson, E. J. Doedel, D. H. Terman. Nonlinearity. Volume 10. 1997. Pages 1231-1255. Abstract: The dynamics of a pair of identical Josephson junctions coupled through a shared purely capacitive load are governed by a two-parameter system of two second-order nonlinear ordinary differential equations. Numerical simulations have shown that this system possesses many different running and periodic solutions. Continuation studies using AUTO indicate that many of these solution branches are generated by a codimension-2 connection which occurs at a particular parameter point. In this paper, we first describe these calculations in detail. We then study a general two-parameter system whose properties reflect some of those found in our numerical studies of the Josephson junction system. In particular, our model system is assumed to possess an appropriate codimension-2 connection, and we prove that its unfolding generates a large variety of codimension-1 connection curves. These results, combined with the particular symmetry and periodicity properties of the junction equations, account for all of the numerically observed solution branches. Indeed, the theoretical analysis predicted the existence of branches which were not initially observed, but which were subsequently found. [Alternate URL: IngentaConnect: A Codimension-Two Point associated with Coupled Josephson Junctions.]
Lecture Notes on "Numerical Analysis of Bifurcation Problems". E. J. Doedel. Survey Lectures on "Nonlinear Systems of Equations". Spring School on Numerical Software. Hamburg, Germany. March 1997. 132 Pages.
AUTO97: Continuation And Bifurcation Software For Ordinary Differential Equations (with HomCont). E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, X.-J. Wang. 1997. 157 Pages. [Alternate URL: AUTO97: Continuation And Bifurcation Software For Ordinary Differential Equations (with HomCont).]
AUTO94P: An Experimental Parallel Version of AUTO. E. J. Doedel, X.-J. Wang. Center for Research on Parallel Computing, California Institute of Technology. Technical Report: CRPC-95-3. 1995. [Center for Research on Parallel Computation, Rice University. Technical Report: CRPC-TR95599.] Abstract. A detailed description is given of the parallel algorithms used in AUTO94P, an experimental parallel version of the software AUTO for the numerical bifurcation analysis of systems of ordinary differential equations. Timing results and user instructions for the Intel Delta are included. The sequential version of the software, AUTO94, is fully described in [8]. For a related tutorial paper see [5, 6]. [Alternate URL: AUTO94P: An Experimental Parallel Version of AUTO.]
AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. E. J. Doedel, T. F. Fairgrieve, X.-J. Wang. Center for Research on Parallel Computing, California Institute of Technology. Technical Report: CRPC-95-2. 1995. [Center for Research on Parallel Computation, Rice University. Technical Report: CRPC-TR94592.] July 1994. Revised: July 1995.
On Locating Connecting Orbits. E. J. Doedel, M. J. Friedman, A. C. Monteiro. Applied Mathematics and Computation. Volume 65. Issues 1-3. September-October 1994. Pages 231-239. Abstract. An extension of a method for computing connecting orbits that locates such orbits in a systematic manner is presented. As an application we compute homoclinic orbits in a singular perturbation problem.
The Dynamics of Coupled Current-Biased Josephson Junctions -- Part II. D. G. Aronson, E. J. Doedel, Hans G. Othmer. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. Volume 1. Number 1. 1991. Pages 51-66. Abstract. A numerical analysis of the dynamics of two coupled current-biased Josephson junctions is presented. The mathematical model of two coupled nonlinear ordinary differential equations can also be interpreted in terms of coupled rotating pendula. We describe the solution manifolds of these equations, particularly the manifolds of rotations of phase gain 4p per period. A variety of homoclinic orbits and heteroclinic cycles is shown to exist when the coupling strength is small. We also discuss the relation between the solution structure of the damped system and the solution structure of the undamped system.
Numerical Analysis and Control of Bifurcation Problems (II): Bifurcation in Infinite Dimensions. E. J. Doedel, H. B. Keller, J.-P. Kernevez. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. Volume 1. Number 4. 1991. Pages 745-772. Abstract. A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for ordinary differential equations. Several illustrative examples computed with the AUTO software package are included. This is Part II of the paper that appeared in the preceding issue [Doedel et al., 1991] and that mainly dealt with algebraic problems.
Numerical Analysis and Control of Bifurcation Problems (I): Bifurcation in Finite Dimensions. E. J. Doedel, H. B. Keller, J.-P. Kernevez. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. Volume 1. Number 3. 1991. Pages 493-520. Abstract. A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for algebraic equations. Several illustrative examples computed with the AUTO software package are included. Part II of this paper deals with ordinary differential equations and will appear in the next issue.
Numerical Computation and Continuation of Invariant Manifolds Connecting Fixed Points. M. J. Friedman and E. J. Doedel. SIAM Journal on Numerical Analysis. Volume 28. Number 3. 1991. Pages 789-808. Abstract. A numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in $\mathbb{R}^n $ is given, extending the results of an earlier paper [Comput. Appl. Math., 26 (1989), Pages 159-170] by the authors. Basically, a boundary value problem on the real line is truncated to a finite interval. The method applies, in particular, to the computation of heteroclinic orbits. The emphasis is on the systematic computation of such orbits by continuation. Using the fact that the linearized operator of our problem is Fredholm in appropriate Banach spaces, the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval. Several applications are considered, including the computation of traveling wave solutions to reaction diffusion problems. Computations were done using the software package AUTO.
On the Resonance Structure in a Forced Excitable System. J. C. Alexander, E. J. Doedel, and H. G. Othmer. SIAM Journal on Applied Mathematics. Volume 50. 1990. Pages 1373-1418. Abstract. The dynamics of forced excitable systems are studied analytically and numerically with a view toward understanding the resonance or phase-locking structure. In a singular limit the system studied reduces to a discontinuous flow on a two-torus, which in turn gives rise to a set-valued circle map. It is shown how to define rotation numbers for such systems and derive properties analogous to those known for smooth flows. The structure of the phase-locking regions for a Fitzhugh–Nagumo system in the singular limit is also analyzed. A singular perturbation argument shows that some of the general results persist for the nonsingularly-perturbed system, and some numerical results on phase-locking in the forced Fitzhugh–Nagumo equations illustrate this fact. The results explain much of the phase-locking behavior seen experimentally and numerically in forced excitable systems, including the existence of threshold stimuli for phase-locking. The results are compared with known results for forced oscillatory systems. [Alternate URL: On the Resonance Structure in a Forced Excitable System. [PDF]]
Numerical Computation of Heteroclinic Orbits. E. J. Doedel and M. J. Friedman. Journal of Computational and Applied Mathematics. Volume 26. Issues 1-2. June 1989. Pages 155-170. Abstract. We give a numerical method for the computation of heteroclinic orbits connecting two saddle points in R^2. These can be computed to very high period due to an integral phase condition and an adaptive discretization. We can also compute entire branches (one-dimensional continua) of such orbits. The method can be extended to compute an invariant manifold that connects two fixed points in n. As an example we compute branches of traveling wave front solutions to the Huxley equation. Using weighted Sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period.
An Analytical and Numerical Study of the Bifurcations in a System of Linearly-Coupled Oscillators. D. G. Aronson, E. J. Doedel, H. G. Othmer. Physica D: Nonlinear Phenomena. Volume 25. Issues 1-3. March-April 1987. Pages 20-104. Abstract. We study a two-parameter family of ordinary differential equations in R4 that governs the dynamics of two coupled planar oscillators. Each oscillator has a unique periodic solution that is attracting and the uncoupled product system has a unique invariant torus that is attracting. The torus persists for weak coupling and contains two periodic solutions when the coupling is linear and conservative. One of these, in which the oscillators are synchronized, persists and is stable for all coupling strengths. The other, in which the oscillators are p radiant out of phase, disappears either in a Hopf bifurcation or when fixed points appear on the orbit at a critical ratio of the coupling strength to the frequency. The out-of-phase oscillation is unstable except on an open set in the frequency-coupling-strength plane which contains moderate values of both parameters. Furthermore, there are tori bifurcating from the out-of-phase solution, which means, according to the Arnol'd theory for Hopf bifurcations in maps, that there may be periodic solutions of arbitrarily large period and chaotic solutions as well. Numerous other bifurcations occur, and there are a number of higher codimension singularities. In a large region of the frequency-coupling parameter plane stable steady states coexist with stable periodic solutions.
Resonance and Bistability in Coupled Oscillators. H. G. Othmer, D. G. Aronson, E. J. Doedel. Physics Letters A. Volume 113. Issue 7. January 6, 1986. Pages 349-354. Abstract. We present results on the bifurcations that occur in a two-parameter family of ordinary differential equations in R^4 that describe a pair of linearly coupled oscillators. In particular, we describe the existence of infinitely many stability zones in parameter space in which two stable periodic orbits coexist.
Dynamics of the Iwan-Blevins Wake Oscillator Model. A. B. Poore, E. J. Doedel, J. E. Cermak. International Journal of Non-Linear Mechanics. Volume 21. Issue 4. 1986. Pages 291-302. Abstract. In this work we explain and utilize recently developed numerical continuation techniques to compute both stable and unstable non-linear oscillatory dynamics present in the Iwan-Blevins wake oscillator model for vortex induced oscillations of a circular cylinder. Frequency lock-in ranges, hysteresis, jump phenomena and double amplitude responses are efficiently computed. Furthermore, the parameter identification problem for this model is discussed and investigated to obtain a good qualitative fit with the experimental results of Feng.
Finite Difference Methods for Singular Two-Point Boundary Value Problems. E. J. Doedel and G. W. Reddien. SIAM Journal on Numerical Analysis. Volume 21. Number 2. 1984. Pages 300-313. Abstract. High order accurate finite difference methods are given for a class of singular two-point boundary value problems. These methods are constructed to adapt to the behavior of the solution being approximated. Stability and convergence theorems are given along with the results of numerical experiments. The results generalize work on these methods for problems without singularities.
Numerical Computation of Periodic Solution Branches and Oscillatory Dynamics of the Stirred Tank Reactor with A -> B -> C Reactions. E. J. Doedel, R. F. Heinemann. Chemical Engineering Science. Volume 38. Issue 9. 1983. Pages 1493-1499. Abstract. We use a continuation technique for branches of periodic solutions to investigate the oscillatory behavior of a continuously stirred tank reactor with consecutive A -> B -> C reactions. This continuation technique allows the computation of entire periodic solution branches, including those with limit points and asymptotically unstable solutions. Our computations reveal dynamic phenomena not seen in previous studies of this reactor. The results include response diagrams exhibiting stable and unstable periodic branches that contain multiple limit points. The presence of these points indicates that the reactor may jump from a steady state to a periodic orbit or from one orbit to another. The computations also illustrate interactions of multiple steady state limit points, Hopf bifurcations and infinite periodic bifurcations.
Numerical Techniques for Bifurcation Problems in Delay Equations. E. J. Doedel, P. C. Leung. Congressus Numerantium. Volume 34. 1982. Pages 225-237.
AUTO: A Program for the Automatic Bifurcation Analysis of Autonomous Systems. E. J. Doedel. Congressus Numerantium. Volume 30. 1981. Pages 265-284.
Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations. W.-J. Beyn and E. J. Doedel. SIAM Journal on Scientific Computing. Volume 2 Number 1. 1981. Pages 107-120. Abstract. 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.
Some Stability Theorems for Finite Difference Collocation Methods on Non-Uniform Meshes. E. J. Doedel. BIT. Volume 20. 1980. Pages 58-66.
Finite Difference Collocation Methods for Nonlinear Two Point Boundary Value Problems. E. J. Doedel. SIAM Journal on Numerical Analysis. Volume 16. Number 2. 1979. Pages 173-185. Abstract. A general class of finite difference methods for solving nonlinear two point boundary value problems is considered. These methods can also be interpreted as collocation methods. A convergence analysis on uniform meshes is given. This analysis is based upon a theorem of H. B. Kelley and a previous paper by the author. A specific example is given in detail and results of some numerical computations are included.
The Construction of Finite Difference Approximations to Ordinary Differential Equations. E. J. Doedel. SIAM Journal on Numerical Analysis. Volume 15 Number 3. 1978. Pages 450-465. Abstract. Finite difference approximations of the form $\Sigma _{i = - r_j }^{s_i } d_{j,i} u_{j + i} = \Sigma _{i = 1}^{m_i } e_{j,i} f(z_{j,i} )$ for the numerical solution of linear $n$th order ordinary differential equations are analyzed. The order of these pproximations 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.
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