2015/06/02 by Mathieu Desroches, Desroches, Mathieu, Mike R. Jeffrey +1 · 1 citation
Computer Science · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1506.00831
openalex publication_date 2015/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The folded node is a singularity associated with loss of normal hyperbolicity in systems where mixtures of slow and fast timescales arise due to singular perturbations. Canards are special solutions that reveal a counteractive feature of the local dynamics, namely trajectories that flow from attractive regions of space into repulsive regions. An alternative way to model switches between timescales is using piecewise-smooth differential equations. There is presently no adequate theory or method for relating slow-fast and piecewise-smooth models. Here we derive the analogous piecewise-smooth system for the folded node by pinching phase space to sharpen the switch between timescales. The corresponding piecewise-smooth system contains a so-called two-fold singularity, and exhibits the same topology and number of canards as the slow-fast system. Thus pinching provides a piecewise-smooth approximation to a slow-fast system.