Numerical Continuation Methods

Numerical Stablity of Path Tracing in Polyhedral Homotopy Continuation Methods

The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotop...

Numerical solution of multivariate polynomial systems by homotopy continuation methods

pn(xi,...,xn) = 0 for x = (x\,... ,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Grobner bas...

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

Numerical Continuation , Andcomputation of Normal

Numerical Continuation of Hamiltonian Relative Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a ...