Computation of Periodic Solution Bifurcations in ODEs Using Bordered Systems

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 of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS.

[1]  G. Moore,et al.  The Calculation of Turning Points of Nonlinear Equations , 1980 .

[2]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[3]  A. Griewank,et al.  Characterization and Computation of Generalized Turning Points , 1984 .

[4]  W. Beyn,et al.  Chapter 4 – Numerical Continuation, and Computation of Normal Forms , 2002 .

[5]  Allan Douglas Jepson.  I. Asymptotic boundary conditions for ordinary differential equations. II. Numerical Hopf bifurcation , 1981 .

[6]  Allan D. Jepson,et al.  The computation and use of Floquet multipliers for bifurcation analysis , 1994 .

[7]  H. Keller,et al.  Multiple limit point bifurcation , 1980 .

[8]  E. Allgower,et al.  Numerical path following , 1997 .

[9]  R. Russell,et al.  Adaptive Mesh Selection Strategies for Solving Boundary Value Problems , 1978 .

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

[11]  E. J. Doedel,et al.  Numerical methods for hope bifurcation and continuation of periodic solution paths , 1985 .

[12]  C. D. Boor,et al.  Collocation at Gaussian Points , 1973 .

[13]  Willy Govaerts,et al.  Numerical methods for bifurcations of dynamical equilibria , 1987 .

[14]  R. Seydel.  Numerical computation of branch points in nonlinear equations , 1979 .

[15]  H. B. Keller,et al.  NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS , 1991 .

[16]  U. Ascher,et al.  A collocation solver for mixed order systems of boundary value problems , 1979 .

[17]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[18]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[19]  Willy Govaerts,et al.  Implementation of Hopf and double-Hopf continuation using bordering methods , 1998, TOMS.

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