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

This work was carried out while the second author was visiting the School of Mathematics of the Georgia Institute of Technology, whose hospitality is gratefully acknowledged

• We consider a smooth planar system having slow-fast motion, where the slow motion takes place near a curve γ. We explore the idea of replacing the original smooth system with a system with discontinuous right-hand side (DRHS system for short), whereby the DRHS system coincides with the smooth one away from a neighborhood of γ. After this reformulation, in the region of phase-space where γ is attracting for the DRHS system, we will obtain sliding motion on γ and numerical methods apt at integrating for sliding motion can be applied. Moreover, we further bypass resolving the sliding motion and monitor entries (transversal) and exits (tangential) on the curve γ, a fact that can be done independently of resolving for the motion itself. The end result is a method free from the need to adopt stiff integrators or to worry about resolving sliding motion for the DRHS system. We illustrate the performance of our method on a few problems, highlighting the feasibility of using simple explicit Runge-Kutta schemes, and that we obtain much the same orbits of the original smooth system.

Mathematics Subject Classification: 65L99, 34A36, 37N30, 34D15.

 Citation: • • 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)

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