# American Institute of Mathematical Sciences

September  2018, 23(7): 2935-2950. doi: 10.3934/dcdsb.2018112

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

 1 School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA 2 Dipartimento di Matematica, Univ. of Bari, I-70100, Bari, Italy

Received  August 2017 Revised  November 2017 Published  September 2018 Early access  March 2018

Fund Project: 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.

Citation: Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
##### References:
 [1] L. Dieci and L. Lopez, A survey of Numerical Methods for IVPs of ODEs with Discontinuous right-hand side, Journal of Computational and Applied Mathematics, 236 (2012), 3967-3991.  doi: 10.1016/j.cam.2012.02.011. [2] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1988. [3] J. K. Hale, Ordinary Differential Equations, Krieger Publishing Co, Malabar, 1980. [4] M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs AMS, 32 (1981), ⅵ+147 pp. [5] N. Levinson, A second order differential equation with singular solutions, Annals of Mathematics, 50 (1949), 127-153.  doi: 10.2307/1969357. [6] S. Natsiavas, Dynamics of piecewise linear oscillators with van der Pol type damping, International Journal of Non Linear Mechanics, 26 (1991), 349-366.  doi: 10.1016/0020-7462(91)90065-2. [7] A. Roberts and P. Glendinning, Canard-like phenomena in piecewise-smooth Van der Pol systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, Chaos, 24 (2014), 023138, 11pp. [8] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1998), 207-223. [9] D. W. Storti and R. H. Rand, A simplified model of coupled relaxation oscillators, Int.l J. Nonlin. Mechanics, 22 (1987), 283-289.  doi: 10.1016/0020-7462(87)90020-5. [10] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, 1987.

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##### References:
 [1] L. Dieci and L. Lopez, A survey of Numerical Methods for IVPs of ODEs with Discontinuous right-hand side, Journal of Computational and Applied Mathematics, 236 (2012), 3967-3991.  doi: 10.1016/j.cam.2012.02.011. [2] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1988. [3] J. K. Hale, Ordinary Differential Equations, Krieger Publishing Co, Malabar, 1980. [4] M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs AMS, 32 (1981), ⅵ+147 pp. [5] N. Levinson, A second order differential equation with singular solutions, Annals of Mathematics, 50 (1949), 127-153.  doi: 10.2307/1969357. [6] S. Natsiavas, Dynamics of piecewise linear oscillators with van der Pol type damping, International Journal of Non Linear Mechanics, 26 (1991), 349-366.  doi: 10.1016/0020-7462(91)90065-2. [7] A. Roberts and P. Glendinning, Canard-like phenomena in piecewise-smooth Van der Pol systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, Chaos, 24 (2014), 023138, 11pp. [8] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1998), 207-223. [9] D. W. Storti and R. H. Rand, A simplified model of coupled relaxation oscillators, Int.l J. Nonlin. Mechanics, 22 (1987), 283-289.  doi: 10.1016/0020-7462(87)90020-5. [10] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, 1987.
Replacing the cubic with piecewise linear
Cubic vs. piecewise linear limit cycles
Tubular (left) and Shift (right) neighborhoods $\mathcal{C}(\rho)$, $\rho = 0.1$
Vector fields inside $\mathcal{C}(\rho)$ for both tubular (left) and shift (right) neighborhoods
Example 3.1. Van der Pol limit cycle: $\beta = 10$
Example 3.1. Modified vector fields for the DRHS reformulation: tubular (left) and shift (right) neighborhoods, $\rho = 0.01$
Example 3.1. Method 2.1: DRHS reformulation and sliding motion, tubular neighborhood
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
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
Example 3.1. Enlargement of entry on $\gamma$ within $\mathcal{C}(\rho)$. Tubular neighborhood, $\rho = 0.01$, $\beta = 10$. Stepsize $\tau = 10^{-4}$
"Corrugated" Van der Pol limit cycle: $\beta = 20$
Modified vector fields, $\rho = 0.01$ (left) and $\rho = 0.001$ (right)
Exit points, case of $\rho = 0.01$
Method 2.2 on (3.1)
Limit cycle of (3.2), and slow manifold
Modified vector fields for (3.2)
Method 2.2 for (3.2)
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