| Abstract: |
| We present a computational framework for locating and continuing reducible, normally hyperbolic invariant tori with quasiperiodic internal dynamics in autonomous ordinary differential equations. The method is based on the parameterization method in a Newton-KAM formulation: the torus embedding, its normal bundle, the associated reducibility data, and selected system parameters are solved for simultaneously. Prescribing the internal frequency vector turns the search for invariant tori into a functional equation with small divisors, which we solve by Fourier-based cohomological equations adapted to dissipative flows. A pseudo-arclength continuation strategy is incorporated to detect and traverse quasiperiodic saddle-node bifurcations of invariant tori, including formulations with a distinguished normal direction. The algorithms are designed for high-dimensional systems, where direct discretizations or Poincare-map approaches become expensive or ill-conditioned. We discuss their implementation in a C++ library and illustrate the approach on benchmark examples that exhibit convergence, continuation through folds, and robust computation of hyperbolic bundles. Although the present work is primarily numerical, the formulation is explicitly tailored to a posteriori validation, making it a natural stepping stone toward computer-assisted proofs for quasiperiodic invariant structures and their bifurcations in dissipative dynamical systems. The same framework also avoids normal-form reductions and works directly with the original vector field equations. |
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