We present a method that we name the constrained invariant manifold method, a visualization tool to construct stable and unstable invariant sets of a map or flow, where the invariant sets are constrained to lie on a slow invariant manifold. The construction of stable and unstable sets constrained to an unstable slow manifold is exemplified in a singularly perturbed model arising from a structural-mechanical system consisting of a pendulum coupled to a viscoelastic rod. Additionally, we extend the step and stagger method [D. Sweet, H. Nusse, and J. Yorke, Phys. Rev. Lett. 86, 2261 (2001)] to calculate a $\delta$ pseudoorbit on a chaotic saddle constrained to the slow manifold in order to be able to compute the Lyapunov exponents of the saddle.

• Received 9 April 2003

DOI:https://doi.org/10.1103/PhysRevE.68.056210