Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics

Figure .  Replacing the cubic with piecewise linear

Figure .  Cubic vs. piecewise linear limit cycles

Figure 1.  Tubular (left) and Shift (right) neighborhoods $\mathcal{C}(\rho)$, $\rho = 0.1$

Figure 2.  Vector fields inside $\mathcal{C}(\rho)$ for both tubular (left) and shift (right) neighborhoods

Figure 3.  Example 3.1. Van der Pol limit cycle: $\beta = 10$

Figure 4.  Example 3.1. Modified vector fields for the DRHS reformulation: tubular (left) and shift (right) neighborhoods, $\rho = 0.01$

Figure 5.  Example 3.1. Method 2.1: DRHS reformulation and sliding motion, tubular neighborhood

Figure 6.  Example 3.1. Method 2.2: DRHS reformulation and no sliding motion. Tubular neighborhood for $C(\rho)$. Stepsize $\tau = 10^{-2}$, and 57 steps are needed for the periodic trajectory

Figure 7.  Example 3.1. Method 2.2: DRHS reformulation and no sliding motion. Shift neighborhood for $C(\rho)$. Stepsize $\tau = 10^{-2}$, and 93 steps are needed for the periodic trajectory

Figure 8.  Example 3.1. Enlargement of entry on $\gamma$ within $\mathcal{C}(\rho)$. Tubular neighborhood, $\rho = 0.01$, $\beta = 10$. Stepsize $\tau = 10^{-4}$

Figure 9.  "Corrugated" Van der Pol limit cycle: $\beta = 20$

Figure 10.  Modified vector fields, $\rho = 0.01$ (left) and $\rho = 0.001$ (right)

Figure .  Exit points, case of $\rho = 0.01$

Figure 11.  Method 2.2 on (3.1)

Figure .  Limit cycle of (3.2), and slow manifold

Figure 12.  Modified vector fields for (3.2)

Figure 13.  Method 2.2 for (3.2)